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Sample records for merged chaotic attractor

  1. Attractor merging crisis in chaotic business cycles

    Chian, Abraham C.-L.; Borotto, Felix A.; Rempel, Erico L.; Rogers, Colin

    2005-01-01

    A numerical study is performed on a forced-oscillator model of nonlinear business cycles. An attractor merging crisis due to a global bifurcation is analyzed using the unstable periodic orbits and their associated stable and unstable manifolds. Characterization of crisis can improve our ability to forecast sudden major changes in economic systems

  2. Dimension of chaotic attractors

    Farmer, J.D.; Ott, E.; Yorke, J.A.

    1982-09-01

    Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on probabilistic properties (that is, they depend on the frequency with which a typical trajectory visits different regions of the attractor). Both our example and the previous work that we review support the conclusion that all of the probabilistic dimensions take on the same value, which we call the dimension of the natural measure, and all of the metric dimensions take on a common value, which we call the fractal dimension. Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.

  3. Chaotic attractors with separated scrolls

    Bouallegue, Kais

    2015-01-01

    This paper proposes a new behavior of chaotic attractors with separated scrolls while combining Julia's process with Chua's attractor and Lorenz's attractor. The main motivation of this work is the ability to generate a set of separated scrolls with different behaviors, which in turn allows us to choose one or many scrolls combined with modulation (amplitude and frequency) for secure communication or synchronization. This set seems a new class of hyperchaos because each element of this set looks like a simple chaotic attractor with one positive Lyapunov exponent, so the cardinal of this set is greater than one. This new approach could be used to generate more general higher-dimensional hyperchaotic attractor for more potential application. Numerical simulations are given to show the effectiveness of the proposed theoretical results

  4. Describing chaotic attractors: Regular and perpetual points

    Dudkowski, Dawid; Prasad, Awadhesh; Kapitaniak, Tomasz

    2018-03-01

    We study the concepts of regular and perpetual points for describing the behavior of chaotic attractors in dynamical systems. The idea of these points, which have been recently introduced to theoretical investigations, is thoroughly discussed and extended into new types of models. We analyze the correlation between regular and perpetual points, as well as their relation with phase space, showing the potential usefulness of both types of points in the qualitative description of co-existing states. The ability of perpetual points in finding attractors is indicated, along with its potential cause. The location of chaotic trajectories and sets of considered points is investigated and the study on the stability of systems is shown. The statistical analysis of the observing desired states is performed. We focus on various types of dynamical systems, i.e., chaotic flows with self-excited and hidden attractors, forced mechanical models, and semiconductor superlattices, exhibiting the universality of appearance of the observed patterns and relations.

  5. Coexisting chaotic attractors in a single neuron model with adapting feedback synapse

    Li Chunguang; Chen Guanrong

    2005-01-01

    In this paper, we consider the nonlinear dynamical behavior of a single neuron model with adapting feedback synapse, and show that chaotic behaviors exist in this model. In some parameter domain, we observe two coexisting chaotic attractors, switching from the coexisting chaotic attractors to a connected chaotic attractor, and then switching back to the two coexisting chaotic attractors. We confirm the chaoticity by simulations with phase plots, waveform plots, and power spectra

  6. Existence of a new three-dimensional chaotic attractor

    Wang Jiezhi; Chen Zengqiang; Yuan Zhuzhi

    2009-01-01

    In this paper, one heteroclinic orbit of a new three-dimensional continuous autonomous chaotic system, whose chaotic attractor belongs to the conjugate Lue attractor, is found. The series expression of the heteroclinic orbit of Shil'nikov type is derived by using the undetermined coefficient method. The uniform convergence of the precise series expansions of this heteroclinic orbits is proved. According to the Shil'nikov theorem, this system clearly has Smale horseshoes and the horseshoe chaos.

  7. A new chaotic attractor with two quadratic nonlinearities, its synchronization and circuit implementation

    Vaidyanathan, S.; Sambas, A.; Sukono; Mamat, M.; Gundara, G.; Mada Sanjaya, W. S.; Subiyanto

    2018-03-01

    A 3-D new chaotic attractor with two quadratic nonlinearities is proposed in this paper. The dynamical properties of the new chaotic system are described in terms of phase portraits, equilibrium points, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. We show that the new chaotic system has three unstable equilibrium points. The new chaotic attractor is dissipative in nature. As an engineering application, adaptive synchronization of identical new chaotic attractors is designed via nonlinear control and Lyapunov stability theory. Furthermore, an electronic circuit realization of the new chaotic attractor is presented in detail to confirm the feasibility of the theoretical chaotic attractor model.

  8. A novel 3D autonomous system with different multilayer chaotic attractors

    Dong Gaogao; Du Ruijin; Tian Lixin; Jia Qiang

    2009-01-01

    This Letter proposes a novel three-dimensional autonomous system which has complex chaotic dynamics behaviors and gives analysis of novel system. More importantly, the novel system can generate three-layer chaotic attractor, four-layer chaotic attractor, five-layer chaotic attractor, multilayer chaotic attractor by choosing different parameters and initial condition. We analyze the new system by means of phase portraits, Lyapunov exponent spectrum, fractional dimension, bifurcation diagram and Poincare maps of the system. The three-dimensional autonomous system is totally different from the well-known systems in previous work. The new multilayer chaotic attractors are also worth causing attention.

  9. Low-dimensional chaotic attractors in drift wave turbulence

    Persson, M.; Nordman, H.

    1991-01-01

    Simulation results of toroidal η i -mode turbulence are analyzed using mathematical tools of nonlinear dynamics. Low-dimensional chaotic attractors are found in the strongly nonlinear regime while in the weakly interacting regime the dynamics is high dimensional. In both regimes, the solutions are found to display sensitive dependence on initial conditions, characterized by a positive largest Liapunov exponent. (au)

  10. A New Chaotic Attractor with Quadratic Exponential Nonlinear Term from Chen’s Attractor

    Iftikhar Ahmed

    2014-02-01

    Full Text Available In this paper a new three-dimensional chaotic system is proposed, which relies on a nonlinear exponential term and a nonlinear quadratic cross term necessary for folding trajectories. Basic dynamical characteristics of the new system are analyzed. Compared with the Chen system, the equilibrium points of the new system does not contain the origin, and has a greater positive Lyapunov index, can produce more complex shaped chaotic attractor.

  11. A novel one equilibrium hyper-chaotic system generated upon Lü attractor

    Hong-Yan, Jia; Zeng-Qiang, Chen; Zhu-Zhi, Yuan

    2010-01-01

    By introducing an additional state feedback into a three-dimensional autonomous chaotic attractor Lü system, this paper presents a novel four-dimensional continuous autonomous hyper-chaotic system which has only one equilibrium. There are only 8 terms in all four equations of the new hyper-chaotic system, which may be less than any other four-dimensional continuous autonomous hyper-chaotic systems generated by three-dimensional (3D) continuous autonomous chaotic systems. The hyper-chaotic system undergoes Hopf bifurcation when parameter c varies, and becomes the 3D modified Lü system when parameter k varies. Although the hyper-chaotic system does not undergo Hopf bifurcation when parameter k varies, many dynamic behaviours such as periodic attractor, quasi periodic attractor, chaotic attractor and hyper-chaotic attractor can be observed. A circuit is also designed when parameter k varies and the results of the circuit experiment are in good agreement with those of simulation. (general)

  12. Determining the flexibility of regular and chaotic attractors

    Marhl, Marko; Perc, Matjaz

    2006-01-01

    We present an overview of measures that are appropriate for determining the flexibility of regular and chaotic attractors. In particular, we focus on those system properties that constitute its responses to external perturbations. We deploy a systematic approach, first introducing the simplest measure given by the local divergence of the system along the attractor, and then develop more rigorous mathematical tools for estimating the flexibility of the system's dynamics. The presented measures are tested on the regular Brusselator and chaotic Hindmarsh-Rose model of an excitable neuron with equal success, thus indicating the overall effectiveness and wide applicability range of the proposed theory. Since responses of dynamical systems to external signals are crucial in several scientific disciplines, and especially in natural sciences, we discuss several important aspects and biological implications of obtained results

  13. Chaotic Attractor Crisis and Climate Sensitivity: a Transfer Operator Approach

    Tantet, A.; Lucarini, V.; Lunkeit, F.; Dijkstra, H. A.

    2015-12-01

    The rough response to a smooth parameter change of some non-chaotic climate models, such as the warm to snowball-Earth transition in energy balance models due to the ice-albedo feedback, can be studied in the framework of bifurcation theory, in particular by analysing the Lyapunov spectrum of fixed points or periodic orbits. However, bifurcation theory is of little help to study the destruction of a chaotic attractor which can occur in high-dimensional General Circulation Models (GCM). Yet, one would expect critical slowing down to occur before the crisis, since, as the system becomes susceptible to the physical instability mechanism responsible for the crisis, it turns out to be less and less resilient to exogenous perturbations and to spontaneous fluctuations due to other types of instabilities on the attractor. The statistical physics framework, extended to nonequilibrium systems, is particularly well suited for the study of global properties of chaotic and stochastic systems. In particular, the semigroup of transfer operators governs the evolution of distributions in phase space and its spectrum characterises both the relaxation rate of distributions to a statistical steady-state and the stability of this steady-state to perturbations. If critical slowing down indeed occurs in the approach to an attractor crisis, the gap in the spectrum of the semigroup of transfer operators is expected to shrink. We show that the chaotic attractor crisis due to the ice-albedo feedback and resulting in a transition from a warm to a snowball-Earth in the Planet Simulator (PlaSim), a GCM of intermediate complexity, is associated with critical slowing down, as observed by the slower decay of correlations before the crisis (cf. left panel). In addition, we demonstrate that this critical slowing down can be traced back to the shrinkage of the gap between the leading eigenvalues of coarse-grained approximations of the transfer operators and that these eigenvalues capture the

  14. Generation and control of multi-scroll chaotic attractors in fractional order systems

    Ahmad, Wajdi M.

    2005-01-01

    The objective of this paper is twofold: on one hand we demonstrate the generation of multi-scroll attractors in fractional order chaotic systems. Then, we design state feedback controllers to eliminate chaos from the system trajectories. It is demonstrated that modifying the underlying nonlinearity of the fractional chaotic system results in the birth of multiple chaotic attractors, thus forming the so called multi-scroll attractors. The presence of chaotic behavior is evidenced by a positive largest Lyapunov exponent computed for the output time series. We investigate generation and control of multi-scroll attractors in two different models, both of which are fractional order and chaotic: an electronic oscillator, and a mechanical 'jerk' model. The current findings extend previously reported results on generation of n-scroll attractors from the domain of integer order to the domain of fractional order chaotic systems, and addresses the issue of controlling such chaotic behaviors. Our investigations are validated through numerical simulations

  15. Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

    Tantet, Alexis; Lucarini, Valerio; Dijkstra, Henk A.

    2018-02-01

    Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems. However, it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle. This behavior is investigated here for a boundary crisis in the Lorenz flow, for which neither the Lyapunov exponents nor the covariant Lyapunov vectors provide a criterion for the crisis. Instead, the convergence of the time evolution of probability densities to the invariant measure, governed by the semigroup of transfer operators, is expected to slow down at the approach of the crisis. Such convergence is described by the eigenvalues of the generator of this semigroup, which can be divided into two families, referred to as the stable and unstable Ruelle-Pollicott resonances, respectively. The former describes the convergence of densities to the attractor (or escape from a repeller) and is estimated from many short time series sampling the state space. The latter is responsible for the decay of correlations, or mixing, and can be estimated from a long times series, invoking ergodicity. It is found numerically for the Lorenz flow that the stable resonances do approach the imaginary axis during the crisis, as is indicative of the loss of global stability of the attractor. On the other hand, the unstable resonances, and a fortiori the decay of correlations, do not flag the proximity of the crisis, thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises.

  16. Lifetime of chaotic attractors in a multidimensional laser system

    Pando L, C.L.; Cerdeira, H.A.

    1995-01-01

    We study the lifetimes of chaotic attractors at crises in a multidimensional laser system. This system describes the CO 2 laser with modulated losses and is known as the four-level model. The critical exponents which are related to the lifetimes of the attractors are estimated in terms of the corresponding eigenvalues and the measured characteristic lifetime in the model. The critical exponents in this model and those of its center manifold version are in good agreement. We conjecture that generically in the four-level model the critical exponents are close to 1/2 at crises. In addition, we compare predictions of a simpler and popular model known as the two-level model with those of the above mentioned models. (author). 21 refs, 2 figs, 3 tabs

  17. Generalized pole inflation: Hilltop, natural, and chaotic inflationary attractors

    Terada, Takahiro, E-mail: takahiro.terada@apctp.org [Department of Physics, The University of Tokyo, Tokyo 113-0033 (Japan); Asia Pacific Center for Theoretical Physics, Pohang 37673 (Korea, Republic of)

    2016-09-10

    A reformulation of inflationary model analyses appeared recently, in which inflationary observables are determined by the structure of a pole in the inflaton kinetic term rather than the shape of the inflaton potential. We comprehensively study this framework with an arbitrary order of the pole taking into account possible additional poles in the kinetic term or in the potential. Depending on the setup, the canonical potential becomes the form of hilltop or plateau models, variants of natural inflation, power-law inflation, or monomial/polynomial chaotic inflation. We demonstrate attractor behaviors of these models and compute corrections from the additional poles to the inflationary observables.

  18. When Darwin meets Lorenz: Evolving new chaotic attractors through genetic programming

    Pan, Indranil; Das, Saptarshi

    2015-01-01

    Highlights: •New 3D continuous time chaotic systems with analytical expressions are obtained. •The multi-gene genetic programming (MGGP) paradigm is employed to achieve this. •Extends earlier works for evolving generalised family of Lorenz attractors. •Over one hundred of new chaotic attractors along with their parameters are reported. •The MGGP method have the potential for finding other similar chaotic attractors. -- Abstract: In this paper, we propose a novel methodology for automatically finding new chaotic attractors through a computational intelligence technique known as multi-gene genetic programming (MGGP). We apply this technique to the case of the Lorenz attractor and evolve several new chaotic attractors based on the basic Lorenz template. The MGGP algorithm automatically finds new nonlinear expressions for the different state variables starting from the original Lorenz system. The Lyapunov exponents of each of the attractors are calculated numerically based on the time series of the state variables using time delay embedding techniques. The MGGP algorithm tries to search the functional space of the attractors by aiming to maximise the largest Lyapunov exponent (LLE) of the evolved attractors. To demonstrate the potential of the proposed methodology, we report over one hundred new chaotic attractor structures along with their parameters, which are evolved from just the Lorenz system alone

  19. Application of fixed point theory to chaotic attractors of forced oscillators

    Stewart, H.B.

    1990-11-01

    A review of the structure of chaotic attractors of periodically forced second order nonlinear oscillators suggests that the theory of fixed points of transformations gives information about the fundamental topological structure of attractors. First a simple extension of the Levinson index formula is proved. Then numerical evidence is used to formulate plausible conjectures about absorbing regions containing chaotic attractors in forced oscillators. Applying the Levinson formula suggests a fundamental relation between the number of fixed points or periodic points in a section of the chaotic attractor on the one hand, and a topological invariant of an absorbing region on the other hand. (author)

  20. Generating two simultaneously chaotic attractors with a switching piecewise-linear controller

    Zheng Zuohuan; Lue Jinhu; Chen Guanrong; Zhou Tianshou; Zhang Suochun

    2004-01-01

    It has been demonstrated that a piecewise-linear system can generate chaos under suitable conditions. This paper proposes a novel method for simultaneously creating two symmetrical chaotic attractor--an upper-attractor and a lower-attractor--in a 3D linear autonomous system. Basically dynamical behaviors of this new chaotic system are further investigated. Especially, the chaos formation mechanism is explored by analyzing the structure of fixed points and the system trajectories

  1. Amplification and displacement of chaotic attractors by means of unidirectional chaotic driving

    González-Miranda, J. M.

    1998-06-01

    Chaotic systems, when used to drive copies of themselves (or parts of themselves) may induce interesting behaviors in the driven system. In case the later exhibits invariance under amplification or translation, they may show amplification (reduction), or displacement of the attractor. It is shown how the behavior to be obtained is implied by the symmetries involved. Two explicit examples are studied to show how these phenomena manifest themselves under perfect and imperfect coupling.

  2. A New Chaotic System with Multiple Attractors: Dynamic Analysis, Circuit Realization and S-Box Design

    Qiang Lai

    2017-12-01

    Full Text Available This paper reports about a novel three-dimensional chaotic system with three nonlinearities. The system has one stable equilibrium, two stable equilibria and one saddle node, two saddle foci and one saddle node for different parameters. One salient feature of this novel system is its multiple attractors caused by different initial values. With the change of parameters, the system experiences mono-stability, bi-stability, mono-periodicity, bi-periodicity, one strange attractor, and two coexisting strange attractors. The complex dynamic behaviors of the system are revealed by analyzing the corresponding equilibria and using the numerical simulation method. In addition, an electronic circuit is given for implementing the chaotic attractors of the system. Using the new chaotic system, an S-Box is developed for cryptographic operations. Moreover, we test the performance of this produced S-Box and compare it to the existing S-Box studies.

  3. Dynamic analyses, FPGA implementation and engineering applications of multi-butterfly chaotic attractors generated from generalised Sprott C system

    Lai, Qiang; Zhao, Xiao-Wen; Rajagopal, Karthikeyan; Xu, Guanghui; Akgul, Akif; Guleryuz, Emre

    2018-01-01

    This paper considers the generation of multi-butterfly chaotic attractors from a generalised Sprott C system with multiple non-hyperbolic equilibria. The system is constructed by introducing an additional variable whose derivative has a switching function to the Sprott C system. It is numerically found that the system creates two-, three-, four-, five-butterfly attractors and any other multi-butterfly attractors. First, the dynamic analyses of multi-butterfly chaotic attractors are presented. Secondly, the field programmable gate array implementation, electronic circuit realisation and random number generator are done with the multi-butterfly chaotic attractors.

  4. A novel double-convection chaotic attractor, its adaptive control and circuit simulation

    Mamat, M.; Vaidyanathan, S.; Sambas, A.; Mujiarto; Sanjaya, W. S. M.; Subiyanto

    2018-03-01

    A 3-D novel double-convection chaotic system with three nonlinearities is proposed in this research work. The dynamical properties of the new chaotic system are described in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, stability analysis of equilibria, etc. Adaptive control and synchronization of the new chaotic system with unknown parameters are achieved via nonlinear controllers and the results are established using Lyapunov stability theory. Furthermore, an electronic circuit realization of the new 3-D novel chaotic system is presented in detail. Finally, the circuit experimental results of the 3-D novel chaotic attractor show agreement with the numerical simulations.

  5. Dynamic analysis, circuit implementation and passive control of a novel four-dimensional chaotic system with multiscroll attractor and multiple coexisting attractors

    Lai, Bang-Cheng; He, Jian-Jun

    2018-03-01

    In this paper, we construct a novel 4D autonomous chaotic system with four cross-product nonlinear terms and five equilibria. The multiple coexisting attractors and the multiscroll attractor of the system are numerically investigated. Research results show that the system has various types of multiple attractors, including three strange attractors with a limit cycle, three limit cycles, two strange attractors with a pair of limit cycles, two coexisting strange attractors. By using the passive control theory, a controller is designed for controlling the chaos of the system. Both analytical and numerical studies verify that the designed controller can suppress chaotic motion and stabilise the system at the origin. Moreover, an electronic circuit is presented for implementing the chaotic system.

  6. Generation of multi-wing chaotic attractor in fractional order system

    Zhang Chaoxia; Yu Simin

    2011-01-01

    Highlights: → We investigate a novel approach for generating multi-wing chaotic attractors. → We introduce a fundamental fractional differential nominal linear system. → A proper nonlinear state feedback controller is designed. → The controlled system can generate fractional-order multi-wing chaotic attractors. - Abstract: In this paper, a novel approach is proposed for generating multi-wing chaotic attractors from the fractional linear differential system via nonlinear state feedback controller equipped with a duality-symmetric multi-segment quadratic function. The main idea is to design a proper nonlinear state feedback controller by using four construction criterions from a fundamental fractional differential nominal linear system, so that the controlled fractional differential system can generate multi-wing chaotic attractors. It is the first time in the literature to report the multi-wing chaotic attractors from an uncoupled fractional differential system. Furthermore, some basic dynamical analysis and numerical simulations are also given, confirming the effectiveness of the proposed method.

  7. On dynamics analysis of a new chaotic attractor

    Zhou Wuneng; Xu Yuhua; Lu Hongqian; Pan Lin

    2008-01-01

    In this Letter, a new chaotic system is discussed. Some basic dynamical properties, such as Lyapunov exponents, Poincare mapping, fractal dimension, bifurcation diagram, continuous spectrum and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed in this Letter is a new chaotic system and deserves a further detailed investigation

  8. Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic.

    Safonov, Leonid A.; Tomer, Elad; Strygin, Vadim V.; Ashkenazy, Yosef; Havlin, Shlomo

    2002-12-01

    We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.

  9. A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems

    Aguirre-Hernández, B.; Campos-Cantón, E.; López-Renteria, J.A.; Díaz González, E.C.

    2015-01-01

    In this paper, we consider characteristic polynomials of n-dimensional systems that determine a segment of polynomials. One parameter is used to characterize this segment of polynomials in order to determine the maximal interval of dissipativity and unstability. Then we apply this result to the generation of a family of attractors based on a class of unstable dissipative systems (UDS) of type affine linear systems. This class of systems is comprised of switched linear systems yielding strange attractors. A family of these chaotic switched systems is determined by the maximal interval of perturbation of the matrix that governs the dynamics for still having scroll attractors

  10. Novel four-wing and eight-wing attractors using coupled chaotic Lorenz systems

    Grassi, Giuseppe

    2008-01-01

    This paper presents the problem of generating four-wing (eight-wing) chaotic attractors. The adopted method consists in suitably coupling two (three) identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings (eight wings) of these novel attractors are located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues. (general)

  11. Implementation of a novel two-attractor grid multi-scroll chaotic system

    Xiao-Hua, Luo; Zheng-Wei, Tu; Xi-Rui, Liu; Chang, Cai; Pu, Gong; Yi-Long, Liang

    2010-01-01

    This paper proposed a method of generating two attractors in a novel grid multi-scroll chaotic system. Based on a newly generated three-dimensional system, a two-attractor grid multi-scroll attractor system can be generated by adding two triangular waves and a sign function. Some basic dynamical properties, such as equilibrium points, bifurcations, and phase diagrams, were studied. Furthermore, the system was experimentally confirmed by an electronic circuit. The circuit simulation results and numerical simulation results verified the feasibility of this method. (general)

  12. Attractors of relaxation discrete-time systems with chaotic dynamics on a fast time scale

    Maslennikov, Oleg V.; Nekorkin, Vladimir I.

    2016-01-01

    In this work, a new type of relaxation systems is considered. Their prominent feature is that they comprise two distinct epochs, one is slow regular motion and another is fast chaotic motion. Unlike traditionally studied slow-fast systems that have smooth manifolds of slow motions in the phase space and fast trajectories between them, in this new type one observes, apart the same geometric objects, areas of transient chaos. Alternating periods of slow regular motions and fast chaotic ones as well as transitions between them result in a specific chaotic attractor with chaos on a fast time scale. We formulate basic properties of such attractors in the framework of discrete-time systems and consider several examples. Finally, we provide an important application of such systems, the neuronal electrical activity in the form of chaotic spike-burst oscillations.

  13. Statistical properties of chaotic dynamical systems which exhibit strange attractors

    Jensen, R.V.; Oberman, C.R.

    1981-07-01

    A path integral method is developed for the calculation of the statistical properties of turbulent dynamical systems. The method is applicable to conservative systems which exhibit a transition to stochasticity as well as dissipative systems which exhibit strange attractors. A specific dissipative mapping is considered in detail which models the dynamics of a Brownian particle in a wave field with a broad frequency spectrum. Results are presented for the low order statistical moments for three turbulent regimes which exhibit strange attractors corresponding to strong, intermediate, and weak collisional damping

  14. A new two-scroll chaotic attractor with three quadratic nonlinearities, its adaptive control and circuit design

    Lien, C.-H.; Vaidyanathan, S.; Sambas, A.; Sukono; Mamat, M.; Sanjaya, W. S. M.; Subiyanto

    2018-03-01

    A 3-D new two-scroll chaotic attractor with three quadratic nonlinearities is investigated in this paper. First, the qualitative and dynamical properties of the new two-scroll chaotic system are described in terms of phase portraits, equilibrium points, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. We show that the new two-scroll dissipative chaotic system has three unstable equilibrium points. As an engineering application, global chaos control of the new two-scroll chaotic system with unknown system parameters is designed via adaptive feedback control and Lyapunov stability theory. Furthermore, an electronic circuit realization of the new chaotic attractor is presented in detail to confirm the feasibility of the theoretical chaotic two-scroll attractor model.

  15. A New Chaotic Flow with Hidden Attractor: The First Hyperjerk System with No Equilibrium

    Ren, Shuili; Panahi, Shirin; Rajagopal, Karthikeyan; Akgul, Akif; Pham, Viet-Thanh; Jafari, Sajad

    2018-02-01

    Discovering unknown aspects of non-equilibrium systems with hidden strange attractors is an attractive research topic. A novel quadratic hyperjerk system is introduced in this paper. It is noteworthy that this non-equilibrium system can generate hidden chaotic attractors. The essential properties of such systems are investigated by means of equilibrium points, phase portrait, bifurcation diagram, and Lyapunov exponents. In addition, a fractional-order differential equation of this new system is presented. Moreover, an electronic circuit is also designed and implemented to verify the feasibility of the theoretical model.

  16. On the New Scenario of Annihilation of the Cross-Well Chaotic Attractor in a Nonlinear Oscillator

    Szemplinska, W.; Zubrzycki, A.; Tyrkiel, E.

    1999-01-01

    The twin-well potential Duffing oscillator is considered and the investigations are focused on a new scenario of destruction of the cross-well chaotic attractor. The new phenomenon belongs to the category of subduction bifurcation and consists in replacement of the cross-well chaotic attractor by a pair of unsymmetric 2T-periodic attractors. It is shown that the new scenario forms a transition zone in the system control parameter plane, the zone, which separates the two known scenarios of annihilation of the cross-well chaotic attractor: the boundary crisis, and the subduction in which the two single-well T-periodic attractors are born in a saddle-node bifurcation. (author)

  17. Architecture of chaotic attractors for flows in the absence of any singular point

    Letellier, Christophe [CORIA-UMR 6614 Normandie Université, CNRS-Université et INSA de Rouen, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray (France); Malasoma, Jean-Marc [Université de Lyon, ENTPE, Laboratoire Génie Civil et Bâtiment, 3 Rue Maurice Audin, F-69518 Vaulx-en-Velin Cedex (France)

    2016-06-15

    Some chaotic attractors produced by three-dimensional dynamical systems without any singular point have now been identified, but explaining how they are structured in the state space remains an open question. We here want to explain—in the particular case of the Wei system—such a structure, using one-dimensional sets obtained by vanishing two of the three derivatives of the flow. The neighborhoods of these sets are made of points which are characterized by the eigenvalues of a 2 × 2 matrix describing the stability of flow in a subspace transverse to it. We will show that the attractor is spiralling and twisted in the neighborhood of one-dimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such one-dimensional sets are also useful in explaining the structure of attractors produced by systems with singular points, by considering the case of the Lorenz system.

  18. On the control of the chaotic attractors of the 2-d Navier-Stokes equations.

    Smaoui, Nejib; Zribi, Mohamed

    2017-03-01

    The control problem of the chaotic attractors of the two dimensional (2-d) Navier-Stokes (N-S) equations is addressed in this paper. First, the Fourier Galerkin method based on a reduced-order modelling approach developed by Chen and Price is applied to the 2-d N-S equations to construct a fifth-order system of nonlinear ordinary differential equations (ODEs). The dynamics of the fifth-order system was studied by analyzing the system's attractor for different values of Reynolds number, R e . Then, control laws are proposed to drive the states of the ODE system to a desired attractor. Finally, an adaptive controller is designed to synchronize two reduced order ODE models having different Reynolds numbers and starting from different initial conditions. Simulation results indicate that the proposed control schemes work well.

  19. Chaotic inflation as an attractor in initial-condition space

    Kung, J.H.; Brandenberger, R.H.

    1990-01-01

    We study the evolution of scalar field inhomogeneities in the preinflationary phase of an inflationary universe. We decompose the scalar field configuration in Fourier modes and consider initial conditions in which more than one mode is excited. We find that the long-wavelength modes are stable against perturbations due to short-wavelength excitations and that chaotic inflation results even if at the initial time the short waves contain most of the energy density

  20. Crisis of the chaotic attractor of a climate model: a transfer operator approach

    Tantet, Alexis; Lucarini, Valerio; Lunkeit, Frank; Dijkstra, Henk A.

    2018-05-01

    The destruction of a chaotic attractor leading to rough changes in the dynamics of a dynamical system is studied. Local bifurcations are known to be characterised by a single or a pair of characteristic exponents crossing the imaginary axis. As a result, the approach of such bifurcations in the presence of noise can be inferred from the slowing down of the decay of correlations (Held and Kleinen 2004 Geophys. Res. Lett. 31 1–4). On the other hand, little is known about global bifurcations involving high-dimensional attractors with several positive Lyapunov exponents. It is known that the global stability of chaotic attractors may be characterised by the spectral properties of the Koopman (Mauroy and Mezić 2016 IEEE Trans. Autom. Control 61 3356–69) or the transfer operators governing the evolution of statistical ensembles. Accordingly, it has recently been shown (Tantet 2017 J. Stat. Phys. 1–33) that a boundary crisis in the Lorenz flow coincides with the approach to the unit circle of the eigenvalues of these operators associated with motions about the attractor, the stable resonances. A second class of resonances, the unstable resonances, are responsible for the decay of correlations and mixing on the attractor. In the deterministic case, these cannot be expected to be affected by general boundary crises. Here, however, we give an example of a chaotic system in which slowing down of the decay of correlations of some observables does occur at the approach of a boundary crisis. The system considered is a high-dimensional, chaotic climate model of physical relevance. Moreover, coarse-grained approximations of the transfer operators on a reduced space, constructed from a long time series of the system, give evidence that this behaviour is due to the approach of unstable resonances to the unit circle. That the unstable resonances are affected by the crisis can be physically understood from the fact that the process responsible for the instability, the ice

  1. Chaotic behaviour of the Rossler model and its analysis by using bifurcations of limit cycles and chaotic attractors

    Ibrahim, K. M.; Jamal, R. K.; Ali, F. H.

    2018-05-01

    The behaviour of certain dynamical nonlinear systems are described in term as chaos, i.e., systems’ variables change with the time, displaying very sensitivity to initial conditions of chaotic dynamics. In this paper, we study archetype systems of ordinary differential equations in two-dimensional phase spaces of the Rössler model. A system displays continuous time chaos and is explained by three coupled nonlinear differential equations. We study its characteristics and determine the control parameters that lead to different behavior of the system output, periodic, quasi-periodic and chaos. The time series, attractor, Fast Fourier Transformation and bifurcation diagram for different values have been described.

  2. Strange attractors in a chaotic coin flip simulation

    Cooper, Crystal

    2006-01-01

    Presented is a computer simulation used to model a variation of the game known as the gambler's ruin. A rich player gambles with a set amount of money m. The poor player starts out with zero capital, and is allowed to flip a coin in order to try to win the money. If the coin is heads, the poor player wins a dollar but if it is tails, the player loses a dollar. The poor player is always allowed to win the first flip, and is allowed to flip n times, even when the amount of money lost reaches zero. The dynamics of this process is chaotic due to fluctuations in the variance of the amount of money

  3. Unraveling chaotic attractors by complex networks and measurements of stock market complexity.

    Cao, Hongduo; Li, Ying

    2014-03-01

    We present a novel method for measuring the complexity of a time series by unraveling a chaotic attractor modeled on complex networks. The complexity index R, which can potentially be exploited for prediction, has a similar meaning to the Kolmogorov complexity (calculated from the Lempel-Ziv complexity), and is an appropriate measure of a series' complexity. The proposed method is used to research the complexity of the world's major capital markets. None of these markets are completely random, and they have different degrees of complexity, both over the entire length of their time series and at a level of detail. However, developing markets differ significantly from mature markets. Specifically, the complexity of mature stock markets is stronger and more stable over time, whereas developing markets exhibit relatively low and unstable complexity over certain time periods, implying a stronger long-term price memory process.

  4. Unraveling chaotic attractors by complex networks and measurements of stock market complexity

    Cao, Hongduo; Li, Ying

    2014-01-01

    We present a novel method for measuring the complexity of a time series by unraveling a chaotic attractor modeled on complex networks. The complexity index R, which can potentially be exploited for prediction, has a similar meaning to the Kolmogorov complexity (calculated from the Lempel–Ziv complexity), and is an appropriate measure of a series' complexity. The proposed method is used to research the complexity of the world's major capital markets. None of these markets are completely random, and they have different degrees of complexity, both over the entire length of their time series and at a level of detail. However, developing markets differ significantly from mature markets. Specifically, the complexity of mature stock markets is stronger and more stable over time, whereas developing markets exhibit relatively low and unstable complexity over certain time periods, implying a stronger long-term price memory process

  5. Reconstruction of chaotic saddles by classification of unstable periodic orbits: Kuramoto-Sivashinsky equation

    Saiki, Yoshitaka; Yamada, Michio; Chian, Abraham C.-L.; Miranda, Rodrigo A.; Rempel, Erico L.

    2015-01-01

    The unstable periodic orbits (UPOs) embedded in a chaotic attractor after an attractor merging crisis (MC) are classified into three subsets, and employed to reconstruct chaotic saddles in the Kuramoto-Sivashinsky equation. It is shown that in the post-MC regime, the two chaotic saddles evolved from the two coexisting chaotic attractors before crisis can be reconstructed from the UPOs embedded in the pre-MC chaotic attractors. The reconstruction also involves the detection of the mediating UPO responsible for the crisis, and the UPOs created after crisis that fill the gap regions of the chaotic saddles. We show that the gap UPOs originate from saddle-node, period-doubling, and pitchfork bifurcations inside the periodic windows in the post-MC chaotic region of the bifurcation diagram. The chaotic attractor in the post-MC regime is found to be the closure of gap UPOs

  6. Reconstruction of chaotic saddles by classification of unstable periodic orbits: Kuramoto-Sivashinsky equation

    Saiki, Yoshitaka, E-mail: yoshi.saiki@r.hit-u.ac.jp [Graduate School of Commerce and Management, Hitotsubashi University, Tokyo 186-8601 (Japan); Yamada, Michio [Research Institute for Mathematical Sciences (RIMS), Kyoto University, Kyoto 606-8502 (Japan); Chian, Abraham C.-L. [Paris Observatory, LESIA, CNRS, 92195 Meudon (France); National Institute for Space Research (INPE), P.O. Box 515, São José dos Campos, São Paulo 12227-010 (Brazil); Institute of Aeronautical Technology (ITA) and World Institute for Space Environment Research (WISER), São José dos Campos, São Paulo 12228-900 (Brazil); School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005 (Australia); Department of Biomedical Engineering, George Washington University, Washington, DC 20052 (United States); Miranda, Rodrigo A. [Faculty UnB-Gama, and Plasma Physics Laboratory, Institute of Physics, University of Brasília (UnB), Brasília DF 70910-900 (Brazil); Rempel, Erico L. [Institute of Aeronautical Technology (ITA) and World Institute for Space Environment Research (WISER), São José dos Campos, São Paulo 12228-900 (Brazil)

    2015-10-15

    The unstable periodic orbits (UPOs) embedded in a chaotic attractor after an attractor merging crisis (MC) are classified into three subsets, and employed to reconstruct chaotic saddles in the Kuramoto-Sivashinsky equation. It is shown that in the post-MC regime, the two chaotic saddles evolved from the two coexisting chaotic attractors before crisis can be reconstructed from the UPOs embedded in the pre-MC chaotic attractors. The reconstruction also involves the detection of the mediating UPO responsible for the crisis, and the UPOs created after crisis that fill the gap regions of the chaotic saddles. We show that the gap UPOs originate from saddle-node, period-doubling, and pitchfork bifurcations inside the periodic windows in the post-MC chaotic region of the bifurcation diagram. The chaotic attractor in the post-MC regime is found to be the closure of gap UPOs.

  7. Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data

    Pathak, Jaideep; Lu, Zhixin; Hunt, Brian R.; Girvan, Michelle; Ott, Edward

    2017-12-01

    We use recent advances in the machine learning area known as "reservoir computing" to formulate a method for model-free estimation from data of the Lyapunov exponents of a chaotic process. The technique uses a limited time series of measurements as input to a high-dimensional dynamical system called a "reservoir." After the reservoir's response to the data is recorded, linear regression is used to learn a large set of parameters, called the "output weights." The learned output weights are then used to form a modified autonomous reservoir designed to be capable of producing an arbitrarily long time series whose ergodic properties approximate those of the input signal. When successful, we say that the autonomous reservoir reproduces the attractor's "climate." Since the reservoir equations and output weights are known, we can compute the derivatives needed to determine the Lyapunov exponents of the autonomous reservoir, which we then use as estimates of the Lyapunov exponents for the original input generating system. We illustrate the effectiveness of our technique with two examples, the Lorenz system and the Kuramoto-Sivashinsky (KS) equation. In the case of the KS equation, we note that the high dimensional nature of the system and the large number of Lyapunov exponents yield a challenging test of our method, which we find the method successfully passes.

  8. The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations

    Simpson, D.J.W., E-mail: d.j.w.simpson@massey.ac.nz

    2016-09-07

    An attractor of a piecewise-smooth continuous system of differential equations can bifurcate from a stable equilibrium to a more complicated invariant set when it collides with a switching manifold under parameter variation. Here numerical evidence is provided to show that this invariant set can be chaotic. The transition occurs locally (in a neighbourhood of a point) and instantaneously (for a single critical parameter value). This phenomenon is illustrated for the normal form of a boundary equilibrium bifurcation in three dimensions using parameter values adapted from of a piecewise-linear model of a chaotic electrical circuit. The variation of a secondary parameter reveals a period-doubling cascade to chaos with windows of periodicity. The dynamics is well approximated by a one-dimensional unimodal map which explains the bifurcation structure. The robustness of the attractor is also investigated by studying the influence of nonlinear terms. - Highlights: • A boundary equilibrium bifurcation involving stable and saddle foci is considered. • A two-dimensional return map is constructed and approximated by a one-dimensional map. • A trapping region and Smale horseshoe are identified for a Rössler-like attractor. • Bifurcation diagrams reveal period-doubling cascades and windows of periodicity.

  9. Transition from complete synchronization to spatio-temporal chaos in coupled chaotic systems with nonhyperbolic and hyperbolic attractors

    Rybalova, Elena; Semenova, Nadezhda; Strelkova, Galina; Anishchenko, Vadim

    2017-06-01

    We study the transition from coherence (complete synchronization) to incoherence (spatio-temporal chaos) in ensembles of nonlocally coupled chaotic maps with nonhyperbolic and hyperbolic attractors. As basic models of a partial element we use the Henon map and the Lozi map. We show that the transition to incoherence in a ring of coupled Henon maps occurs through the appearance of phase and amplitude chimera states. An ensemble of coupled Lozi maps demonstrates the coherence-incoherence transition via solitary states and no chimera states are observed in this case.

  10. Modelling and prediction for chaotic fir laser attractor using rational function neural network.

    Cho, S

    2001-02-01

    Many real-world systems such as irregular ECG signal, volatility of currency exchange rate and heated fluid reaction exhibit highly complex nonlinear characteristic known as chaos. These chaotic systems cannot be retreated satisfactorily using linear system theory due to its high dimensionality and irregularity. This research focuses on prediction and modelling of chaotic FIR (Far InfraRed) laser system for which the underlying equations are not given. This paper proposed a method for prediction and modelling a chaotic FIR laser time series using rational function neural network. Three network architectures, TDNN (Time Delayed Neural Network), RBF (radial basis function) network and the RF (rational function) network, are also presented. Comparisons between these networks performance show the improvements introduced by the RF network in terms of a decrement in network complexity and better ability of predictability.

  11. Chaotic attractors in tumor growth and decay: a differential equation model.

    Harney, Michael; Yim, Wen-sau

    2015-01-01

    Tumorigenesis can be modeled as a system of chaotic nonlinear differential equations. A simulation of the system is realized by converting the differential equations to difference equations. The results of the simulation show that an increase in glucose in the presence of low oxygen levels decreases tumor growth.

  12. A Novel Type of Chaotic Attractor for Quadratic Systems Without Equilibriums

    Dantsev, Danylo

    In this paper, a new chaotic dynamic system without equilibriums is presented. A conducted research of the qualitative properties of the discovered system reveals a noncompliance between the bifurcation behavior of the system and the Feigenbaum-Sharkovskii-Magnitsky theory. Additional research of known systems confirms the discrepancy.

  13. Multiple steady states, limit cycles and chaotic attractors in evolutionary games with Logit Dynamics

    Hommes, C.H.; Ochea, M.I.

    2010-01-01

    This paper investigates, by means of simple, three and four strategy games, the occurrence of periodic and chaotic behaviour in a smooth version of the Best Response Dynamics, the Logit Dynamics. The main finding is that, unlike Replicator Dynamics, generic Hopf bifurcation and thus, stable limit

  14. Multi-piecewise quadratic nonlinearity memristor and its 2N-scroll and 2N + 1-scroll chaotic attractors system.

    Wang, Chunhua; Liu, Xiaoming; Xia, Hu

    2017-03-01

    In this paper, two kinds of novel ideal active flux-controlled smooth multi-piecewise quadratic nonlinearity memristors with multi-piecewise continuous memductance function are presented. The pinched hysteresis loop characteristics of the two memristor models are verified by building a memristor emulator circuit. Using the two memristor models establish a new memristive multi-scroll Chua's circuit, which can generate 2N-scroll and 2N+1-scroll chaotic attractors without any other ordinary nonlinear function. Furthermore, coexisting multi-scroll chaotic attractors are found in the proposed memristive multi-scroll Chua's circuit. Phase portraits, Lyapunov exponents, bifurcation diagrams, and equilibrium point analysis have been used to research the basic dynamics of the memristive multi-scroll Chua's circuit. The consistency of circuit implementation and numerical simulation verifies the effectiveness of the system design.

  15. Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability.

    Zhang, Sen; Zeng, Yicheng; Li, Zhijun; Wang, Mengjiao; Xiong, Le

    2018-01-01

    By using a simple state feedback controller in a three-dimensional chaotic system, a novel 4D chaotic system is derived in this paper. The system state equations are composed of nine terms including only one constant term. Depending on the different values of the constant term, this new proposed system has a line of equilibrium points or no equilibrium points. Compared with other similar chaotic systems, the newly presented system owns more abundant and complicated dynamic properties. What interests us is the observation that if the value of the constant term of the system is nonzero, it has no equilibria, and therefore, the Shil'nikov theorem is not suitable to verify the existence of chaos for the lack of heteroclinic or homoclinic trajectory. However, one-wing, two-wing, three-wing, and four-wing hidden attractors can be obtained from this new system. In addition, various coexisting hidden attractors are obtained and the complex transient transition behaviors are also observed. More interestingly, the unusual and striking dynamic behavior of the coexistence of infinitely many hidden attractors is revealed by selecting the different initial values of the system, which means that extreme multistability arises. The rich and complex hidden dynamic characteristics of this system are investigated by phase portraits, bifurcation diagrams, Lyapunov exponents, and so on. Finally, the new system is implemented by an electronic circuit. A very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform.

  16. Horseshoes in modified Chen's attractors

    Huang Yan; Yang Xiaosong

    2005-01-01

    In this paper we study dynamics of a class of modified Chen's attractors, we show that these attractors are chaotic by giving a rigorous verification for existence of horseshoes in these systems. We prove that the Poincare maps derived from these modified Chen's attractors are semi-conjugate to the 2-shift map

  17. Polynomial law for controlling the generation of n-scroll chaotic attractors in an optoelectronic delayed oscillator

    Márquez, Bicky A., E-mail: bmarquez@ivic.gob.ve; Suárez-Vargas, José J., E-mail: jjsuarez@ivic.gob.ve; Ramírez, Javier A. [Centro de Física, Instituto Venezolano de Investigaciones Científicas, km. 11 Carretera Panamericana, Caracas 1020-A (Venezuela, Bolivarian Republic of)

    2014-09-01

    Controlled transitions between a hierarchy of n-scroll attractors are investigated in a nonlinear optoelectronic oscillator. Using the system's feedback strength as a control parameter, it is shown experimentally the transition from Van der Pol-like attractors to 6-scroll, but in general, this scheme can produce an arbitrary number of scrolls. The complexity of every state is characterized by Lyapunov exponents and autocorrelation coefficients.

  18. Complex economic dynamics: Chaotic saddle, crisis and intermittency

    Chian, Abraham C.-L.; Rempel, Erico L.; Rogers, Colin

    2006-01-01

    Complex economic dynamics is studied by a forced oscillator model of business cycles. The technique of numerical modeling is applied to characterize the fundamental properties of complex economic systems which exhibit multiscale and multistability behaviors, as well as coexistence of order and chaos. In particular, we focus on the dynamics and structure of unstable periodic orbits and chaotic saddles within a periodic window of the bifurcation diagram, at the onset of a saddle-node bifurcation and of an attractor merging crisis, and in the chaotic regions associated with type-I intermittency and crisis-induced intermittency, in non-linear economic cycles. Inside a periodic window, chaotic saddles are responsible for the transient motion preceding convergence to a periodic or a chaotic attractor. The links between chaotic saddles, crisis and intermittency in complex economic dynamics are discussed. We show that a chaotic attractor is composed of chaotic saddles and unstable periodic orbits located in the gap regions of chaotic saddles. Non-linear modeling of economic chaotic saddle, crisis and intermittency can improve our understanding of the dynamics of financial intermittency observed in stock market and foreign exchange market. Characterization of the complex dynamics of economic systems is a powerful tool for pattern recognition and forecasting of business and financial cycles, as well as for optimization of management strategy and decision technology

  19. Controlling Strange Attractor in Dynamics

    2000-01-01

    A nonlinear system which exhibits a strange attractor is considered, with the goal of illustrating how to control the chaotic dynamical system and to obtain a desired attracting periodic orbit by the OGY control algorithm.

  20. Role of multistability in the transition to chaotic phase synchronization

    Postnov, D.E.; Vadivasova, T.E.; Sosnovtseva, Olga

    1999-01-01

    In this paper we describe the transition to phase synchronization for systems of coupled nonlinear oscillators that individually follow the Feigenbaum route to chaos. A nested structure of phase synchronized regions of different attractor families is observed. With this structure, the transition...... to nonsynchronous behavior is determined by the loss of stability for the most stable synchronous mode. It is shown that the appearance of hyperchaos and the transition from lag synchronization to phase synchronization are related to the merging of chaotic attractors from different families. Numerical examples...

  1. A chaotic system with a single unstable node

    Sprott, J.C. [Department of Physics, University of Wisconsin, Madison, WI 53706 (United States); Jafari, Sajad, E-mail: sajadjafari@aut.ac.ir [Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413 (Iran, Islamic Republic of); Pham, Viet-Thanh [School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi (Viet Nam); Hosseini, Zahra Sadat [Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413 (Iran, Islamic Republic of)

    2015-09-25

    This paper describes an unusual example of a three-dimensional dissipative chaotic flow with quadratic nonlinearities in which the only equilibrium is an unstable node. The region of parameter space with bounded solutions is relatively small as is the basin of attraction, which accounts for the difficulty of its discovery. Furthermore, for some values of the parameters, the system has an attracting torus, which is uncommon in three-dimensional systems, and this torus can coexist with a strange attractor or with a limit cycle. The limit cycle and strange attractor exhibit symmetry breaking and attractor merging. All the attractors appear to be hidden in that they cannot be found by starting with initial conditions in the vicinity of the equilibrium, and thus they represent a new type of hidden attractor with important and potentially problematic engineering consequences. - Highlights: • An unusual example of a three-dimensional dissipative chaotic flow is introduced. • In this system the only equilibrium is an unstable node. • For some values of the parameters, the system has an attracting torus. • This torus can coexist with a strange attractor or with a limit cycle. • These properties are uncommon in three-dimensional systems.

  2. The Lorentz Attractor and Other Attractors in the Economic System of a Firm

    Shapovalov, V I; Kazakov, N V

    2015-01-01

    A nonlinear model of the economic system of ''a firm'' is offered. It is shown that this model has several chaotic attractors, including the Lorentz attractor and a new attractor that, in our opinion, has not yet been described in the scientific literature. The chaotic nature of the attractors that were found was confirmed by computing the Lyapunov indicators. The functioning of our economic model is demonstrated with examples of firm behaviour that change the control parameters; these are well known in practice. In particular, it is shown that changes in the specific control parameters may change the system and avoid bankruptcy for the firm

  3. Attractors and basins of dynamical systems

    Attila Dénes

    2011-03-01

    Full Text Available There are several programs for studying dynamical systems, but none of them is very useful for investigating basins and attractors of higher dimensional systems. Our goal in this paper is to show a new algorithm for finding even chaotic attractors and their basins for these systems. We present an implementation and examples for the use of this program.

  4. On the Dynamics of a Model with Coexistence of Three Attractors: A Point, a Periodic Orbit and a Strange Attractor

    Llibre, Jaume, E-mail: jllibre@mat.uab.cat [Universitat Autònoma de Barcelona, Departament de Matemàtiques (Spain); Valls, Claudia, E-mail: cvalls@math.ist.utl.pt [Universidade de Lisboa, Departamento de Matemática, Instituto Superior Técnico (Portugal)

    2017-06-15

    For a dynamical system described by a set of autonomous differential equations, an attractor can be either a point, or a periodic orbit, or even a strange attractor. Recently a new chaotic system with only one parameter has been presented where besides a point attractor and a chaotic attractor, it also has a coexisting attractor limit cycle which makes evident the complexity of such a system. We study using analytic tools the dynamics of such system. We describe its global dynamics near the infinity, and prove that it has no Darboux first integrals.

  5. Complex dynamics of a new 3D Lorenz-type autonomous chaotic ...

    Newautonomous chaotic system; chaotic attractors; Lyapunov stability theory; ultimate ... College of Mathematics and Statistics, Chongqing Technology and Business ... College of Electronic and Information Engineering, Southwest University, ...

  6. Hypogenetic chaotic jerk flows

    Li, Chunbiao; Sprott, Julien Clinton; Xing, Hongyan

    2016-01-01

    Removing the amplitude or polarity information in the feedback loop of a jerk structure shows that special nonlinearities with partial information in the variable can also lead to chaos. Some striking properties are found for this kind of hypogenetic chaotic jerk flow, including multistability of symmetric coexisting attractors from an asymmetric structure, hidden attractors with respect to equilibria but with global attraction, easy amplitude control, and phase reversal which is convenient for chaos applications. - Highlights: • Hypogenetic chaotic jerk flows with incomplete feedback of amplitude or polarity are obtained. • Multistability of symmetric coexisting attractors from an asymmetric structure is found. • Some jerk systems have hidden attractors with respect to equilibria but have global attraction. • These chaotic jerk flows have the properties of amplitude control and phase reversal.

  7. Initial conditions for chaotic inflation

    Brandenberger, R.; Kung, J.; Feldman, H.

    1991-01-01

    In contrast to many other inflationary Universe models, chaotic inflation does not depend on fine tuning initial conditions. Within the context of linear perturbation theory, it is shown that chaotic inflation is stable towards both metric and matter perturbations. Neglecting gravitational perturbations, it is shown that chaotic inflation is an attractor in initial condition space. (orig.)

  8. Attractor comparisons based on density

    Carroll, T. L.

    2015-01-01

    Recognizing a chaotic attractor can be seen as a problem in pattern recognition. Some feature vector must be extracted from the attractor and used to compare to other attractors. The field of machine learning has many methods for extracting feature vectors, including clustering methods, decision trees, support vector machines, and many others. In this work, feature vectors are created by representing the attractor as a density in phase space and creating polynomials based on this density. Density is useful in itself because it is a one dimensional function of phase space position, but representing an attractor as a density is also a way to reduce the size of a large data set before analyzing it with graph theory methods, which can be computationally intensive. The density computation in this paper is also fast to execute. In this paper, as a demonstration of the usefulness of density, the density is used directly to construct phase space polynomials for comparing attractors. Comparisons between attractors could be useful for tracking changes in an experiment when the underlying equations are too complicated for vector field modeling

  9. Coexisting multiple attractors and riddled basins of a memristive system.

    Wang, Guangyi; Yuan, Fang; Chen, Guanrong; Zhang, Yu

    2018-01-01

    In this paper, a new memristor-based chaotic system is designed, analyzed, and implemented. Multistability, multiple attractors, and complex riddled basins are observed from the system, which are investigated along with other dynamical behaviors such as equilibrium points and their stabilities, symmetrical bifurcation diagrams, and sustained chaotic states. With different sets of system parameters, the system can also generate various multi-scroll attractors. Finally, the system is realized by experimental circuits.

  10. Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap

    Qu Shixian; Lu Yongzhi; Zhang Lin; He Daren

    2008-01-01

    Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-11, period-6, chaotic band-12 and band-6 attractors. They are induced by different mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically. (general)

  11. Recurrence quantification analysis in Liu's attractor

    Balibrea, Francisco; Caballero, M. Victoria; Molera, Lourdes

    2008-01-01

    Recurrence Quantification Analysis is used to detect transitions chaos to periodical states or chaos to chaos in a new dynamical system proposed by Liu et al. This system contains a control parameter in the second equation and was originally introduced to investigate the forming mechanism of the compound structure of the chaotic attractor which exists when the control parameter is zero

  12. Unstable periodic orbits and chaotic economic growth

    Ishiyama, K.; Saiki, Y.

    2005-01-01

    We numerically find many unstable periodic solutions embedded in a chaotic attractor in a macroeconomic growth cycle model of two countries with different fiscal policies, and we focus on a special type of the unstable periodic solutions. It is confirmed that chaotic behavior represented by the model is qualitatively and quantitatively related to the unstable periodic solutions. We point out that the structure of a chaotic solution is dissolved into a class of finite unstable periodic solutions picked out among a large number of periodic solutions. In this context it is essential for the unstable periodic solutions to be embedded in the chaotic attractor

  13. Global bifurcations in fractional-order chaotic systems with an extended generalized cell mapping method

    Liu, Xiaojun [State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi' an Jiaotong University, Xi' an 710049 (China); School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001 (China); Hong, Ling, E-mail: hongling@mail.xjtu.edu.cn; Jiang, Jun [State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi' an Jiaotong University, Xi' an 710049 (China)

    2016-08-15

    Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.

  14. New robust chaotic system with exponential quadratic term

    Bao Bocheng; Li Chunbiao; Liu Zhong; Xu Jianping

    2008-01-01

    This paper proposes a new robust chaotic system of three-dimensional quadratic autonomous ordinary differential equations by introducing an exponential quadratic term. This system can display a double-scroll chaotic attractor with only two equilibria, and can be found to be robust chaotic in a very wide parameter domain with positive maximum Lyapunov exponent. Some basic dynamical properties and chaotic behaviour of novel attractor are studied. By numerical simulation, this paper verifies that the three-dimensional system can also evolve into periodic and chaotic behaviours by a constant controller. (general)

  15. Repetitive learning control of continuous chaotic systems

    Chen Maoyin; Shang Yun; Zhou Donghua

    2004-01-01

    Combining a shift method and the repetitive learning strategy, a repetitive learning controller is proposed to stabilize unstable periodic orbits (UPOs) within chaotic attractors in the sense of least mean square. If nonlinear parts in chaotic systems satisfy Lipschitz condition, the proposed controller can be simplified into a simple proportional repetitive learning controller

  16. Noise-Induced Riddling in Chaotic Systems

    Lai, Y.; Grebogi, C.

    1996-01-01

    Recent works have considered the situation of riddling where, when a chaotic attractor lying in an invariant subspace is transversely stable, the basin of the attractor can be riddled with holes that belong to the basin of another attractor. We show that riddling can be induced by arbitrarily small random noise even if the attractor is transversely unstable, and we obtain universal scaling laws for noise-induced riddling. Our results imply that the phenomenon of riddling can be more prevalent than expected before, as noise is practically inevitable in dynamical systems. copyright 1996 The American Physical Society

  17. A time-delayed method for controlling chaotic maps

    Chen Maoyin; Zhou Donghua; Shang Yun

    2005-01-01

    Combining the repetitive learning strategy and the optimality principle, this Letter proposes a time-delayed method to control chaotic maps. This method can effectively stabilize unstable periodic orbits within chaotic attractors in the sense of least mean square. Numerical simulations of some chaotic maps verify the effectiveness of this method

  18. Renormalization group structure for sums of variables generated by incipiently chaotic maps

    Fuentes, Miguel Angel; Robledo, Alberto

    2010-01-01

    We look at the limit distributions of sums of deterministic chaotic variables in unimodal maps and find a remarkable renormalization group (RG) structure associated with the operation of increment of summands and rescaling. In this structure—where the only relevant variable is the difference in control parameter from its value at the transition to chaos—the trivial fixed point is the Gaussian distribution and a novel nontrivial fixed point is a multifractal distribution that emulates the Feigenbaum attractor, and is universal in the sense of the latter. The crossover between the two fixed points is explained and the flow toward the trivial fixed point is seen to be comparable to the chaotic band merging sequence. We discuss the nature of the central limit theorem for deterministic variables

  19. Applications of chaotic neurodynamics in pattern recognition

    Baird, Bill; Freeman, Walter J.; Eeckman, Frank H.; Yao, Yong

    1991-08-01

    Network algorithms and architectures for pattern recognition derived from neural models of the olfactory system are reviewed. These span a range from highly abstract to physiologically detailed, and employ the kind of dynamical complexity observed in olfactory cortex, ranging from oscillation to chaos. A simple architecture and algorithm for analytically guaranteed associative memory storage of analog patterns, continuous sequences, and chaotic attractors in the same network is described. A matrix inversion determines network weights, given prototype patterns to be stored. There are N units of capacity in an N node network with 3N2 weights. It costs one unit per static attractor, two per Fourier component of each sequence, and three to four per chaotic attractor. There are no spurious attractors, and for sequences there is a Liapunov function in a special coordinate system which governs the approach of transient states to stored trajectories. Unsupervised or supervised incremental learning algorithms for pattern classification, such as competitive learning or bootstrap Widrow-Hoff can easily be implemented. The architecture can be ''folded'' into a recurrent network with higher order weights that can be used as a model of cortex that stores oscillatory and chaotic attractors by a Hebb rule. Network performance is demonstrated by application to the problem of real-time handwritten digit recognition. An effective system with on-line learning has been written by Eeckman and Baird for the Macintosh. It utilizes static, oscillatory, and/or chaotic attractors of two kinds--Lorenze attractors, or attractors resulting from chaotically interacting oscillatory modes. The successful application to an industrial pattern recognition problem of a network architecture of considerable physiological and dynamical complexity, developed by Freeman and Yao, is described. The data sets of the problem come in three classes of difficulty, and performance of the biological network is

  20. Lorenz-like attractors in a nonholonomic model of a rattleback

    Gonchenko, A S; Gonchenko, S V

    2015-01-01

    We study chaotic dynamics in a nonholonomic model of a rattleback stone. We show that, for certain values of parameters that characterise geometrical and physical properties of the stone, a strange Lorenz-like attractor is observed in the model. We also study bifurcation scenarios for the appearance and break-down of this attractor. (paper)

  1. Using periodic modulation to control coexisting attractors induced by delayed feedback

    Martinez-Zerega, B.E.; Pisarchik, A.N.; Tsimring, L.S.

    2003-01-01

    A delay in feedback can stabilize simultaneously several unstable periodic orbits embedded in a chaotic attractor. We show that by modulating the feedback variable it is possible to lock one of these states and eliminate other coexisting periodic attractors. The method is demonstrated with both a logistic map and a CO 2 laser model

  2. Coupled flare attractors – a discrete prototype for economic modelling

    Georg C. Hartmann

    1999-01-01

    Full Text Available A chaotic environment can give rise to “flares” if an autocatalytic variable responds in a multiplicative, threshold-type fashion to the environmental forcing. An “economic unit” similarly depends in its growth behavior on the unpredictable (chaotic? buying habits of its customers, say. It turns out that coupled flare attractors are surprisingly robust in the sense that the resulting “economy” is largely independent of the extent of diffusive coupling used. Some simulations are presented.

  3. Reduction of Dietrich-Ruina attractors to unimodal maps

    S. Shkoller

    1997-01-01

    Full Text Available We present a geometric analysis of a quasi-static single degree of freedom elastic slider with a state and rate dependent friction law. In particular, we examine and characterize the regime of chaotic motions displayed by the Dieterich-Ruina model. We do so by numerically reducing the chaotic attractors to a family of unimodal maps and discuss why this suggests complex behaviour in the dynamical system.

  4. A snapshot attractor view of the advection of inertial particles in the presence of history force

    Guseva, Ksenia; Daitche, Anton; Tél, Tamás

    2017-06-01

    We analyse the effect of the Basset history force on the sedimentation or rising of inertial particles in a two-dimensional convection flow. We find that the concept of snapshot attractors is useful to understand the extraordinary slow convergence due to long-term memory: an ensemble of particles converges exponentially fast towards a snapshot attractor, and this attractor undergoes a slow drift for long times. We demonstrate for the case of a periodic attractor that the drift of the snapshot attractor can be well characterized both in the space of the fluid and in the velocity space. For the case of quasiperiodic and chaotic dynamics we propose the use of the average settling velocity of the ensemble as a distinctive measure to characterize the snapshot attractor and the time scale separation corresponding to the convergence towards the snapshot attractor and its own slow dynamics.

  5. Forward and adjoint sensitivity computation of chaotic dynamical systems

    Wang, Qiqi, E-mail: qiqi@mit.edu [Department of Aeronautics and Astronautics, MIT, 77 Mass Ave., Cambridge, MA 02139 (United States)

    2013-02-15

    This paper describes a forward algorithm and an adjoint algorithm for computing sensitivity derivatives in chaotic dynamical systems, such as the Lorenz attractor. The algorithms compute the derivative of long time averaged “statistical” quantities to infinitesimal perturbations of the system parameters. The algorithms are demonstrated on the Lorenz attractor. We show that sensitivity derivatives of statistical quantities can be accurately estimated using a single, short trajectory (over a time interval of 20) on the Lorenz attractor.

  6. On reliability of singular-value decomposition in attractor reconstruction

    Palus, M.; Dvorak, I.

    1990-12-01

    Applicability of singular-value decomposition for reconstructing the strange attractor from one-dimensional chaotic time series, proposed by Broomhead and King, is extensively tested and discussed. Previously published doubts about its reliability are confirmed: singular-value decomposition, by nature a linear method, is only of a limited power when nonlinear structures are studied. (author). 29 refs, 9 figs

  7. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics

    Kuznetsov, Sergei P [Saratov Branch, Kotel' nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Saratov (Russian Federation)

    2011-02-28

    Research is reviewed on the identification and construction of physical systems with chaotic dynamics due to uniformly hyperbolic attractors (such as the Plykin attraction or the Smale-Williams solenoid). Basic concepts of the mathematics involved and approaches proposed in the literature for constructing systems with hyperbolic attractors are discussed. Topics covered include periodic pulse-driven models; dynamics models consisting of periodically repeated stages, each described by its own differential equations; the construction of systems of alternately excited coupled oscillators; the use of parametrically excited oscillations; and the introduction of delayed feedback. Some maps, differential equations, and simple mechanical and electronic systems exhibiting chaotic dynamics due to the presence of uniformly hyperbolic attractors are presented as examples. (reviews of topical problems)

  8. Chaotic interactions of self-replicating RNA.

    Forst, C V

    1996-03-01

    A general system of high-order differential equations describing complex dynamics of replicating biomolecules is given. Symmetry relations and coordinate transformations of general replication systems leading to topologically equivalent systems are derived. Three chaotic attractors observed in Lotka-Volterra equations of dimension n = 3 are shown to represent three cross-sections of one and the same chaotic regime. Also a fractal torus in a generalized three-dimensional Lotka-Volterra Model has been linked to one of the chaotic attractors. The strange attractors are studied in the equivalent four-dimensional catalytic replicator network. The fractal torus has been examined in adapted Lotka-Volterra equations. Analytic expressions are derived for the Lyapunov exponents of the flow in the replicator system. Lyapunov spectra for different pathways into chaos has been calculated. In the generalized Lotka-Volterra system a second inner rest point--coexisting with (quasi)-periodic orbits--can be observed; with an abundance of different bifurcations. Pathways from chaotic tori, via quasi-periodic tori, via limit cycles, via multi-periodic orbits--emerging out of periodic doubling bifurcations--to "simple" chaotic attractors can be found.

  9. Attractors near grazing–sliding bifurcations

    Glendinning, P; Kowalczyk, P; Nordmark, A B

    2012-01-01

    In this paper we prove, for the first time, that multistability can occur in three-dimensional Fillipov type flows due to grazing–sliding bifurcations. We do this by reducing the study of the dynamics of Filippov type flows around a grazing–sliding bifurcation to the study of appropriately defined one-dimensional maps. In particular, we prove the presence of three qualitatively different types of multiple attractors born in grazing–sliding bifurcations. Namely, a period-two orbit with a sliding segment may coexist with a chaotic attractor, two stable, period-two and period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three orbit with two sliding segments may coexist

  10. Coupled chaotic fluctuations in a model of international trade and innovation: Some preliminary results

    Sushko, Iryna; Gardini, Laura; Matsuyama, Kiminori

    2018-05-01

    We consider a two-dimensional continuous noninvertible piecewise smooth map, which characterizes the dynamics of innovation activities in the two-country model of trade and product innovation proposed in [7]. This two-dimensional map can be viewed as a coupling of two one-dimensional skew tent maps, each of which characterizes the innovation dynamics in each country in the absence of trade, and the coupling parameter depends inversely on the trade cost between the two countries. Hence, this model offers a laboratory for studying how a decline in the trade cost, or globalization, might synchronize endogenous fluctuations of innovation activities in the two countries. In this paper, we focus on the bifurcation scenarios, how the phase portrait of the two-dimensional map changes with a gradual decline of the trade cost, leading to border collision, merging, expansion and final bifurcations of the coexisting chaotic attractors. An example of peculiar border collision bifurcation leading to an increase of dimension of the chaotic attractor is also presented.

  11. Lectures on chaotic dynamical systems

    Afraimovich, Valentin

    2002-01-01

    This book is devoted to chaotic nonlinear dynamics. It presents a consistent, up-to-date introduction to the field of strange attractors, hyperbolic repellers, and nonlocal bifurcations. The authors keep the highest possible level of "physical" intuition while staying mathematically rigorous. In addition, they explain a variety of important nonstandard algorithms and problems involving the computation of chaotic dynamics. The book will help readers who are not familiar with nonlinear dynamics to understand and appreciate sophisticated modern dynamical systems and chaos. Intended for courses in either mathematics, physics, or engineering, prerequisites are calculus, differential equations, and functional analysis.

  12. Design of Threshold Controller Based Chaotic Circuits

    Mohamed, I. Raja; Murali, K.; Sinha, Sudeshna

    2010-01-01

    We propose a very simple implementation of a second-order nonautonomous chaotic oscillator, using a threshold controller as the only source of nonlinearity. We demonstrate the efficacy and simplicity of our design through numerical and experimental results. Further, we show that this approach...... of using a threshold controller as a nonlinear element, can be extended to obtain autonomous and multiscroll chaotic attractor circuits as well....

  13. Localization of hidden Chua's attractors

    Leonov, G.A.; Kuznetsov, N.V.; Vagaitsev, V.I.

    2011-01-01

    The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. In the present Letter for localization of hidden attractors of Chua's circuit it is suggested to use a special analytical-numerical algorithm. -- Highlights: → There are hidden attractors: basin doesn't contain neighborhoods of equilibria. → Hidden attractors cannot be reached by trajectory from neighborhoods of equilibria. → We suggested special procedure for localization of hidden attractors. → We discovered hidden attractor in Chua's system, L. Chua in his work didn't expect this.

  14. Strange Attractors in Drift Wave Turbulence

    Lewandowski, Jerome L.V.

    2003-01-01

    There are growing experimental, numerical and theoretical evidences that the anomalous transport observed in tokamaks and stellarators is caused by slow, drift-type modes (such as trapped electron modes and ion-temperature gradient-driven modes). Although typical collision frequencies in hot, magnetized fusion plasmas can be quite low in absolute values, collisional effects are nevertheless important since they act as dissipative sinks. As it is well known, dissipative systems with many (strictly speaking more than two) degrees of freedom are often chaotic and may evolve towards a so-called attractor

  15. Attractors of the periodically forced Rayleigh system

    Petre Bazavan

    2011-07-01

    Full Text Available The autonomous second order nonlinear ordinary differential equation(ODE introduced in 1883 by Lord Rayleigh, is the equation whichappears to be the closest to the ODE of the harmonic oscillator withdumping.In this paper we present a numerical study of the periodic andchaotic attractors in the dynamical system associated with the generalized Rayleigh equation. Transition between periodic and quasiperiodic motion is also studied. Numerical results describe the system dynamics changes (in particular bifurcations, when the forcing frequency is varied and thus, periodic, quasiperiodic or chaotic behaviour regions are predicted.

  16. Characterizing chaotic melodies in automatic music composition

    Coca, Andrés E.; Tost, Gerard O.; Zhao, Liang

    2010-09-01

    In this paper, we initially present an algorithm for automatic composition of melodies using chaotic dynamical systems. Afterward, we characterize chaotic music in a comprehensive way as comprising three perspectives: musical discrimination, dynamical influence on musical features, and musical perception. With respect to the first perspective, the coherence between generated chaotic melodies (continuous as well as discrete chaotic melodies) and a set of classical reference melodies is characterized by statistical descriptors and melodic measures. The significant differences among the three types of melodies are determined by discriminant analysis. Regarding the second perspective, the influence of dynamical features of chaotic attractors, e.g., Lyapunov exponent, Hurst coefficient, and correlation dimension, on melodic features is determined by canonical correlation analysis. The last perspective is related to perception of originality, complexity, and degree of melodiousness (Euler's gradus suavitatis) of chaotic and classical melodies by nonparametric statistical tests.

  17. Anisotropic nonequilibrium hydrodynamic attractor

    Strickland, Michael; Noronha, Jorge; Denicol, Gabriel S.

    2018-02-01

    We determine the dynamical attractors associated with anisotropic hydrodynamics (aHydro) and the DNMR equations for a 0 +1 d conformal system using kinetic theory in the relaxation time approximation. We compare our results to the nonequilibrium attractor obtained from the exact solution of the 0 +1 d conformal Boltzmann equation, the Navier-Stokes theory, and the second-order Mueller-Israel-Stewart theory. We demonstrate that the aHydro attractor equation resums an infinite number of terms in the inverse Reynolds number. The resulting resummed aHydro attractor possesses a positive longitudinal-to-transverse pressure ratio and is virtually indistinguishable from the exact attractor. This suggests that an optimized hydrodynamic treatment of kinetic theory involves a resummation not only in gradients (Knudsen number) but also in the inverse Reynolds number. We also demonstrate that the DNMR result provides a better approximation of the exact kinetic theory attractor than the Mueller-Israel-Stewart theory. Finally, we introduce a new method for obtaining approximate aHydro equations which relies solely on an expansion in the inverse Reynolds number. We then carry this expansion out to the third order, and compare these third-order results to the exact kinetic theory solution.

  18. Chaotic universe model.

    Aydiner, Ekrem

    2018-01-15

    In this study, we consider nonlinear interactions between components such as dark energy, dark matter, matter and radiation in the framework of the Friedman-Robertson-Walker space-time and propose a simple interaction model based on the time evolution of the densities of these components. By using this model we show that these interactions can be given by Lotka-Volterra type equations. We numerically solve these coupling equations and show that interaction dynamics between dark energy-dark matter-matter or dark energy-dark matter-matter-radiation has a strange attractor for 0 > w de  >-1, w dm  ≥ 0, w m  ≥ 0 and w r  ≥ 0 values. These strange attractors with the positive Lyapunov exponent clearly show that chaotic dynamics appears in the time evolution of the densities. These results provide that the time evolution of the universe is chaotic. The present model may have potential to solve some of the cosmological problems such as the singularity, cosmic coincidence, big crunch, big rip, horizon, oscillation, the emergence of the galaxies, matter distribution and large-scale organization of the universe. The model also connects between dynamics of the competing species in biological systems and dynamics of the time evolution of the universe and offers a new perspective and a new different scenario for the universe evolution.

  19. NARX prediction of some rare chaotic flows: Recurrent fuzzy functions approach

    Goudarzi, Sobhan [Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413 (Iran, Islamic Republic of); Jafari, Sajad, E-mail: sajadjafari@aut.ac.ir [Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413 (Iran, Islamic Republic of); Moradi, Mohammad Hassan [Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413 (Iran, Islamic Republic of); Sprott, J.C. [Department of Physics, University of Wisconsin–Madison, Madison, WI 53706 (United States)

    2016-02-15

    The nonlinear and dynamic accommodating capability of time domain models makes them a useful representation of chaotic time series for analysis, modeling and prediction. This paper is devoted to the modeling and prediction of chaotic time series with hidden attractors using a nonlinear autoregressive model with exogenous inputs (NARX) based on a novel recurrent fuzzy functions (RFFs) approach. Case studies of recently introduced chaotic systems with hidden attractors plus classical chaotic systems demonstrate that the proposed modeling methodology exhibits better prediction performance from different viewpoints (short term and long term) compared to some other existing methods. - Highlights: • A new method is proposed for prediction of chaotic time series. • This method is based on novel recurrent fuzzy functions (RFFs) approach. • Some rare chaotic flows are used as test systems. • The new method shows proper performance in short-term prediction. • It also shows proper performance in prediction of attractor's topology.

  20. NARX prediction of some rare chaotic flows: Recurrent fuzzy functions approach

    Goudarzi, Sobhan; Jafari, Sajad; Moradi, Mohammad Hassan; Sprott, J.C.

    2016-01-01

    The nonlinear and dynamic accommodating capability of time domain models makes them a useful representation of chaotic time series for analysis, modeling and prediction. This paper is devoted to the modeling and prediction of chaotic time series with hidden attractors using a nonlinear autoregressive model with exogenous inputs (NARX) based on a novel recurrent fuzzy functions (RFFs) approach. Case studies of recently introduced chaotic systems with hidden attractors plus classical chaotic systems demonstrate that the proposed modeling methodology exhibits better prediction performance from different viewpoints (short term and long term) compared to some other existing methods. - Highlights: • A new method is proposed for prediction of chaotic time series. • This method is based on novel recurrent fuzzy functions (RFFs) approach. • Some rare chaotic flows are used as test systems. • The new method shows proper performance in short-term prediction. • It also shows proper performance in prediction of attractor's topology.

  1. Chaotic behavior learning of Chua's circuit

    Sun Jian-Cheng

    2012-01-01

    Least-square support vector machines (LS-SVM) are applied for learning the chaotic behavior of Chua's circuit. The system is divided into three multiple-input single-output (MISO) structures and the LS-SVM are trained individually. Comparing with classical approaches, the proposed one reduces the structural complexity and the selection of parameters is avoided. Some parameters of the attractor are used to compare the chaotic behavior of the reconstructed and the original systems for model validation. Results show that the LS-SVM combined with the MISO can be trained to identify the underlying link among Chua's circuit state variables, and exhibit the chaotic attractors under the autonomous working mode

  2. Co-existing hidden attractors in a radio-physical oscillator system

    Kuznetsov, A. P.; Kuznetsov, S. P.; Mosekilde, Erik

    2015-01-01

    The term `hidden attractor' relates to a stable periodic, quasiperiodic or chaotic state whose basin of attraction does not overlap with the neighborhood of an unstable equilibrium point. Considering a three-dimensional oscillator system that does not allow for the existence of an equilibrium point...... frequency, describe the bifurcations through which hidden attractors of different type arise and disappear, and illustrate the form of the basins of attraction....

  3. Extreme multistability in a memristor-based multi-scroll hyper-chaotic system

    Yuan, Fang, E-mail: yf210yf@163.com; Wang, Guangyi, E-mail: wanggyi@163.com [Institute of Modern Circuits and Intelligent Information, Hangzhou Dianzi University, Hangzhou 310018 (China); Wang, Xiaowei [Department of Automation, Shanghai University, Shanghai 200072 (China)

    2016-07-15

    In this paper, a new memristor-based multi-scroll hyper-chaotic system is designed. The proposed memristor-based system possesses multiple complex dynamic behaviors compared with other chaotic systems. Various coexisting attractors and hidden coexisting attractors are observed in this system, which means extreme multistability arises. Besides, by adjusting parameters of the system, this chaotic system can perform single-scroll attractors, double-scroll attractors, and four-scroll attractors. Basic dynamic characteristics of the system are investigated, including equilibrium points and stability, bifurcation diagrams, Lyapunov exponents, and so on. In addition, the presented system is also realized by an analog circuit to confirm the correction of the numerical simulations.

  4. Extreme multistability in a memristor-based multi-scroll hyper-chaotic system.

    Yuan, Fang; Wang, Guangyi; Wang, Xiaowei

    2016-07-01

    In this paper, a new memristor-based multi-scroll hyper-chaotic system is designed. The proposed memristor-based system possesses multiple complex dynamic behaviors compared with other chaotic systems. Various coexisting attractors and hidden coexisting attractors are observed in this system, which means extreme multistability arises. Besides, by adjusting parameters of the system, this chaotic system can perform single-scroll attractors, double-scroll attractors, and four-scroll attractors. Basic dynamic characteristics of the system are investigated, including equilibrium points and stability, bifurcation diagrams, Lyapunov exponents, and so on. In addition, the presented system is also realized by an analog circuit to confirm the correction of the numerical simulations.

  5. Plykin type attractor in electronic device simulated in MULTISIM

    Kuznetsov, Sergey P.

    2011-12-01

    An electronic device is suggested representing a non-autonomous dynamical system with hyperbolic chaotic attractor of Plykin type in the stroboscopic map, and the results of its simulation with software package NI MULTISIM are considered in comparison with numerical integration of the underlying differential equations. A main practical advantage of electronic devices of this kind is their structural stability that means insensitivity of the chaotic dynamics in respect to variations of functions and parameters of elements constituting the system as well as to interferences and noises.

  6. Multistability and hidden attractors in a relay system with hysteresis

    Zhusubaliyev, Zhanybai T.; Mosekilde, Erik; Rubanov, Vasily G.

    2015-01-01

    with the neighborhood of that cycle. We show how the equilibrium point of a relay system disappears in a boundary-equilibrium bifurcation as the system enters the region of autonomous switching dynamics and demonstrate experimentally how a relay system can exhibit large amplitude chaotic oscillations at high values...... of the supply voltage. By investigating a four-dimensional model of the experimental relay system we finally show how a variety of hidden periodic, quasiperiodic and chaotic attractors arise, transform and disappear through different bifurcations. (C) 2015 Elsevier B.V. All rights reserved....

  7. Generating multi-double-scroll attractors via nonautonomous approach

    Hong, Qinghui; Xie, Qingguo, E-mail: qgxie@mail.hust.edu.cn [Wuhan National Laboratory for Optoelectronics, Wuhan 430074 (China); Shen, Yi; Wang, Xiaoping [School of Automation, Huazhong University of Science and Technology, Wuhan 430074 (China)

    2016-08-15

    It is a common phenomenon that multi-scroll attractors are realized by introducing the various nonlinear functions with multiple breakpoints in double scroll chaotic systems. Differently, we present a nonautonomous approach for generating multi-double-scroll attractors (MDSA) without changing the original nonlinear functions. By using the multi-level-logic pulse excitation technique in double scroll chaotic systems, MDSA can be generated. A Chua's circuit, a Jerk circuit, and a modified Lorenz system are given as designed example and the Matlab simulation results are presented. Furthermore, the corresponding realization circuits are designed. The Pspice results are in agreement with numerical simulation results, which verify the availability and feasibility of this method.

  8. Generating multi-double-scroll attractors via nonautonomous approach.

    Hong, Qinghui; Xie, Qingguo; Shen, Yi; Wang, Xiaoping

    2016-08-01

    It is a common phenomenon that multi-scroll attractors are realized by introducing the various nonlinear functions with multiple breakpoints in double scroll chaotic systems. Differently, we present a nonautonomous approach for generating multi-double-scroll attractors (MDSA) without changing the original nonlinear functions. By using the multi-level-logic pulse excitation technique in double scroll chaotic systems, MDSA can be generated. A Chua's circuit, a Jerk circuit, and a modified Lorenz system are given as designed example and the Matlab simulation results are presented. Furthermore, the corresponding realization circuits are designed. The Pspice results are in agreement with numerical simulation results, which verify the availability and feasibility of this method.

  9. Attractors in complex networks

    Rodrigues, Alexandre A. P.

    2017-10-01

    In the framework of the generalized Lotka-Volterra model, solutions representing multispecies sequential competition can be predictable with high probability. In this paper, we show that it occurs because the corresponding "heteroclinic channel" forms part of an attractor. We prove that, generically, in an attracting heteroclinic network involving a finite number of hyperbolic and non-resonant saddle-equilibria whose linearization has only real eigenvalues, the connections corresponding to the most positive expanding eigenvalues form part of an attractor (observable in numerical simulations).

  10. Hidden attractors in dynamical systems

    Dudkowski, Dawid; Jafari, Sajad; Kapitaniak, Tomasz; Kuznetsov, Nikolay V.; Leonov, Gennady A.; Prasad, Awadhesh

    2016-06-01

    Complex dynamical systems, ranging from the climate, ecosystems to financial markets and engineering applications typically have many coexisting attractors. This property of the system is called multistability. The final state, i.e., the attractor on which the multistable system evolves strongly depends on the initial conditions. Additionally, such systems are very sensitive towards noise and system parameters so a sudden shift to a contrasting regime may occur. To understand the dynamics of these systems one has to identify all possible attractors and their basins of attraction. Recently, it has been shown that multistability is connected with the occurrence of unpredictable attractors which have been called hidden attractors. The basins of attraction of the hidden attractors do not touch unstable fixed points (if exists) and are located far away from such points. Numerical localization of the hidden attractors is not straightforward since there are no transient processes leading to them from the neighborhoods of unstable fixed points and one has to use the special analytical-numerical procedures. From the viewpoint of applications, the identification of hidden attractors is the major issue. The knowledge about the emergence and properties of hidden attractors can increase the likelihood that the system will remain on the most desirable attractor and reduce the risk of the sudden jump to undesired behavior. We review the most representative examples of hidden attractors, discuss their theoretical properties and experimental observations. We also describe numerical methods which allow identification of the hidden attractors.

  11. β-expansion attractors observed in A/D converters

    Kohda, Tohru; Horio, Yoshihiko; Aihara, Kazuyuki

    2012-12-01

    The recently proposed β-encoders, analog-to-digital converters using an amplifier with a factor β and a flaky quantizer with threshold ν, have proven to be explained by the deterministic dynamics of multi-valued Rényi-Parry maps. Such a map is locally eventually onto [ν-1, ν), which is topologically conjugate to Parry's (β,α)-map with α =(β-1)(ν-1). This implies that β-encoders have a closed subinterval [ν-1,ν), which includes an attractor. Thus, the iteration of the multi-valued Rényi-Parry map performs the β-expansion of x while quantization errors in β-encoders behave chaotically and do not converge to a fixed point. This β-expansion attractor is relatively simpler than previously reported attractors. The object of this paper is twofold: to observe the embedded attractors in the β-encoder and to identify attractors that are useful for spread-spectrum codes and optimization techniques using pseudo-random numbers.

  12. Pitchfork bifurcation and circuit implementation of a novel Chen hyper-chaotic system

    Dong En-Zeng; Chen Zeng-Qiang; Chen Zai-Ping; Ni Jian-Yun

    2012-01-01

    In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-chaotic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-chaotic system is useful in the secure communication. (general)

  13. Attractors under discretisation

    Han, Xiaoying

    2017-01-01

    This work focuses on the preservation of attractors and saddle points of ordinary differential equations under discretisation. In the 1980s, key results for autonomous ordinary differential equations were obtained – by Beyn for saddle points and by Kloeden & Lorenz for attractors. One-step numerical schemes with a constant step size were considered, so the resulting discrete time dynamical system was also autonomous. One of the aims of this book is to present new findings on the discretisation of dissipative nonautonomous dynamical systems that have been obtained in recent years, and in particular to examine the properties of nonautonomous omega limit sets and their approximations by numerical schemes – results that are also of importance for autonomous systems approximated by a numerical scheme with variable time steps, thus by a discrete time nonautonomous dynamical system.

  14. Multiple single-centered attractors

    Dominic, Pramod; Mandal, Taniya; Tripathy, Prasanta K.

    2014-01-01

    In this paper we study spherically symmetric single-centered attractors in N=2 supergravity in four dimensions. The attractor points are obtained by extremising the effective black hole potential in the moduli space. Both supersymmetric as well as non-supersymmetric attractors exist in mutually exclusive domains of the charge lattice. We construct axion free supersymmetric as well as non-supersymmetric multiple attractors in a simple two parameter model. We further obtain explicit examples of two distinct non-supersymmetric attractors in type IIA string theory compactified on K3×T"2 carrying D0−D4−D6 charges. We compute the entropy of these attractors and analyse their stability in detail.

  15. Chaotic Patterns in Aeroelastic Signals

    F. D. Marques

    2009-01-01

    patterns. With the reconstructed state spaces, qualitative analyses may be done, and the attractors evolutions with parametric variation are presented. Overall results reveal complex system dynamics associated with highly separated flow effects together with nonlinear coupling between aeroelastic modes. Bifurcations to the nonlinear aeroelastic system are observed for two investigations, that is, considering oscillations-induced aeroelastic evolutions with varying freestream speed, and aeroelastic evolutions at constant freestream speed and varying oscillations. Finally, Lyapunov exponent calculation is proceeded in order to infer on chaotic behavior. Poincaré mappings also suggest bifurcations and chaos, reinforced by the attainment of maximum positive Lyapunov exponents.

  16. Searching of Chaotic Elements in Hydrology

    Sorin VLAD

    2014-03-01

    Full Text Available Chaos theory offers new means of understanding and prediction of phenomena otherwise considered random and unpredictable. The signatures of chaos can be isolated by performing nonlinear analysis of the time series available. The paper presents the results obtained by conducting a nonlinear analysis of the time series of daily Siret river flow (located in the North-Eastern part of Romania. The time series analysis is recorded starting with January 1999 to July 2009. The attractor is embedded in the reconstructed phase space then the chaotic dynamics is revealed computing the chaotic invariants - correlation dimension and the maximum Lyapunov Exponent.

  17. A novel four-wing non-equilibrium chaotic system and its circuit ...

    Abstract. In this paper, we construct a novel, 4D smooth autonomous system. Compared to the existing chaotic systems, the most attractive point is that this system does not display any equilib- ria, but can still exhibit four-wing chaotic attractors. The proposed system is investigated through numerical simulations and ...

  18. Periodic or chaotic bursting dynamics via delayed pitchfork bifurcation in a slow-varying controlled system

    Yu, Yue; Zhang, Zhengdi; Han, Xiujing

    2018-03-01

    In this work, we aim to demonstrate the novel routes to periodic and chaotic bursting, i.e., the different bursting dynamics via delayed pitchfork bifurcations around stable attractors, in the classical controlled Lü system. First, by computing the corresponding characteristic polynomial, we determine where some critical values about bifurcation behaviors appear in the Lü system. Moreover, the transition mechanism among different stable attractors has been introduced including homoclinic-type connections or chaotic attractors. Secondly, taking advantage of the above analytical results, we carry out a study of the mechanism for bursting dynamics in the Lü system with slowly periodic variation of certain control parameter. A distinct delayed supercritical pitchfork bifurcation behavior can be discussed when the control item passes through bifurcation points periodically. This delayed dynamical behavior may terminate at different parameter areas, which leads to different spiking modes around different stable attractors (equilibriums, limit cycles, or chaotic attractors). In particular, the chaotic attractor may appear by Shilnikov connections or chaos boundary crisis, which leads to the occurrence of impressive chaotic bursting oscillations. Our findings enrich the study of bursting dynamics and deepen the understanding of some similar sorts of delayed bursting phenomena. Finally, some numerical simulations are included to illustrate the validity of our study.

  19. Attractors, universality, and inflation

    Downes, Sean; Dutta, Bhaskar; Sinha, Kuver

    2012-11-01

    Studies of the initial conditions for inflation have conflicting predictions from exponential suppression to inevitability. At the level of phase space, this conflict arises from the competing intuitions of CPT invariance and thermodynamics. After reviewing this conflict, we enlarge the ensemble beyond phase space to include scalar potential data. We show how this leads to an important contribution from inflection point inflation, enhancing the likelihood of inflation to a power law, 1/Ne3. In the process, we emphasize the attractor dynamics of the gravity-scalar system and the existence of universality classes from inflection point inflation. Finally, we comment on the predictivity of inflation in light of these results.

  20. Fractional order control and synchronization of chaotic systems

    Vaidyanathan, Sundarapandian; Ouannas, Adel

    2017-01-01

    The book reports on the latest advances in and applications of fractional order control and synchronization of chaotic systems, explaining the concepts involved in a clear, matter-of-fact style. It consists of 30 original contributions written by eminent scientists and active researchers in the field that address theories, methods and applications in a number of research areas related to fractional order control and synchronization of chaotic systems, such as: fractional chaotic systems, hyperchaotic systems, complex systems, fractional order discrete chaotic systems, chaos control, chaos synchronization, jerk circuits, fractional chaotic systems with hidden attractors, neural network, fuzzy logic controllers, behavioral modeling, robust and adaptive control, sliding mode control, different types of synchronization, circuit realization of chaotic systems, etc. In addition to providing readers extensive information on chaos fundamentals, fractional calculus, fractional differential equations, fractional contro...

  1. Hidden Attractors in a Model of a Bubble Contrast Agent Oscillating Near an Elastic Wall

    Garashchuk, Ivan; Sinelshchikov, Dmitry; Kudryashov, Nikolay

    2018-02-01

    A model describing the dynamics of a spherical gas bubble in a compressible viscous liquid is studied. The bubble is oscillating close to an elastic wall of finite thickness under the influence of an external pressure field which simulates a contrast agent oscillating close to a blood vessel wall. Here we investigate numerically the coexistence of chaotic and periodic attractors in this model. One of the tools applied for seeking coexisting attractors is the perpetual points method. This method can be helpful for localizing coexisting attractors, occurring in various physically realistic ranges of variation of the control parameters. We provide some examples of coexisting attractors to demonstrate the importance of the multistability problem for the applications.

  2. Multiple attractors and crisis route to chaos in a model food-chain

    Upadhyay, Ranjit Kumar

    2003-01-01

    An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a 1 =1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable

  3. Chaotic spectroscopy

    Doron, E.; Smilanski, U.

    1991-11-01

    We discuss the spectra of quantized chaotic billiards from the point of view of scattering theory. We show that the spectral and resonance density functions both fluctuate about a common mean. A semiclassical treatment explains this in terms of classical scattering trajectories and periodic orbits of the poincare scattering map. This formalism is used to interpret recent experiments where the spectra of chaotic cavities where measured by microwave scattering. (author)

  4. A simple time-delayed method to control chaotic systems

    Chen Maoyin; Zhou Donghua; Shang Yun

    2004-01-01

    Based on the adaptive iterative learning strategy, a simple time-delayed controller is proposed to stabilize unstable periodic orbits (UPOs) embedded in chaotic attractors. This controller includes two parts: one is a linear feedback part; the other is an adaptive iterative learning estimation part. Theoretical analysis and numerical simulation show the effectiveness of this controller

  5. Attractors for discrete periodic dynamical systems

    John E. Franke; James F. Selgrade

    2003-01-01

    A mathematical framework is introduced to study attractors of discrete, nonautonomous dynamical systems which depend periodically on time. A structure theorem for such attractors is established which says that the attractor of a time-periodic dynamical system is the unin of attractors of appropriate autonomous maps. If the nonautonomous system is a perturbation of an...

  6. Modified scaling function projective synchronization of chaotic systems

    Xu Yu-Hua; Zhou Wu-Neng; Fang Jian-An

    2011-01-01

    This paper investigates a kind of modified scaling function projective synchronization of uncertain chaotic systems using an adaptive controller. The given scaling function in the new method can be an equilibrium point, a periodic orbit, or even a chaotic attractor in the phase space. Based on LaSalle's invariance set principle, the adaptive control law is derived to make the states of two chaotic systems function projective synchronized. Some numerical examples are also given to show the effectiveness of the proposed method. (general)

  7. Some statistical properties of strange attractors: engineering view

    Mijangos, M; Kontorovich, V; Aguilar-Torrentera, J

    2008-01-01

    In this paper, the statistical characterization of strange attractors is investigated via the so-called 'model distribution' approach. It is shown that in order to calculate the first four cumulants, which are necessary to create a model distribution of kurtosis approximation, a systematic method for the calculus of the variance needs to be considered. Correspondently, an analytical method based on the Kolmogorov-Sinai (K-S) entropy for variance approximation is herein proposed. The methodology is of interest for its application in the statistical analysis of chaotic systems that model physical phenomena found in some areas of electrical (communication) engineering

  8. Denoising of chaotic signal using independent component analysis and empirical mode decomposition with circulate translating

    Wang Wen-Bo; Zhang Xiao-Dong; Chang Yuchan; Wang Xiang-Li; Wang Zhao; Chen Xi; Zheng Lei

    2016-01-01

    In this paper, a new method to reduce noises within chaotic signals based on ICA (independent component analysis) and EMD (empirical mode decomposition) is proposed. The basic idea is decomposing chaotic signals and constructing multidimensional input vectors, firstly, on the base of EMD and its translation invariance. Secondly, it makes the independent component analysis on the input vectors, which means that a self adapting denoising is carried out for the intrinsic mode functions (IMFs) of chaotic signals. Finally, all IMFs compose the new denoised chaotic signal. Experiments on the Lorenz chaotic signal composed of different Gaussian noises and the monthly observed chaotic sequence on sunspots were put into practice. The results proved that the method proposed in this paper is effective in denoising of chaotic signals. Moreover, it can correct the center point in the phase space effectively, which makes it approach the real track of the chaotic attractor. (paper)

  9. Shape synchronization control for three-dimensional chaotic systems

    Huang, Yuanyuan; Wang, Yinhe; Chen, Haoguang; Zhang, Siying

    2016-01-01

    This paper aims to the three-dimensional continuous chaotic system and shape of the chaotic attractor by utilizing the basic theory of plane curves in classical differential geometry, the continuous controller is synthesized for the master–slave synchronization in shape. This means that the slave system can possess the same shape of state trajectory with the master system via the continuous controller. The continuous controller is composed of three sub-controllers, which respectively correspond to the master–slave synchronization in shape for the three projective curves of the chaotic attractor onto the three coordinate planes. Moreover, the proposed shape synchronization technique as well as application of control scheme to secure communication is also demonstrated in this paper, where numerical simulation results show the proposed control method works well.

  10. Attractor behaviour in ELKO cosmology

    Basak, Abhishek; Bhatt, Jitesh R.; Shankaranarayanan, S.; Varma, K.V. Prasantha

    2013-01-01

    We study the dynamics of ELKO in the context of accelerated phase of our universe. To avoid the fine tuning problem associated with the initial conditions, it is required that the dynamical equations lead to an early-time attractor. In the earlier works, it was shown that the dynamical equations containing ELKO fields do not lead to early-time stable fixed points. In this work, using redefinition of variables, we show that ELKO cosmology admits early-time stable fixed points. More interestingly, we show that ELKO cosmology admit two sets of attractor points corresponding to slow and fast-roll inflation. The fast-roll inflation attractor point is unique for ELKO as it is independent of the form of the potential. We also discuss the plausible choice of interaction terms in these two sets of attractor points and constraints on the coupling constant

  11. Is Cygus X-1 a chaotic dynamical system?

    Unno, Wasaburo; Yoneyama, Tadaoki; Urata, Kenji; Masaki, Isao; Kondo, Masa-aki; Inoue, Hajime.

    1990-01-01

    X-ray data of Cyg X-1 observed by the Tenma satellite were analyzed to determine whether Cyg X-1 is a chaotic dynamical system of low dimension. Since Poisson noise disturbs the determination of the attractor dimension of the system, comparative studies were carried out for the Cyg X-1 data relative to artificial data of purely stochastic Poisson noise and to a Lorenz attractor plus noise. The attractor dimension was searched using trajectories of time series data in phase space, the dimension of which was varied up to 21. The relation between the attractor dimension and the phase-space dimension for the Cyg X-1 data starts to deviate from that of noise data from a phase-space dimension of about 7, showing the presence of an attractor with a dimension of about 7 or less. Though three positive Lyapunov exponents were calculated, they are too small (∼10 -2 ) to prove with certainty that the Cyg X-1 attractor should be a strange attractor. (author)

  12. Coupled chaotic attractors and driving-induced bistability: A brief ...

    2015-02-04

    Feb 4, 2015 ... In the 'drive–response' scenario, a system is unidirectionally coupled to another sys- tem. Here, the ... where R1 and R2 are the dynamical variables of the drive and response systems, respec- tively. F and G ..... [3] K Kaneko, Theory and applications of coupled map lattices (John Wiley and Sons, NY, 1993).

  13. Synchronization of Rikitake chaotic attractor using active control

    Vincent, U.E.

    2005-01-01

    Using synchronization technique based on control theory, we design an active controller which enables the synchronization of two identical Rikitake two-disc dynamo systems. Numerical simulations are used to show the robustness of the active control scheme in synchronizing coupled Rikitake dynamical systems. On the sequential application of the active control, transitions from temporary phase locking (TPL) state to complete synchronization state were found

  14. Rank One Strange Attractors in Periodically Kicked Predator-Prey System with Time-Delay

    Yang, Wenjie; Lin, Yiping; Dai, Yunxian; Zhao, Huitao

    2016-06-01

    This paper is devoted to the study of the problem of rank one strange attractor in a periodically kicked predator-prey system with time-delay. Our discussion is based on the theory of rank one maps formulated by Wang and Young. Firstly, we develop the rank one chaotic theory to delayed systems. It is shown that strange attractors occur when the delayed system undergoes a Hopf bifurcation and encounters an external periodic force. Then we use the theory to the periodically kicked predator-prey system with delay, deriving the conditions for Hopf bifurcation and rank one chaos along with the results of numerical simulations.

  15. Multistability and hidden attractors in a multilevel DC/DC converter

    Zhusubaliyev, Zhanybai T.; Mosekilde, Erik

    2015-01-01

    An attracting periodic, quasiperiodic or chaotic set of a smooth, autonomous system may be referred to as a "hidden attractor" if its basin of attraction does not overlap with the neighborhood of an unstable equilibrium point. Historically, this condition has implied that the basin of attraction...... produce complicated structures of attracting and repelling states organized around the basic switching cycle. This leads us to suggest the existence of hidden attractors in such systems as well. In this case, the condition will be that the basin of attraction does not overlap with the fixed point...

  16. A New Simple Chaotic Circuit Based on Memristor

    Wu, Renping; Wang, Chunhua

    In this paper, a new memristor is proposed, and then an emulator built from off-the-shelf solid state components imitating the behavior of the proposed memristor is presented. Multisim simulation and breadboard experiment are done on the emulator, exhibiting a pinched hysteresis loop in the voltage-current plane when the emulator is driven by a periodic excitation voltage. In addition, a new simple chaotic circuit is designed by using the proposed memristor and other circuit elements. It is exciting that this circuit with only a linear negative resistor, a capacitor, an inductor and a memristor can generate a chaotic attractor. The dynamical behaviors of the proposed chaotic system are analyzed by Lyapunov exponents, phase portraits and bifurcation diagrams. Finally, an electronic circuit is designed to implement the chaotic system. For the sake of simple circuit topology, the proposed chaotic circuit can be easily manufactured at low cost.

  17. Chaotic examination

    Bildirici, Melike; Sonustun, Fulya Ozaksoy; Sonustun, Bahri

    2018-01-01

    In the regards of chaos theory, new concepts such as complexity, determinism, quantum mechanics, relativity, multiple equilibrium, complexity, (continuously) instability, nonlinearity, heterogeneous agents, irregularity were widely questioned in economics. It is noticed that linear models are insufficient for analyzing unpredictable, irregular and noncyclical oscillations of economies, and for predicting bubbles, financial crisis, business cycles in financial markets. Therefore, economists gave great consequence to use appropriate tools for modelling non-linear dynamical structures and chaotic behaviors of the economies especially in macro and the financial economy. In this paper, we aim to model the chaotic structure of exchange rates (USD-TL and EUR-TL). To determine non-linear patterns of the selected time series, daily returns of the exchange rates were tested by BDS during the period from January 01, 2002 to May 11, 2017 which covers after the era of the 2001 financial crisis. After specifying the non-linear structure of the selected time series, it was aimed to examine the chaotic characteristic for the selected time period by Lyapunov Exponents. The findings verify the existence of the chaotic structure of the exchange rate returns in the analyzed time period.

  18. Supersymmetry and attractors

    Ferrara, S.; Kallosh, R.

    1996-01-01

    We find a general principle which allows one to compute the area of the horizon of N=2 extremal black holes as an extremum of the central charge. One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the moduli space (a minimum corresponds to a fixed point of attraction). The extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges. The doubling of unbroken supersymmetry at the fixed point of attraction for N=2 black holes near the horizon is derived via conformal flatness of the Bertotti-Robinson-type geometry. These results provide an explicit model-independent expression for the macroscopic Bekenstein-Hawking entropy of N=2 black holes which is manifestly duality invariant. The presence of hypermultiplets in the solution does not affect the area formula. Various examples of the general formula are displayed. We outline the attractor mechanism in N=4,8 supersymmetries and the relation to the N=2 case. The entropy-area formula in five dimensions, recently discussed in the literature, is also seen to be obtained by extremizing the 5d central charge. copyright 1996 The American Physical Society

  19. Applications of modularized circuit designs in a new hyper-chaotic system circuit implementation

    Wang Rui; Sun Hui; Wang Jie-Zhi; Wang Lu; Wang Yan-Chao

    2015-01-01

    Modularized circuit designs for chaotic systems are introduced in this paper. Especially, a typical improved modularized design strategy is proposed and applied to a new hyper-chaotic system circuit implementation. In this paper, the detailed design procedures are described. Multisim simulations and physical experiments are conducted, and the simulation results are compared with Matlab simulation results for different system parameter pairs. These results are consistent with each other and they verify the existence of the hyper-chaotic attractor for this new hyper-chaotic system. (paper)

  20. A 3D Fractional-Order Chaotic System with Only One Stable Equilibrium and Controlling Chaos

    Shiyun Shen

    2017-01-01

    Full Text Available One 3D fractional-order chaotic system with only one locally asymptotically stable equilibrium is reported. To verify the chaoticity, the maximum Lyapunov exponent (MAXLE with respect to the fractional-order and chaotic attractors are obtained by numerical calculation for this system. Furthermore, by linear scalar controller consisting of a single state variable, one control scheme for stabilization of the 3D fractional-order chaotic system is suggested. The numerical simulations show the feasibility of the control scheme.

  1. Cosmological attractors in massive gravity

    Dubovsky, S; Tkachev, I I

    2005-01-01

    We study Lorentz-violating models of massive gravity which preserve rotations and are invariant under time-dependent shifts of the spatial coordinates. In the linear approximation the Newtonian potential in these models has an extra ``confining'' term proportional to the distance from the source. We argue that during cosmological expansion the Universe may be driven to an attractor point with larger symmetry which includes particular simultaneous dilatations of time and space coordinates. The confining term in the potential vanishes as one approaches the attractor. In the vicinity of the attractor the extra contribution is present in the Friedmann equation which, in a certain range of parameters, gives rise to the cosmic acceleration.

  2. Moduli backreaction on inflationary attractors

    Roest, Diederik; Werkman, Pelle

    2016-07-01

    We investigate the interplay between moduli dynamics and inflation, focusing on the KKLT- scenario and cosmological α-attractors. General couplings between these sectors can induce a significant backreaction and potentially destroy the inflationary regime; however, we demonstrate that this generically does not happen for α-attractors. Depending on the details of the superpotential, the volume modulus can either be stable during the entire inflationary trajectory, or become tachyonic at some point and act as a waterfall field, resulting in a sudden end of inflation. In the latter case there is a universal supersymmetric minimum where the scalars end up, preventing the decompactification scenario. The gravitino mass is independent from the inflationary scale with no fine-tuning of the parameters. The observational predictions conform to the universal value of attractors, fully compatible with the Planck data, with possibly a capped number of e-folds due to the interplay with moduli.

  3. Moduli Backreaction on Inflationary Attractors

    Roest, Diederik; Werkman, Pelle

    2016-01-01

    We investigate the interplay between moduli dynamics and inflation, focusing on the KKLT-scenario and cosmological $\\alpha$-attractors. General couplings between these sectors can induce a significant backreaction and potentially destroy the inflationary regime; however, we demonstrate that this generically does not happen for $\\alpha$-attractors. Depending on the details of the superpotential, the volume modulus can either be stable during the entire inflationary trajectory, or become tachyonic at some point and act as a waterfall field, resulting in a sudden end of inflation. In the latter case there is a universal supersymmetric minimum where the scalars end up, preventing the decompactification scenario. The observational predictions conform to the universal value of attractors, fully compatible with the Planck data, with possibly a capped number of e-folds due to the interplay with moduli.

  4. Approximate convex hull of affine iterated function system attractors

    Mishkinis, Anton; Gentil, Christian; Lanquetin, Sandrine; Sokolov, Dmitry

    2012-01-01

    Highlights: ► We present an iterative algorithm to approximate affine IFS attractor convex hull. ► Elimination of the interior points significantly reduces the complexity. ► To optimize calculations, we merge the convex hull images at each iteration. ► Approximation by ellipses increases speed of convergence to the exact convex hull. ► We present a method of the output convex hull simplification. - Abstract: In this paper, we present an algorithm to construct an approximate convex hull of the attractors of an affine iterated function system (IFS). We construct a sequence of convex hull approximations for any required precision using the self-similarity property of the attractor in order to optimize calculations. Due to the affine properties of IFS transformations, the number of points considered in the construction is reduced. The time complexity of our algorithm is a linear function of the number of iterations and the number of points in the output approximate convex hull. The number of iterations and the execution time increases logarithmically with increasing accuracy. In addition, we introduce a method to simplify the approximate convex hull without loss of accuracy.

  5. Chaotic behavior in the Henon mapping

    Marotto, F R [Drexel Univ., Philadelphia, PA (USA). Dept. of Mathematics

    1979-01-01

    In a previous work Henon investigated a two-dimensional difference equation which was motivated by a hydrodynamical system of Lorenz. Numerically solving this equation indicated for certain parameter values the existence of a 'strange attractor', i.e., a region in the plane which attracts bounded solutions and in which solutions wander erratically. In the present work it is shown that this behavior is related to the mathematical concept of 'chaos'. Using general methods previously developed, it is proven analytically that for some parameter values the mapping has a transversal homoclinic orbit, which implies the existence of the chaotic behavior observed by Henon.

  6. Chaotic combustion in spark ignition engines

    Wendeker, Miroslaw; Czarnigowski, Jacek; Litak, Grzegorz; Szabelski, Kazimierz

    2003-01-01

    We analyse the combustion process in a spark ignition engine using the experimental data of an internal pressure during the combustion process and show that the system can be driven to chaotic behaviour. Our conclusion is based on the observation of unperiodicity in the time series, suitable stroboscopic maps and a complex structure of a reconstructed strange attractor. This analysis can explain that in some circumstances the level of noise in spark ignition engines increases considerably due to nonlinear dynamics of a combustion process

  7. Heteroclinic cycles between unstable attractors

    Broer, Henk; Efstathiou, Konstantinos; Subramanian, Easwar

    2008-01-01

    We consider networks of pulse coupled linear oscillators with non-zero delay where the coupling between the oscillators is given by the Mirollo–Strogatz function. We prove the existence of heteroclinic cycles between unstable attractors for a network of four oscillators and for an open set of parameter values

  8. Heteroclinic cycles between unstable attractors

    Broer, Henk; Efstathiou, Konstantinos; Subramanian, Easwar

    We consider networks of pulse coupled linear oscillators with non-zero delay where the coupling between the oscillators is given by the Mirollo-Strogatz function. We prove the existence of heteroclinic cycles between unstable attractors for a network of four oscillators and for an open set of

  9. Hyperbolic geometry of cosmological attractors

    Carrasco, John Joseph M.; Kallosh, Renata; Linde, Andrei; Roest, Diederik

    2015-01-01

    Cosmological alpha attractors give a natural explanation for the spectral index n(s) of inflation as measured by Planck while predicting a range for the tensor-to-scalar ratio r, consistent with all observations, to be measured more precisely in future B-mode experiments. We highlight the crucial

  10. Chaotic Traversal (CHAT): Very Large Graphs Traversal Using Chaotic Dynamics

    Changaival, Boonyarit; Rosalie, Martin; Danoy, Grégoire; Lavangnananda, Kittichai; Bouvry, Pascal

    2017-12-01

    Graph Traversal algorithms can find their applications in various fields such as routing problems, natural language processing or even database querying. The exploration can be considered as a first stepping stone into knowledge extraction from the graph which is now a popular topic. Classical solutions such as Breadth First Search (BFS) and Depth First Search (DFS) require huge amounts of memory for exploring very large graphs. In this research, we present a novel memoryless graph traversal algorithm, Chaotic Traversal (CHAT) which integrates chaotic dynamics to traverse large unknown graphs via the Lozi map and the Rössler system. To compare various dynamics effects on our algorithm, we present an original way to perform the exploration of a parameter space using a bifurcation diagram with respect to the topological structure of attractors. The resulting algorithm is an efficient and nonresource demanding algorithm, and is therefore very suitable for partial traversal of very large and/or unknown environment graphs. CHAT performance using Lozi map is proven superior than the, commonly known, Random Walk, in terms of number of nodes visited (coverage percentage) and computation time where the environment is unknown and memory usage is restricted.

  11. A new transiently chaotic flow with ellipsoid equilibria

    Panahi, Shirin; Aram, Zainab; Jafari, Sajad; Pham, Viet-Thanh; Volos, Christos; Rajagopal, Karthikeyan

    2018-03-01

    In this article, a simple autonomous transiently chaotic flow with cubic nonlinearities is proposed. This system represents some unusual features such as having a surface of equilibria. We shall describe some dynamical properties and behaviours of this system in terms of eigenvalue structures, bifurcation diagrams, time series, and phase portraits. Various behaviours of this system such as periodic and transiently chaotic dynamics can be shown by setting special parameters in proper values. Our system belongs to a newly introduced category of transiently chaotic systems: systems with hidden attractors. Transiently chaotic behaviour of our proposed system has been implemented and tested by the OrCAD-PSpise software. We have found a proper qualitative similarity between circuit and simulation results.

  12. New Chaotic Dynamical System with a Conic-Shaped Equilibrium Located on the Plane Structure

    Jiri Petrzela

    2017-09-01

    Full Text Available This paper presents a new autonomous deterministic dynamical system with equilibrium degenerated into a plane-oriented hyperbolic geometrical structure. It is demonstrated via numerical analysis and laboratory experiments that the discovered system has both a structurally stable strange attractor and experimentally measurable chaotic behavior. It is shown that the evolution of complex dynamics can be associated with a single parameter of a mathematical model and, due to one-to-one correspondence, to a single circuit parameter. Two-dimensional high resolution plots of the largest Lyapunov exponent and basins of attraction expressed in terms of final state energy are calculated and put into the context of the discovered third-order mathematical model and real chaotic oscillator. Both voltage- and current-mode analog chaotic oscillators are presented and verified by visualization of the typical chaotic attractor in a different fashion.

  13. Chaotic convection of viscoelastic fluids in porous media

    Sheu, L.-J. [Department of Mechanical Engineering, Chung Hua University, Hsinchu, Taiwan (China)], E-mail: ljsheu@chu.edu.tw; Tam, L.-M. [Department of Electromechanical Engineering, University of Macau, Macau (China)], E-mail: fstlmt@umac.mo; Chen, J.-H. [Department of Mechanical Engineering, Chung Hua University, Hsinchu, Taiwan (China)], E-mail: chen@chu.edu.tw; Chen, H.-K. [Department of Industrial Engineering and Management, Hsiuping Institute of Technology, Taichung, Taiwan (China)], E-mail: kanechen@giga.net.tw; Lin, K.-T. [Department of Mechanical Engineering, Chung Yuan Christian University, Chung Li, Taiwan (China)], E-mail: willie@nanya.edu.tw; Kang Yuan [Department of Mechanical Engineering, Chung Yuan Christian University, Chung Li, Taiwan (China)], E-mail: yk@cycu.edu.tw

    2008-07-15

    Buoyancy-induced convection in a viscoelastic fluid-saturated porous medium was analyzed using an Oldroydian-type constitutive relation. An autonomous system with four differential equations was deduced by applying the truncated Galerkin expansion to the momentum and heat transfer equations. The four-dimensional system can be reduced to many systems provided in the literature such as the Lorenz system, Vadasz system, Khayat system, and Akhatov system. Depending on the flow parameters, the asymptotic behavior can be stationary, periodic, or chaotic. Generation of a four-scroll, or two-'butterfly', chaotic attractor was observed. Results also show that stress relaxation tends to precipitate the onset of chaos.

  14. Unstable Periodic Orbit Analysis of Histograms of Chaotic Time Series

    Zoldi, S.M.

    1998-01-01

    Using the Lorenz equations, we have investigated whether unstable periodic orbits (UPOs) associated with a strange attractor may predict the occurrence of the robust sharp peaks in histograms of some experimental chaotic time series. Histograms with sharp peaks occur for the Lorenz parameter value r=60.0 but not for r=28.0 , and the sharp peaks for r=60.0 do not correspond to a histogram derived from any single UPO. However, we show that histograms derived from the time series of a non-Axiom-A chaotic system can be accurately predicted by an escape-time weighting of UPO histograms. copyright 1998 The American Physical Society

  15. Analysis, synchronisation and circuit design of a new highly nonlinear chaotic system

    Mobayen, Saleh; Kingni, Sifeu Takougang; Pham, Viet-Thanh; Nazarimehr, Fahimeh; Jafari, Sajad

    2018-02-01

    This paper investigates a three-dimensional autonomous chaotic flow without linear terms. Dynamical behaviour of the proposed system is investigated through eigenvalue structures, phase portraits, bifurcation diagram, Lyapunov exponents and basin of attraction. For a suitable choice of the parameters, the proposed system can exhibit anti-monotonicity, periodic oscillations and double-scroll chaotic attractor. Basin of attraction of the proposed system shows that the chaotic attractor is self-excited. Furthermore, feasibility of double-scroll chaotic attractor in the real word is investigated by using the OrCAD-PSpice software via an electronic implementation of the proposed system. A good qualitative agreement is illustrated between the numerical simulations and the OrCAD-PSpice results. Finally, a finite-time control method based on dynamic sliding surface for the synchronisation of master and slave chaotic systems in the presence of external disturbances is performed. Using the suggested control technique, the superior master-slave synchronisation is attained. Illustrative simulation results on the studied chaotic system are presented to indicate the effectiveness of the suggested scheme.

  16. Cortical computations via transient attractors.

    Oliver L C Rourke

    Full Text Available The ability of sensory networks to transiently store information on the scale of seconds can confer many advantages in processing time-varying stimuli. How a network could store information on such intermediate time scales, between typical neurophysiological time scales and those of long-term memory, is typically attributed to persistent neural activity. An alternative mechanism which might allow for such information storage is through temporary modifications to the neural connectivity which decay on the same second-long time scale as the underlying memories. Earlier work that has explored this method has done so by emphasizing one attractor from a limited, pre-defined set. Here, we describe an alternative, a Transient Attractor network, which can learn any pattern presented to it, store several simultaneously, and robustly recall them on demand using targeted probes in a manner reminiscent of Hopfield networks. We hypothesize that such functionality could be usefully embedded within sensory cortex, and allow for a flexibly-gated short-term memory, as well as conferring the ability of the network to perform automatic de-noising, and separation of input signals into distinct perceptual objects. We demonstrate that the stored information can be refreshed to extend storage time, is not sensitive to noise in the system, and can be turned on or off by simple neuromodulation. The diverse capabilities of transient attractors, as well as their resemblance to many features observed in sensory cortex, suggest the possibility that their actions might underlie neural processing in many sensory areas.

  17. Cortical computations via transient attractors.

    Rourke, Oliver L C; Butts, Daniel A

    2017-01-01

    The ability of sensory networks to transiently store information on the scale of seconds can confer many advantages in processing time-varying stimuli. How a network could store information on such intermediate time scales, between typical neurophysiological time scales and those of long-term memory, is typically attributed to persistent neural activity. An alternative mechanism which might allow for such information storage is through temporary modifications to the neural connectivity which decay on the same second-long time scale as the underlying memories. Earlier work that has explored this method has done so by emphasizing one attractor from a limited, pre-defined set. Here, we describe an alternative, a Transient Attractor network, which can learn any pattern presented to it, store several simultaneously, and robustly recall them on demand using targeted probes in a manner reminiscent of Hopfield networks. We hypothesize that such functionality could be usefully embedded within sensory cortex, and allow for a flexibly-gated short-term memory, as well as conferring the ability of the network to perform automatic de-noising, and separation of input signals into distinct perceptual objects. We demonstrate that the stored information can be refreshed to extend storage time, is not sensitive to noise in the system, and can be turned on or off by simple neuromodulation. The diverse capabilities of transient attractors, as well as their resemblance to many features observed in sensory cortex, suggest the possibility that their actions might underlie neural processing in many sensory areas.

  18. Chaotic synchronization of two complex nonlinear oscillators

    Mahmoud, Gamal M.; Mahmoud, Emad E.; Farghaly, Ahmed A.; Aly, Shaban A.

    2009-01-01

    Synchronization is an important phenomenon commonly observed in nature. It is also often artificially induced because it is desirable for a variety of applications in physics, applied sciences and engineering. In a recent paper [Mahmoud GM, Mohamed AA, Aly SA. Strange attractors and chaos control in periodically forced complex Duffing's oscillators. Physica A 2001;292:193-206], a system of periodically forced complex Duffing's oscillators was introduced and shown to display chaotic behavior and possess strange attractors. Such complex oscillators appear in many problems of physics and engineering, as, for example, nonlinear optics, deep-water wave theory, plasma physics and bimolecular dynamics. Their connection to solutions of the nonlinear Schroedinger equation has also been pointed out. In this paper, we study the remarkable phenomenon of chaotic synchronization on these oscillator systems, using active control and global synchronization techniques. We derive analytical expressions for control functions and show that the dynamics of error evolution is globally stable, by constructing appropriate Lyapunov functions. This means that, for a relatively large set initial conditions, the differences between the drive and response systems vanish exponentially and synchronization is achieved. Numerical results are obtained to test the validity of the analytical expressions and illustrate the efficiency of these techniques for inducing chaos synchronization in our nonlinear oscillators.

  19. Generalized Attractor Points in Gauged Supergravity

    Kachru, Shamit; /Stanford U., Phys. Dept. /SLAC; Kallosh, Renata; /Stanford U., Phys. Dept.; Shmakova, Marina; /KIPAC, Menlo Park /SLAC /Stanford U., Phys. Dept.

    2011-08-15

    The attractor mechanism governs the near-horizon geometry of extremal black holes in ungauged 4D N=2 supergravity theories and in Calabi-Yau compactifications of string theory. In this paper, we study a natural generalization of this mechanism to solutions of arbitrary 4D N=2 gauged supergravities. We define generalized attractor points as solutions of an ansatz which reduces the Einstein, gauge field, and scalar equations of motion to algebraic equations. The simplest generalized attractor geometries are characterized by non-vanishing constant anholonomy coefficients in an orthonormal frame. Basic examples include Lifshitz and Schroedinger solutions, as well as AdS and dS vacua. There is a generalized attractor potential whose critical points are the attractor points, and its extremization explains the algebraic nature of the equations governing both supersymmetric and non-supersymmetric attractors.

  20. Driving-induced multistability in coupled chaotic oscillators: Symmetries and riddled basins

    Ujjwal, Sangeeta Rani; Ramaswamy, Ram [School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067 (India); Punetha, Nirmal; Prasad, Awadhesh [Department of Physics and Astrophysics, University of Delhi, Delhi 110007 (India); Agrawal, Manish [Department of Physics, Sri Aurobindo College, University of Delhi, New Delhi 110017 (India)

    2016-06-15

    We study the multistability that results when a chaotic response system that has an invariant symmetry is driven by another chaotic oscillator. We observe that there is a transition from a desynchronized state to a situation of multistability. In the case considered, there are three coexisting attractors, two of which are synchronized and one is desynchronized. For large coupling, the asynchronous attractor disappears, leaving the system bistable. We study the basins of attraction of the system in the regime of multistability. The three attractor basins are interwoven in a complex manner, with extensive riddling within a sizeable region of (but not the entire) phase space. A quantitative characterization of the riddling behavior is made via the so–called uncertainty exponent, as well as by evaluating the scaling behavior of tongue–like structures emanating from the synchronization manifold.

  1. From Wang-Chen System with Only One Stable Equilibrium to a New Chaotic System Without Equilibrium

    Pham, Viet-Thanh; Wang, Xiong; Jafari, Sajad; Volos, Christos; Kapitaniak, Tomasz

    2017-06-01

    Wang-Chen system with only one stable equilibrium as well as the coexistence of hidden attractors has attracted increasing interest due to its striking features. In this work, the effect of state feedback on Wang-Chen system is investigated by introducing a further state variable. It is worth noting that a new chaotic system without equilibrium is obtained. We believe that the system is an interesting example to illustrate the conversion of hidden attractors with one stable equilibrium to hidden attractors without equilibrium.

  2. A new chaotic cryptosystem

    Wei Jun; Liao Xiaofeng; Wong, Kwok-wo; Xiang Tao

    2006-01-01

    Based on the study of some previously proposed chaotic encryption algorithms, we found that it is dangerous to mix chaotic state or iteration number of the chaotic system with ciphertext. In this paper, a new chaotic cryptosystem is proposed. Instead of simply mixing the chaotic signal of the proposed chaotic cryptosystem with the ciphertext, a noise-like variable is utilized to govern the encryption and decryption processes. This adds statistical sense to the new cryptosystem. Numerical simulations show that the new cryptosystem is practical whenever efficiency, ciphertext length or security is concerned

  3. Hypercrater Bifurcations, Attractor Coexistence, and Unfolding in a 5D Model of Economic Dynamics

    Toichiro Asada

    2011-01-01

    Full Text Available Complex dynamical features are explored in a discrete interregional macrodynamic model proposed by Asada et al., using numerical methods. The model is five-dimensional with four parameters. The results demonstrate patterns of dynamical behaviour, such as bifurcation processes and coexistence of attractors, generated by high-dimensional discrete systems. In three cases of two-dimensional parameter subspaces the stability of equilibrium region is determined and its boundaries, the flip and Neimark-Hopf bifurcation curves, are identified by means of necessary coefficient criteria. In the first case closed invariant curves (CICs are found to occur through 5D-crater-type bifurcations, and for certain ranges of parameter values a stable equilibrium coexists with an unstable CIC associated with the subcritical bifurcation, as well as with an outer stable CIC. A remarkable feature of the second case is the coexistence of two attracting CICs outside the stability region. In both these cases the related hysteresis effects are illustrated by numerical simulations. In the third case a remarkable feature is the apparent unfolding of an attracting CIC before it evolves to a chaotic attractor. Examples of CICs and chaotic attractors are given in subspaces of phase space.

  4. Merge of terminological resources

    Henriksen, Lina; Braasch, Anna

    2012-01-01

    In our globalized world, the amount of cross-national communication increases rapidly, which also calls for easy access to multi-lingual high quality terminological resources. Sharing of terminology resources is currently becoming common practice, and efficient strategies for integration...... – or merging – of terminology resources are strongly needed. This paper discusses prerequisites for successful merging with the focus on identification of candidate duplicates of a subject domain found in the resources to be merged, and it describes automatic merging strategies to be applied to such duplicates...... in electronic terminology resources. Further, some perspectives of manual, supplementary assessment methods supporting the automatic procedures are sketched. Our considerations are primarily based on experience gained in the IATE and EuroTermBank projects, as merging was a much discussed issue in both projects....

  5. A novel grid multiwing chaotic system with only non-hyperbolic equilibria

    Zhang, Sen; Zeng, Yicheng; Li, Zhijun; Wang, Mengjiao; Xiong, Le

    2018-05-01

    The structure of the chaotic attractor of a system is mainly determined by the nonlinear functions in system equations. By using a new saw-tooth wave function and a new stair function, a novel complex grid multiwing chaotic system which belongs to non-Shil'nikov chaotic system with non-hyperbolic equilibrium points is proposed in this paper. It is particularly interesting that the complex grid multiwing attractors are generated by increasing the number of non-hyperbolic equilibrium points, which are different from the traditional methods of realising multiwing attractors by adding the index-2 saddle-focus equilibrium points in double-wing chaotic systems. The basic dynamical properties of the new system, such as dissipativity, phase portraits, the stability of the equilibria, the time-domain waveform, power spectrum, bifurcation diagram, Lyapunov exponents, and so on, are investigated by theoretical analysis and numerical simulations. Furthermore, the corresponding electronic circuit is designed and simulated on the Multisim platform. The Multisim simulation results and the hardware experimental results are in good agreement with the numerical simulations of the same system on Matlab platform, which verify the feasibility of this new grid multiwing chaotic system.

  6. Finite connectivity attractor neural networks

    Wemmenhove, B; Coolen, A C C

    2003-01-01

    We study a family of diluted attractor neural networks with a finite average number of (symmetric) connections per neuron. As in finite connectivity spin glasses, their equilibrium properties are described by order parameter functions, for which we derive an integral equation in replica symmetric approximation. A bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin-glass transition lines in the phase diagram. The line separating the retrieval phase from the spin-glass phase is calculated at zero temperature. All phase transitions are found to be continuous

  7. Analysis, Adaptive Control and Adaptive Synchronization of a Nine-Term Novel 3-D Chaotic System with Four Quadratic Nonlinearities and its Circuit Simulation

    S. Vaidyanathan

    2014-11-01

    Full Text Available This research work describes a nine-term novel 3-D chaotic system with four quadratic nonlinearities and details its qualitative properties. The phase portraits of the 3-D novel chaotic system simulated using MATLAB, depict the strange chaotic attractor of the system. For the parameter values chosen in this work, the Lyapunov exponents of the novel chaotic system are obtained as L1 = 6.8548, L2 = 0 and L3 = −32.8779. Also, the Kaplan-Yorke dimension of the novel chaotic system is obtained as DKY = 2.2085. Next, an adaptive controller is design to achieve global stabilization of the 3-D novel chaotic system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global chaos synchronization of two identical novel chaotic systems with unknown system parameters. Finally, an electronic circuit realization of the novel chaotic system is presented using SPICE to confirm the feasibility of the theoretical model.

  8. A New Chaotic System with Positive Topological Entropy

    Zhonglin Wang

    2015-08-01

    Full Text Available This paper introduces a new simple system with a butterfly chaotic attractor. This system has rich and complex dynamics. With some typical parameters, its Lyapunov dimension is greater than other known three dimensional chaotic systems. It exhibits chaotic behavior over a large range of parameters, and the divergence of flow of this system is not a constant. The dynamics of this new system are analyzed via Lyapunov exponent spectrum, bifurcation diagrams, phase portraits and the Poincaré map. The compound structures of this new system are also analyzed. By means of topological horseshoe theory and numerical computation, the Poincaré map defined for the system is proved to be semi-conjugate to 3-shift map, and thus the system has positive topological entropy.

  9. Tetrapterous butterfly attractors in modified Lorenz systems

    Yu Simin; Tang, Wallace K.S.

    2009-01-01

    In this paper, the Lorenz-type tetrapterous butterfly attractors are firstly reported. With the introduction of multiple segment piecewise linear functions, these interesting and complex attractors are obtained from two different modified Lorenz models. This approach are verified in both simulations and experiments.

  10. Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system

    Vaidyanathan Sundarapandian

    2015-09-01

    Full Text Available First, this paper announces a seven-term novel 3-D conservative chaotic system with four quadratic nonlinearities. The conservative chaotic systems are characterized by the important property that they are volume conserving. The phase portraits of the novel conservative chaotic system are displayed and the mathematical properties are discussed. An important property of the proposed novel chaotic system is that it has no equilibrium point. Hence, it displays hidden chaotic attractors. The Lyapunov exponents of the novel conservative chaotic system are obtained as L1 = 0.0395,L2 = 0 and L3 = −0.0395. The Kaplan-Yorke dimension of the novel conservative chaotic system is DKY =3. Next, an adaptive controller is designed to globally stabilize the novel conservative chaotic system with unknown parameters. Moreover, an adaptive controller is also designed to achieve global chaos synchronization of the identical conservative chaotic systems with unknown parameters. MATLAB simulations have been depicted to illustrate the phase portraits of the novel conservative chaotic system and also the adaptive control results.

  11. The chaotic environment

    Cook, A.

    1990-09-01

    An elementary account of the origin of chaotic behaviour in classical dynamics is given with examples from geophysics, and in conclusion some thoughts about what can be predicted of chaotic behaviour and what sorts of arguments can be used to guide human behaviour in chaotic conditions are presented. 4 refs

  12. Holonomy Attractor Connecting Spaces of Different Curvature Responsible for ``Anomalies''

    Binder, Bernd

    2009-03-01

    In this lecture paper we derive Magic Angle Precession (MAP) from first geometric principles. MAP can arise in situations, where precession is multiply related to spin, linearly by time or distance (dynamic phase, rolling, Gauss law) and transcendentally by the holonomy loop path (geometric phase). With linear spin-precession coupling, gyroscopes can be spun up and down to very high frequencies via low frequency holonomy control induced by external accelerations, which provides for extreme coupling strengths or "anomalies" that can be tested by the powerball or gyrotwister device. Geometrically, a gyroscopic manifold with spherical metric is tangentially aligned to a precession wave channel with conic or hyperbolic metric (like the relativistic Thomas precession). Transporting triangular spin/precession vector relations across the tangential boundary of contact with SO(3) Lorentz symmetry, we get extreme vector currents near the attractor fixed points in precession phase space, where spin currents remain intact while crossing the contact boundaries between regions of different curvature signature (-1, 0, +1). The problem can be geometrically solved by considering a curvature invariant triangular condition, which holds on surfaces with different curvature that are in contact and locally parallel. In this case two out of three angles are identical, whereas the third angle is different due to holonomy. If we require that the side length ratio corresponding to these angles are invariant we get a geodesic chaotic attractor, which is a cosine map cos(x)˜Mx in parameter space providing for fixed points, limit cycle bifurcations, and singularities. The situation could be quite natural and common in the context of vector currents in curved spacetime and gauge theories. MAP could even be part of the electromagnetic interaction, where the electric charge is the geometric U(1) precession spin current and gauge potential with magnetic effects given by extra rotations under the

  13. Chaotic behaviour of Zeeman machines at introductory course of mechanics

    Nagy, Péter; Tasnádi, Péter

    2016-05-01

    Investigation of chaotic motions and cooperative systems offers a magnificent opportunity to involve modern physics into the basic course of mechanics taught to engineering students. In the present paper it will be demonstrated that Zeeman Machine can be a versatile and motivating tool for students to get introductory knowledge about chaotic motion via interactive simulations. It works in a relatively simple way and its properties can be understood very easily. Since the machine can be built easily and the simulation of its movement is also simple the experimental investigation and the theoretical description can be connected intuitively. Although Zeeman Machine is known mainly for its quasi-static and catastrophic behaviour, its dynamic properties are also of interest with its typical chaotic features. By means of a periodically driven Zeeman Machine a wide range of chaotic properties of the simple systems can be demonstrated such as bifurcation diagrams, chaotic attractors, transient chaos and so on. The main goal of this paper is the presentation of an interactive learning material for teaching the basic features of the chaotic systems through the investigation of the Zeeman Machine.

  14. Chaotic behaviour of Zeeman machines at introductory course of mechanics

    Nagy, P.; Tasnádi, P.

    2015-01-01

    Investigation of chaotic motions and cooperative systems offers a magnificent opportunity to involve modern physics into the basic course of mechanics taught to engineering students. In the present paper it will be demonstrated that Zeeman Machine can be a versatile and motivating tool for students to get introductory knowledge about chaotic motion via interactive simulations. It works in a relatively simple way and its properties can be understood very easily. Since the machine can be built easily and the simulation of its movement is also simple the experimental investigation and the theoretical description can be connected intuitively. Although Zeeman Machine is known mainly for its quasi-static and catastrophic behaviour, its dynamic properties are also of interest with its typical chaotic features. By means of a periodically driven Zeeman Machine a wide range of chaotic properties of the simple systems can be demonstrated such as bifurcation diagrams, chaotic attractors, transient chaos and so on. The main goal of this paper is the presentation of an interactive learning material for teaching the basic features of the chaotic systems through the investigation of the Zeeman Machine. 1. –

  15. Period-doubling cascades and strange attractors in the triple-well Φ6-Van der Pol oscillator

    Yu Jun; Zhang Rongbo; Pan Weizhen; Schimansky-Geier, L

    2008-01-01

    Duffing-Van der Pol equation with the fifth nonlinear-restoring force is investigated. The bifurcation structure and chaotic motion under the periodic perturbation are obtained by numerical simulations. Numerical simulations, including bifurcation diagrams, Lyapunov exponents, phase portraits and Poincare maps, exhibit some new complex dynamical behaviors of the system. Different routes to chaos, such as period doubling and quasi-periodic routes, and various kinds of strange attractors are also demonstrated

  16. Merging {DBMs} Efficiently

    David, Alexandre

    2005-01-01

    In this paper we present different algorithms to reduce the number of DBMs in federations by merging them. Federations are unions of DBMs and are used to represent non-convex zones. Inclusion checking between DBMs is a limited technique to reduce the size of federations and how to choose some DBMs...... to merge them into a larger one is a combi-natorial problem. We present a number of simple but efficient techniques to avoid searching the combinations while still being able to merge any number of DBMs...

  17. Transient chaotic transport in dissipative drift motion

    Oyarzabal, R.S. [Pós-Graduação em Ciências/Física, Universidade Estadual de Ponta Grossa, 84030-900, Ponta Grossa, PR (Brazil); Szezech, J.D. [Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, 84030-900, Ponta Grossa, PR (Brazil); Batista, A.M., E-mail: antoniomarcosbatista@gmail.com [Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, 84030-900, Ponta Grossa, PR (Brazil); Souza, S.L.T. de [Departamento de Física e Matemática, Universidade Federal de São João del Rei, 36420-000, Ouro Branco, MG (Brazil); Caldas, I.L. [Instituto de Física, Universidade de São Paulo, 05315-970, São Paulo, SP (Brazil); Viana, R.L. [Departamento de Física, Universidade Federal do Paraná, 81531-990, Curitiba, PR (Brazil); Sanjuán, M.A.F. [Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid (Spain)

    2016-04-22

    Highlights: • We consider a situation for which a chaotic transient is present in the dynamics of the two-wave model with damping. • The damping in plasma models can be a way for study a realistic behavior of confinement due the collisional effect. • The escape time as a function of the damping obey a power-law scaling. • We have made a qualitative transport analysis with a simple model that can be useful for more complete models. • We have shown that the pattern of the basin of attraction depends on the damping parameter. - Abstract: We investigate chaotic particle transport in magnetised plasmas with two electrostatic drift waves. Considering dissipation in the drift motion, we verify that the removed KAM surfaces originate periodic attractors with their corresponding basins of attraction. We show that the properties of the basins depend on the dissipation and the space-averaged escape time decays exponentially when the dissipation increases. We find positive finite time Lyapunov exponents in dissipative drift motion, consequently the trajectories exhibit transient chaotic transport. These features indicate how the transient plasma transport depends on the dissipation.

  18. Periodicity, chaos, and multiple attractors in a memristor-based Shinriki's circuit

    Kengne, J. [Laboratory of Automation and Applied Computer (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang (Cameroon); Njitacke Tabekoueng, Z.; Kamdoum Tamba, V.; Nguomkam Negou, A. [Laboratory of Automation and Applied Computer (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang (Cameroon); Department of Physics, Laboratory of Electronics and Signal Processing (LETS), Faculty of Science, University of Dschang, Dschang (Cameroon)

    2015-10-15

    In this contribution, a novel memristor-based oscillator, obtained from Shinriki's circuit by substituting the nonlinear positive conductance with a first order memristive diode bridge, is introduced. The model is described by a continuous time four-dimensional autonomous system with smooth nonlinearities. The basic dynamical properties of the system are investigated including equilibria and stability, phase portraits, frequency spectra, bifurcation diagrams, and Lyapunov exponents' spectrum. It is found that in addition to the classical period-doubling and symmetry restoring crisis scenarios reported in the original circuit, the memristor-based oscillator experiences the unusual and striking feature of multiple attractors (i.e., coexistence of a pair of asymmetric periodic attractors with a pair of asymmetric chaotic ones) over a broad range of circuit parameters. Results of theoretical analyses are verified by laboratory experimental measurements.

  19. Chaotic Behavior of a Generalized Sprott E Differential System

    Oliveira, Regilene; Valls, Claudia

    A chaotic system with only one equilibrium, a stable node-focus, was introduced by Wang and Chen [2012]. This system was found by adding a nonzero constant b to the Sprott E system [Sprott, 1994]. The coexistence of three types of attractors in this autonomous system was also considered by Braga and Mello [2013]. Adding a second parameter to the Sprott E differential system, we get the autonomous system ẋ = ayz + b,ẏ = x2 - y,ż = 1 - 4x, where a,b ∈ ℝ are parameters and a≠0. In this paper, we consider theoretically some global dynamical aspects of this system called here the generalized Sprott E differential system. This polynomial differential system is relevant because it is the first polynomial differential system in ℝ3 with two parameters exhibiting, besides the point attractor and chaotic attractor, coexisting stable limit cycles, demonstrating that this system is truly complicated and interesting. More precisely, we show that for b sufficiently small this system can exhibit two limit cycles emerging from the classical Hopf bifurcation at the equilibrium point p = (1/4, 1/16, 0). We also give a complete description of its dynamics on the Poincaré sphere at infinity by using the Poincaré compactification of a polynomial vector field in ℝ3, and we show that it has no first integrals in the class of Darboux functions.

  20. Attractor hopping between polarization dynamical states in a vertical-cavity surface-emitting laser subject to parallel optical injection

    Denis-le Coarer, Florian; Quirce, Ana; Valle, Angel; Pesquera, Luis; Rodríguez, Miguel A.; Panajotov, Krassimir; Sciamanna, Marc

    2018-03-01

    We present experimental and theoretical results of noise-induced attractor hopping between dynamical states found in a single transverse mode vertical-cavity surface-emitting laser (VCSEL) subject to parallel optical injection. These transitions involve dynamical states with different polarizations of the light emitted by the VCSEL. We report an experimental map identifying, in the injected power-frequency detuning plane, regions where attractor hopping between two, or even three, different states occur. The transition between these behaviors is characterized by using residence time distributions. We find multistability regions that are characterized by heavy-tailed residence time distributions. These distributions are characterized by a -1.83 ±0.17 power law. Between these regions we find coherence enhancement of noise-induced attractor hopping in which transitions between states occur regularly. Simulation results show that frequency detuning variations and spontaneous emission noise play a role in causing switching between attractors. We also find attractor hopping between chaotic states with different polarization properties. In this case, simulation results show that spontaneous emission noise inherent to the VCSEL is enough to induce this hopping.

  1. Extreme multistability analysis of memristor-based chaotic system and its application in image decryption

    Chuang Li

    2017-12-01

    Full Text Available An active charge-controlled memristive Chua’s circuit is implemented, and its basic properties are analyzed. Firstly, with the system trajectory starting from an equilibrium point, the dynamic behavior of multiple coexisting attractors depending on the memristor initial value and the system parameter is studied, which shows the coexisting behaviors of point, period, chaos, and quasic-period. Secondly, with the system motion starting from a non-equilibrium point, the dynamics of extreme multistability in a wide initial value domain are easily conformed by new analytical methods. Furthermore, the simulation results indicate that some strange chaotic attractors like multi-wing type and multi-scroll type are observed when the observed signals are extended from voltage and current to power and energy, respectively. Specially, when different initial conditions are taken, the coexisting strange chaotic attractors between the power and energy signals are exhibited. Finally, the chaotic sequences of the new system are used for encrypting color image to protect image information security. The encryption performance is analyzed by statistic histogram, correlation, key spaces and key sensitivity. Simulation results show that the new memristive chaotic system has high security in color image encryption.

  2. Extreme multistability analysis of memristor-based chaotic system and its application in image decryption

    Li, Chuang; Min, Fuhong; Jin, Qiusen; Ma, Hanyuan

    2017-12-01

    An active charge-controlled memristive Chua's circuit is implemented, and its basic properties are analyzed. Firstly, with the system trajectory starting from an equilibrium point, the dynamic behavior of multiple coexisting attractors depending on the memristor initial value and the system parameter is studied, which shows the coexisting behaviors of point, period, chaos, and quasic-period. Secondly, with the system motion starting from a non-equilibrium point, the dynamics of extreme multistability in a wide initial value domain are easily conformed by new analytical methods. Furthermore, the simulation results indicate that some strange chaotic attractors like multi-wing type and multi-scroll type are observed when the observed signals are extended from voltage and current to power and energy, respectively. Specially, when different initial conditions are taken, the coexisting strange chaotic attractors between the power and energy signals are exhibited. Finally, the chaotic sequences of the new system are used for encrypting color image to protect image information security. The encryption performance is analyzed by statistic histogram, correlation, key spaces and key sensitivity. Simulation results show that the new memristive chaotic system has high security in color image encryption.

  3. Noise induced stabilization of chaotic free-running laser diode

    Virte, Martin, E-mail: mvirte@b-phot.org [Brussels Photonics Team, Department of Applied Physics and Photonics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel (Belgium)

    2016-05-15

    In this paper, we investigate theoretically the stabilization of a free-running vertical-cavity surface-emitting laser exhibiting polarization chaos dynamics. We report the existence of a boundary isolating the chaotic attractor on one side and a steady-state on the other side and identify the unstable periodic orbit playing the role of separatrix. In addition, we highlight a small range of parameters where the chaotic attractor passes through this boundary, and therefore where chaos only appears as a transient behaviour. Then, including the effect of spontaneous emission noise in the laser, we demonstrate that, for realistic levels of noise, the system is systematically pushed over the separating solution. As a result, we show that the chaotic dynamics cannot be sustained unless the steady-state on the other side of the separatrix becomes unstable. Finally, we link the stability of this steady-state to a small value of the birefringence in the laser cavity and discuss the significance of this result on future experimental work.

  4. Exponential attractors for a nonclassical diffusion equation

    Qiaozhen Ma

    2009-01-01

    Full Text Available In this article, we prove the existence of exponential attractors for a nonclassical diffusion equation in ${H^{2}(Omega}cap{H}^{1}_{0}(Omega$ when the space dimension is less than 4.

  5. Feigenbaum attractor and intermittency in particle collisions

    Batunin, A.V.

    1992-01-01

    The hypothesis is proposed that the Feigenbaum attractor arising as a limit set in an infinite pichfork bifurcation sequence for unimodal one-dimensional maps underlies the intermittency phenomena in particle collisions. 23 refs.; 8 figs

  6. Efficient chaotic based satellite power supply subsystem

    Ramos Turci, Luiz Felipe; Macau, Elbert E.N.; Yoneyama, Takashi

    2009-01-01

    In this work, we investigate the use of the Dynamical System Theory to increase the efficiency of the satellite power supply subsystems. The core of a satellite power subsystem relies on its DC/DC converter. This is a very nonlinear system that presents a multitude of phenomena ranging from bifurcations, quasi-periodicity, chaos, coexistence of attractors, among others. The traditional power subsystem design techniques try to avoid these nonlinear phenomena so that it is possible to use linear system theory in small regions about the equilibrium points. Here, we show that more efficiency can be drawn from a power supply subsystem if the DC/DC converter operates in regions of high nonlinearity. In special, if it operates in a chaotic regime, is has an intrinsic sensitivity that can be exploited to efficiently drive the power subsystem over high ranges of power requests by using control of chaos techniques.

  7. Efficient chaotic based satellite power supply subsystem

    Ramos Turci, Luiz Felipe [Technological Institute of Aeronautics (ITA), Sao Jose dos Campos, SP (Brazil)], E-mail: felipeturci@yahoo.com.br; Macau, Elbert E.N. [National Institute of Space Research (Inpe), Sao Jose dos Campos, SP (Brazil)], E-mail: elbert@lac.inpe.br; Yoneyama, Takashi [Technological Institute of Aeronautics (ITA), Sao Jose dos Campos, SP (Brazil)], E-mail: takashi@ita.br

    2009-10-15

    In this work, we investigate the use of the Dynamical System Theory to increase the efficiency of the satellite power supply subsystems. The core of a satellite power subsystem relies on its DC/DC converter. This is a very nonlinear system that presents a multitude of phenomena ranging from bifurcations, quasi-periodicity, chaos, coexistence of attractors, among others. The traditional power subsystem design techniques try to avoid these nonlinear phenomena so that it is possible to use linear system theory in small regions about the equilibrium points. Here, we show that more efficiency can be drawn from a power supply subsystem if the DC/DC converter operates in regions of high nonlinearity. In special, if it operates in a chaotic regime, is has an intrinsic sensitivity that can be exploited to efficiently drive the power subsystem over high ranges of power requests by using control of chaos techniques.

  8. A Chaotic Oscillator Based on HP Memristor Model

    Guangyi Wang

    2015-01-01

    Full Text Available This paper proposes a simple autonomous memristor-based oscillator for generating periodic signals. Applying an external sinusoidal excitation to the autonomous system, a nonautonomous oscillator is obtained, which contains HP memristor model and four linear circuit elements. This memristor-based oscillator can generate periodic, chaotic, and hyperchaotic signals under the periodic excitation and an appropriate set of circuit parameters. It also shows that the system exhibits alternately a hidden attractor with no equilibrium and a self-excited attractor with a line equilibrium as time goes on. Furthermore, some specialties including burst chaos, irregular periodic bifurcations, and nonintermittence chaos of the circuit are found by theoretical analysis and numerical simulations. Finally, a discrete model for the HP memristor is given and the main statistical properties of this memristor-based oscillator are verified via DSP chip experiments and NIST (National Institute of Standards and Technology tests.

  9. Strange Attractors in Drift Wave Turbulence

    Lewandowski, J.L.V.

    2003-01-01

    A multi-grid part-in-cell algorithm for a shearless slab drift wave model with kinetic electrons is presented. The algorithm, which is based on an exact separation of adiabatic and nonadiabatic electron responses, is used to investigate the presence of strange attractors in drift wave turbulence. Although the simulation model has a large number of degrees of freedom, it is found that the strange attractor is low-dimensional and that it is strongly affected by dissipative (collisional) effects

  10. Cusps enable line attractors for neural computation

    Xiao, Zhuocheng; Zhang, Jiwei; Sornborger, Andrew T.; Tao, Louis

    2017-01-01

    Here, line attractors in neuronal networks have been suggested to be the basis of many brain functions, such as working memory, oculomotor control, head movement, locomotion, and sensory processing. In this paper, we make the connection between line attractors and pulse gating in feed-forward neuronal networks. In this context, because of their neutral stability along a one-dimensional manifold, line attractors are associated with a time-translational invariance that allows graded information to be propagated from one neuronal population to the next. To understand how pulse-gating manifests itself in a high-dimensional, nonlinear, feedforward integrate-and-fire network, we use a Fokker-Planck approach to analyze system dynamics. We make a connection between pulse-gated propagation in the Fokker-Planck and population-averaged mean-field (firing rate) models, and then identify an approximate line attractor in state space as the essential structure underlying graded information propagation. An analysis of the line attractor shows that it consists of three fixed points: a central saddle with an unstable manifold along the line and stable manifolds orthogonal to the line, which is surrounded on either side by stable fixed points. Along the manifold defined by the fixed points, slow dynamics give rise to a ghost. We show that this line attractor arises at a cusp catastrophe, where a fold bifurcation develops as a function of synaptic noise; and that the ghost dynamics near the fold of the cusp underly the robustness of the line attractor. Understanding the dynamical aspects of this cusp catastrophe allows us to show how line attractors can persist in biologically realistic neuronal networks and how the interplay of pulse gating, synaptic coupling, and neuronal stochasticity can be used to enable attracting one-dimensional manifolds and, thus, dynamically control the processing of graded information.

  11. Cusps enable line attractors for neural computation

    Xiao, Zhuocheng; Zhang, Jiwei; Sornborger, Andrew T.; Tao, Louis

    2017-11-01

    Line attractors in neuronal networks have been suggested to be the basis of many brain functions, such as working memory, oculomotor control, head movement, locomotion, and sensory processing. In this paper, we make the connection between line attractors and pulse gating in feed-forward neuronal networks. In this context, because of their neutral stability along a one-dimensional manifold, line attractors are associated with a time-translational invariance that allows graded information to be propagated from one neuronal population to the next. To understand how pulse-gating manifests itself in a high-dimensional, nonlinear, feedforward integrate-and-fire network, we use a Fokker-Planck approach to analyze system dynamics. We make a connection between pulse-gated propagation in the Fokker-Planck and population-averaged mean-field (firing rate) models, and then identify an approximate line attractor in state space as the essential structure underlying graded information propagation. An analysis of the line attractor shows that it consists of three fixed points: a central saddle with an unstable manifold along the line and stable manifolds orthogonal to the line, which is surrounded on either side by stable fixed points. Along the manifold defined by the fixed points, slow dynamics give rise to a ghost. We show that this line attractor arises at a cusp catastrophe, where a fold bifurcation develops as a function of synaptic noise; and that the ghost dynamics near the fold of the cusp underly the robustness of the line attractor. Understanding the dynamical aspects of this cusp catastrophe allows us to show how line attractors can persist in biologically realistic neuronal networks and how the interplay of pulse gating, synaptic coupling, and neuronal stochasticity can be used to enable attracting one-dimensional manifolds and, thus, dynamically control the processing of graded information.

  12. Quantum-chaotic cryptography

    de Oliveira, G. L.; Ramos, R. V.

    2018-03-01

    In this work, it is presented an optical scheme for quantum key distribution employing two synchronized optoelectronic oscillators (OEO) working in the chaotic regime. The produced key depends on the chaotic dynamic, and the synchronization between Alice's and Bob's OEOs uses quantum states. An attack on the synchronization signals will disturb the synchronization of the chaotic systems increasing the error rate in the final key.

  13. Stages of chaotic synchronization.

    Tang, D. Y.; Dykstra, R.; Hamilton, M. W.; Heckenberg, N. R.

    1998-09-01

    In an experimental investigation of the response of a chaotic system to a chaotic driving force, we have observed synchronization of chaos of the response system in the forms of generalized synchronization, phase synchronization, and lag synchronization to the driving signal. In this paper we compare the features of these forms of synchronized chaos and study their relations and physical origins. We found that different forms of chaotic synchronization could be interpreted as different stages of nonlinear interaction between the coupled chaotic systems. (c) 1998 American Institute of Physics.

  14. Merged neutral beams

    Osterwalder, Andreas [Ecole Polytechnique Federale de Lausanne (EPFL), Institute for Chemical Sciences and Engineering, Lausanne (Switzerland)

    2015-12-15

    A detailed description of a merged beam apparatus for the study of low energy molecular scattering is given. This review is intended to guide any scientist who plans to construct a similar experiment, and to provide some inspiration in describing the approach we chose to our goal. In our experiment a supersonic expansion of paramagnetic particles is merged with one of polar molecules. A magnetic and an electric multipole guide are used to bend the two beams onto the same axis. We here describe in detail how the apparatus is designed, characterised, and operated. (orig.)

  15. Supersymmetry, attractors and cosmic censorship

    Bellorin, Jorge [Instituto de Fisica Teorica UAM/CSIC, Facultad de Ciencias C-XVI, C.U. Cantoblanco, E-28049 Madrid (Spain)]. E-mail: jorge.bellorin@uam.es; Meessen, Patrick [Instituto de Fisica Teorica UAM/CSIC, Facultad de Ciencias C-XVI, C.U. Cantoblanco, E-28049 Madrid (Spain)]. E-mail: patrick.meessen@cern.ch; Ortin, Tomas [Instituto de Fisica Teorica UAM/CSIC, Facultad de Ciencias C-XVI, C.U. Cantoblanco, E-28049 Madrid (Spain)]. E-mail: tomas.ortin@cern.ch

    2007-01-29

    We show that requiring unbroken supersymmetry everywhere in black-hole-type solutions of N=2, d=4 supergravity coupled to vector supermultiplets ensures in most cases absence of naked singularities. We formulate three specific conditions which we argue are equivalent to the requirement of global supersymmetry. These three conditions can be related to the absence of sources for NUT charge, angular momentum, scalar hair and negative energy, although the solutions can still have globally defined angular momentum and non-trivial scalar fields, as we show in an explicit example. Furthermore, only the solutions satisfying these requirements seem to have a microscopic interpretation in string theory since only they have supersymmetric sources. These conditions exclude, for instance, singular solutions such as the Kerr-Newman with M=|q|, which fails to be everywhere supersymmetric. We also present a re-derivation of several results concerning attractors in N=2, d=4 theories based on the explicit knowledge of the most general solutions in the timelike class.

  16. Merged reality for everyone

    Carvalho, Fausto de; Morgado, Leonel

    2017-01-01

    This article addresses some interesting challenges and business opportunities within the promising merged reality ecosystem, which offers the vision of bringing together virtual, augmented and physical realities, seamlessly. The article also links the current status of this field with exploratory research and development work carried out by Altice Labs. Altice Labs info:eu-repo/semantics/publishedVersion

  17. Plan Merging : Experimental results

    De Weerdt, M.M.; Van der Krogt, R.P.J.; Zutt, J.

    2003-01-01

    In this paper we discuss the results of a plan merging algorithm. This algorithm coordinates the plans of multiple, autonomous agents, each able to independently find a plan. This algorithm is evaluated using realistic data from a taxi company. We show that when we allow passengers to be a few

  18. Optimizing Markovian modeling of chaotic systems with recurrent neural networks

    Cechin, Adelmo L.; Pechmann, Denise R.; Oliveira, Luiz P.L. de

    2008-01-01

    In this paper, we propose a methodology for optimizing the modeling of an one-dimensional chaotic time series with a Markov Chain. The model is extracted from a recurrent neural network trained for the attractor reconstructed from the data set. Each state of the obtained Markov Chain is a region of the reconstructed state space where the dynamics is approximated by a specific piecewise linear map, obtained from the network. The Markov Chain represents the dynamics of the time series in its statistical essence. An application to a time series resulted from Lorenz system is included

  19. A New Feigenbaum-Like Chaotic 3D System

    Huitao Zhao

    2014-01-01

    Full Text Available Based on Sprott N system, a new three-dimensional autonomous system is reported. It is demonstrated to be chaotic in the sense of having positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping, and period-doubling route to chaos are analyzed with careful numerical simulations. The obtained results also show that the period-doubling sequence of bifurcations leads to a Feigenbaum-like strange attractor.

  20. Disordered chaotic strings

    Schäfer, Mirko; Greiner, Martin

    2011-01-01

    to chaotic strings. Inhomogeneous coupling weights as well as small-world perturbations of the ring-network structure are discussed. It is found that certain combinations of coupling and network disorder preserve the empirical relationship between chaotic strings and the weak and strong sector...

  1. Temporal chaotic behaviour of vortex motion in a type-II superconductors with periodically-distributed pinning centres

    Lin, H.T.; Ke, C.; Cheng, C.H.

    2010-01-01

    Temporal chaotic character of vortex motion in systems where defects are arranged in periodic arrays has been investigated by computer simulation. Due to the high nonlinearity of the vortex-defect interaction, the temporal evolution of the vortex motion is chaotic with a power spectrum similar to what have been observed in the experiments. It is found that the strength of both the vortex-vortex and vortex-defect interactions have no significant effects on the chaotic motion of the vortices, however, the mismatch between these two interactions causes attractor crisis of the system. Different from them, the Lorentz force is not the origin of the attractor crisis, but it causes a divergent motion of the vortex (i.e., the flux flow).

  2. Temporal chaotic behaviour of vortex motion in a type-II superconductors with periodically-distributed pinning centres

    Lin, H.T. [Faculty of Information Management, Cheng Shiu University, Kaoshuing, Taiwan (China); Ke, C. [Key Laboratory of Magnetic Levitation and Maglev Trains (Ministry of Education of China), Superconductivity R and D Center (SRDC), Mail Stop 165, Southwest Jiaotong University, Chengdu, Sichuan 610031 (China); Cheng, C.H., E-mail: c.cheng@unsw.edu.a [Key Laboratory of Magnetic Levitation and Maglev Trains (Ministry of Education of China), Superconductivity R and D Center (SRDC), Mail Stop 165, Southwest Jiaotong University, Chengdu, Sichuan 610031 (China); School of Materials Science and Engineering, University of New South Wale, Sydney, 2052 NSW (Australia)

    2010-11-01

    Temporal chaotic character of vortex motion in systems where defects are arranged in periodic arrays has been investigated by computer simulation. Due to the high nonlinearity of the vortex-defect interaction, the temporal evolution of the vortex motion is chaotic with a power spectrum similar to what have been observed in the experiments. It is found that the strength of both the vortex-vortex and vortex-defect interactions have no significant effects on the chaotic motion of the vortices, however, the mismatch between these two interactions causes attractor crisis of the system. Different from them, the Lorentz force is not the origin of the attractor crisis, but it causes a divergent motion of the vortex (i.e., the flux flow).

  3. Denoising of chaotic signal using independent component analysis and empirical mode decomposition with circulate translating

    Wen-Bo, Wang; Xiao-Dong, Zhang; Yuchan, Chang; Xiang-Li, Wang; Zhao, Wang; Xi, Chen; Lei, Zheng

    2016-01-01

    In this paper, a new method to reduce noises within chaotic signals based on ICA (independent component analysis) and EMD (empirical mode decomposition) is proposed. The basic idea is decomposing chaotic signals and constructing multidimensional input vectors, firstly, on the base of EMD and its translation invariance. Secondly, it makes the independent component analysis on the input vectors, which means that a self adapting denoising is carried out for the intrinsic mode functions (IMFs) of chaotic signals. Finally, all IMFs compose the new denoised chaotic signal. Experiments on the Lorenz chaotic signal composed of different Gaussian noises and the monthly observed chaotic sequence on sunspots were put into practice. The results proved that the method proposed in this paper is effective in denoising of chaotic signals. Moreover, it can correct the center point in the phase space effectively, which makes it approach the real track of the chaotic attractor. Project supported by the National Science and Technology, China (Grant No. 2012BAJ15B04), the National Natural Science Foundation of China (Grant Nos. 41071270 and 61473213), the Natural Science Foundation of Hubei Province, China (Grant No. 2015CFB424), the State Key Laboratory Foundation of Satellite Ocean Environment Dynamics, China (Grant No. SOED1405), the Hubei Provincial Key Laboratory Foundation of Metallurgical Industry Process System Science, China (Grant No. Z201303), and the Hubei Key Laboratory Foundation of Transportation Internet of Things, Wuhan University of Technology, China (Grant No.2015III015-B02).

  4. Synchronization of chaotic systems

    Pecora, Louis M.; Carroll, Thomas L.

    2015-01-01

    We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators

  5. An Eight-Term Novel Four-Scroll Chaotic System with Cubic Nonlinearity and its Circuit Simulation

    S. Sampath

    2014-11-01

    Full Text Available This research work proposes an eight-term novel four-scroll chaotic system with cubic nonlinearity and analyses its fundamental properties such as dissipativity, equilibria, symmetry and invariance, Lyapunov exponents and KaplanYorke dimension. The phase portraits of the novel chaotic system, which are obtained in this work by using MATLAB, depict the four-scroll attractor of the system. For the parameter values and initial conditions chosen in this work, the Lyapunov exponents of the novel four-scroll chaotic system are obtained as L1 = 0.75335, L2 = 0 and L3 = −22.43304. Also, the Kaplan-Yorke dimension of the novel four-scroll chaotic system is obtained as DKY = 2.0336. Finally, an electronic circuit realization of the novel four-scroll chaotic system is presented by using SPICE to confirm the feasibility of the theoretical model.

  6. Encoding and decoding messages with chaotic lasers

    Alsing, P.M.; Gavrielides, A.; Kovanis, V.; Roy, R.; Thornburg, K.S. Jr.

    1997-01-01

    We investigate the structure of the strange attractor of a chaotic loss-modulated solid-state laser utilizing return maps based on a combination of intensity maxima and interspike intervals, as opposed to those utilizing Poincare sections defined by the intensity maxima of the laser (I=0,Ie<0) alone. We find both experimentally and numerically that a simple, intrinsic relationship exists between an intensity maximum and the pair of preceding and succeeding interspike intervals. In addition, we numerically investigate encoding messages on the output of a chaotic transmitter laser and its subsequent decoding by a similar receiver laser. By exploiting the relationship between the intensity maxima and the interspike intervals, we demonstrate that the method utilized to encode the message is vital to the system close-quote s ability to hide the signal from unwanted deciphering. In this work alternative methods are studied in order to encode messages by modulating the magnitude of pumping of the transmitter laser and also by driving its loss modulation with more than one frequency. copyright 1997 The American Physical Society

  7. Stochastic Resonance in a System of Coupled Chaotic Oscillators

    Krawiecki, A.

    1999-01-01

    Noise-free stochastic resonance is investigated numerically in a system of two coupled chaotic Roessler oscillators. Periodic signal is applied either additively or multiplicatively to the coupling term. When the coupling constant is varied the oscillators lose synchronization via attractor bubbling or on-off intermittency. Properly chosen signals are analyzed which reflect the sequence of synchronized (laminar) phases and non-synchronized bursts in the time evolution of the oscillators. Maximum of the signal-to-noise ratio as a function of the coupling constant is observed. Dependence of the signal-to-noise ratio on the frequency of the periodic signal and parameter mismatch between the oscillators is investigated. Possible applications of stochastic resonance in the recovery of signals in secure communication systems based on chaotic synchronization are briefly discussed. (author)

  8. Black hole attractors and pure spinors

    Hsu, Jonathan P.; Maloney, Alexander; Tomasiello, Alessandro

    2006-01-01

    We construct black hole attractor solutions for a wide class of N = 2 compactifications. The analysis is carried out in ten dimensions and makes crucial use of pure spinor techniques. This formalism can accommodate non-Kaehler manifolds as well as compactifications with flux, in addition to the usual Calabi-Yau case. At the attractor point, the charges fix the moduli according to Σf k = Im(CΦ), where Φ is a pure spinor of odd (even) chirality in IIB (A). For IIB on a Calabi-Yau, Φ = Ω and the equation reduces to the usual one. Methods in generalized complex geometry can be used to study solutions to the attractor equation

  9. Black Hole Attractors and Pure Spinors

    Hsu, Jonathan P.; Maloney, Alexander; Tomasiello, Alessandro

    2006-01-01

    We construct black hole attractor solutions for a wide class of N = 2 compactifications. The analysis is carried out in ten dimensions and makes crucial use of pure spinor techniques. This formalism can accommodate non-Kaehler manifolds as well as compactifications with flux, in addition to the usual Calabi-Yau case. At the attractor point, the charges fix the moduli according to Σf k = Im(CΦ), where Φ is a pure spinor of odd (even) chirality in IIB (A). For IIB on a Calabi-Yau, Φ = (Omega) and the equation reduces to the usual one. Methods in generalized complex geometry can be used to study solutions to the attractor equation

  10. Non-linguistic Conditions for Causativization as a Linguistic Attractor

    Johanna Nichols; Johanna Nichols; Johanna Nichols

    2018-01-01

    An attractor, in complex systems theory, is any state that is more easily or more often entered or acquired than departed or lost; attractor states therefore accumulate more members than non-attractors, other things being equal. In the context of language evolution, linguistic attractors include sounds, forms, and grammatical structures that are prone to be selected when sociolinguistics and language contact make it possible for speakers to choose between competing forms. The reasons why an e...

  11. Chaotic Boltzmann machines

    Suzuki, Hideyuki; Imura, Jun-ichi; Horio, Yoshihiko; Aihara, Kazuyuki

    2013-01-01

    The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented. PMID:23558425

  12. Determining the Lyapunov Spectrum of Continuous-Time 1D and 2D Multiscroll Chaotic Oscillators via the Solution of m-PWL Variational Equations

    Jesus Manuel Munoz-Pacheco

    2013-01-01

    Full Text Available An algorithm to compute the Lyapunov exponents of piecewise linear function-based multidirectional multiscroll chaotic oscillators is reported. Based on the m regions in the piecewise linear functions, the suggested algorithm determines the individual expansion rate of Lyapunov exponents from m-piecewise linear variational equations and their associated m-Jacobian matrices whose entries remain constant during all computation cycles. Additionally, by considering OpAmp-based chaotic oscillators, we study the impact of two analog design procedures on the magnitude of Lyapunov exponents. We focus on analyzing variations of both frequency bandwidth and voltage/current dynamic range of the chaotic signals at electronic system level. As a function of the design parameters, a renormalization factor is proposed to estimate correctly the Lyapunov spectrum. Numerical simulation results in a double-scroll type chaotic oscillator and complex chaotic oscillators generating multidirectional multiscroll chaotic attractors on phase space confirm the usefulness of the reported algorithm.

  13. Connecting coherent structures and strange attractors

    Keefe, Laurence R.

    1990-01-01

    A concept of turbulence derived from nonlinear dynamical systems theory suggests that turbulent solutions to the Navier-Stokes equations are restricted to strange attractors, and, by implication, that turbulent phenomenology must find some expression or source in the structure of these mathematical objects. Examples and discussions are presented to link coherent structures to some of the commonly known characteristics of strange attractors. Basic to this link is a geometric interpretation of conditional sampling techniques employed to educe coherent structures that offers an explanation for their appearance in measurements as well as their size.

  14. Applying Chaos Theory to Careers: Attraction and Attractors

    Pryor, Robert G. L.; Bright, Jim E. H.

    2007-01-01

    This article presents the Chaos Theory of Careers with particular reference to the concepts of "attraction" and "attractors". Attractors are defined in terms of characteristic trajectories, feedback mechanisms, end states, ordered boundedness, reality visions and equilibrium and fluctuation. The identified types of attractors (point, pendulum,…

  15. Modeling and Analysis of a Fractional-Order Generalized Memristor-Based Chaotic System and Circuit Implementation

    Yang, Ningning; Xu, Cheng; Wu, Chaojun; Jia, Rong; Liu, Chongxin

    2017-12-01

    Memristor is a nonlinear “missing circuit element”, that can easily achieve chaotic oscillation. Memristor-based chaotic systems have received more and more attention. Research shows that fractional-order systems are more close to real systems. As an important parameter, the order can increase the flexibility and degree of freedom of the system. In this paper, a fractional-order generalized memristor, which consists of a diode bridge and a parallel circuit with an equivalent unit circuit and a linear resistance, is proposed. Frequency and electrical characteristics of the fractional-order memristor are analyzed. A chain structure circuit is used to implement the fractional-order unit circuit. Then replacing the conventional Chua’s diode by the fractional-order generalized memristor, a fractional-order memristor-based chaotic circuit is proposed. A large amount of research work has been done to investigate the influence of the order on the dynamical behaviors of the fractional-order memristor-based chaotic circuit. Varying with the order, the system enters the chaotic state from the periodic state through the Hopf bifurcation and period-doubling bifurcation. The chaotic state of the system has two types of attractors: single-scroll and double-scroll attractor. The stability theory of fractional-order systems is used to determine the minimum order occurring Hopf bifurcation. And the influence of the initial value on the system is analyzed. Circuit simulations are designed to verify the results of theoretical analysis and numerical simulation.

  16. Elementary chaotic snap flows

    Munmuangsaen, Buncha; Srisuchinwong, Banlue

    2011-01-01

    Highlights: → Five new elementary chaotic snap flows and a generalization of an existing chaotic snap flow have been presented. → Three of all are conservative systems whilst three others are dissipative systems. → Four cases need only a single control parameter and a single nonlinearity. → A cubic case in a jerk representation requires only two terms and a single nonlinearity. - Abstract: Hyperjerk systems with 4th-order derivative of the form x .... =f(x ... ,x .. ,x . ,x) have been referred to as snap systems. Five new elementary chaotic snap flows and a generalization of an existing flow are presented through an extensive numerical search. Four of these flows demonstrate elegant simplicity of a single control parameter based on a single nonlinearity of a quadratic, a piecewise-linear or an exponential type. Two others demonstrate elegant simplicity of all unity-in-magnitude parameters based on either a single cubic nonlinearity or three cubic nonlinearities. The chaotic snap flow with a single cubic nonlinearity requires only two terms and can be transformed to its equivalent dynamical form of only five terms which have a single nonlinearity. An advantage is that such a chaotic flow offers only five terms even though the (four) dimension is high. Three of the chaotic snap flows are characterized as conservative systems whilst three others are dissipative systems. Basic dynamical properties are described.

  17. Atypical transistor-based chaotic oscillators: Design, realization, and diversity

    Minati, Ludovico; Frasca, Mattia; OświÈ©cimka, Paweł; Faes, Luca; DroŻdŻ, Stanisław

    2017-07-01

    In this paper, we show that novel autonomous chaotic oscillators based on one or two bipolar junction transistors and a limited number of passive components can be obtained via random search with suitable heuristics. Chaos is a pervasive occurrence in these circuits, particularly after manual adjustment of a variable resistor placed in series with the supply voltage source. Following this approach, 49 unique circuits generating chaotic signals when physically realized were designed, representing the largest collection of circuits of this kind to date. These circuits are atypical as they do not trivially map onto known topologies or variations thereof. They feature diverse spectra and predominantly anti-persistent monofractal dynamics. Notably, we recurrently found a circuit comprising one resistor, one transistor, two inductors, and one capacitor, which generates a range of attractors depending on the parameter values. We also found a circuit yielding an irregular quantized spike-train resembling some aspects of neural discharge and another one generating a double-scroll attractor, which represent the smallest known transistor-based embodiments of these behaviors. Through three representative examples, we additionally show that diffusive coupling of heterogeneous oscillators of this kind may give rise to complex entrainment, such as lag synchronization with directed information transfer and generalized synchronization. The replicability and reproducibility of the experimental findings are good.

  18. Trajectory attractors of equations of mathematical physics

    Vishik, Marko I; Chepyzhov, Vladimir V

    2011-01-01

    In this survey the method of trajectory dynamical systems and trajectory attractors is described, and is applied in the study of the limiting asymptotic behaviour of solutions of non-linear evolution equations. This method is especially useful in the study of dissipative equations of mathematical physics for which the corresponding Cauchy initial-value problem has a global (weak) solution with respect to the time but the uniqueness of this solution either has not been established or does not hold. An important example of such an equation is the 3D Navier-Stokes system in a bounded domain. In such a situation one cannot use directly the classical scheme of construction of a dynamical system in the phase space of initial conditions of the Cauchy problem of a given equation and find a global attractor of this dynamical system. Nevertheless, for such equations it is possible to construct a trajectory dynamical system and investigate a trajectory attractor of the corresponding translation semigroup. This universal method is applied for various types of equations arising in mathematical physics: for general dissipative reaction-diffusion systems, for the 3D Navier-Stokes system, for dissipative wave equations, for non-linear elliptic equations in cylindrical domains, and for other equations and systems. Special attention is given to using the method of trajectory attractors in approximation and perturbation problems arising in complicated models of mathematical physics. Bibliography: 96 titles.

  19. Synchronization of identical chaotic systems through external chaotic driving

    Patidar, V.; Sud, K.K.

    2005-11-01

    In recent years, the study of synchronization of identical chaotic systems subjected to a common fluctuating random driving signal has drawn considerable interest. In this communication, we report that it is possible to achieve synchronization between two identical chaotic systems, which are not coupled directly but subjected to an external chaotic signal. The external chaotic signal may be obtained from any chaotic system identical or non-identical to both identical chaotic systems. Results of numerical simulations on well known Roessler and jerk dynamical systems have been presented. (author)

  20. Why healthcare providers merge.

    Postma, Jeroen; Roos, Anne-Fleur

    2016-04-01

    In many OECD countries, healthcare sectors have become increasingly concentrated as a result of mergers. However, detailed empirical insight into why healthcare providers merge is lacking. Also, we know little about the influence of national healthcare policies on mergers. We fill this gap in the literature by conducting a survey study on mergers among 848 Dutch healthcare executives, of which 35% responded (resulting in a study sample of 239 executives). A total of 65% of the respondents was involved in at least one merger between 2005 and 2012. During this period, Dutch healthcare providers faced a number of policy changes, including increasing competition, more pressure from purchasers, growing financial risks, de-institutionalisation of long-term care and decentralisation of healthcare services to municipalities. Our empirical study shows that healthcare providers predominantly merge to improve the provision of healthcare services and to strengthen their market position. Also efficiency and financial reasons are important drivers of merger activity in healthcare. We find that motives for merger are related to changes in health policies, in particular to the increasing pressure from competitors, insurers and municipalities.

  1. Cascade Chaotic System With Applications.

    Zhou, Yicong; Hua, Zhongyun; Pun, Chi-Man; Chen, C L Philip

    2015-09-01

    Chaotic maps are widely used in different applications. Motivated by the cascade structure in electronic circuits, this paper introduces a general chaotic framework called the cascade chaotic system (CCS). Using two 1-D chaotic maps as seed maps, CCS is able to generate a huge number of new chaotic maps. Examples and evaluations show the CCS's robustness. Compared with corresponding seed maps, newly generated chaotic maps are more unpredictable and have better chaotic performance, more parameters, and complex chaotic properties. To investigate applications of CCS, we introduce a pseudo-random number generator (PRNG) and a data encryption system using a chaotic map generated by CCS. Simulation and analysis demonstrate that the proposed PRNG has high quality of randomness and that the data encryption system is able to protect different types of data with a high-security level.

  2. COSMOS-e'-soft Higgsotic attractors

    Choudhury, Sayantan

    2017-07-01

    In this work, we have developed an elegant algorithm to study the cosmological consequences from a huge class of quantum field theories (i.e. superstring theory, supergravity, extra dimensional theory, modified gravity, etc.), which are equivalently described by soft attractors in the effective field theory framework. In this description we have restricted our analysis for two scalar fields - dilaton and Higgsotic fields minimally coupled with Einstein gravity, which can be generalized for any arbitrary number of scalar field contents with generalized non-canonical and non-minimal interactions. We have explicitly used R^2 gravity, from which we have studied the attractor and non-attractor phases by exactly computing two point, three point and four point correlation functions from scalar fluctuations using the In-In (Schwinger-Keldysh) and the δ N formalisms. We have also presented theoretical bounds on the amplitude, tilt and running of the primordial power spectrum, various shapes (equilateral, squeezed, folded kite or counter-collinear) of the amplitude as obtained from three and four point scalar functions, which are consistent with observed data. Also the results from two point tensor fluctuations and the field excursion formula are explicitly presented for the attractor and non-attractor phase. Further, reheating constraints, scale dependent behavior of the couplings and the dynamical solution for the dilaton and Higgsotic fields are also presented. New sets of consistency relations between two, three and four point observables are also presented, which shows significant deviation from canonical slow-roll models. Additionally, three possible theoretical proposals have presented to overcome the tachyonic instability at the time of late time acceleration. Finally, we have also provided the bulk interpretation from the three and four point scalar correlation functions for completeness.

  3. COSMOS-e"'-soft Higgsotic attractors

    Choudhury, Sayantan

    2017-01-01

    In this work, we have developed an elegant algorithm to study the cosmological consequences from a huge class of quantum field theories (i.e. superstring theory, supergravity, extra dimensional theory, modified gravity, etc.), which are equivalently described by soft attractors in the effective field theory framework. In this description we have restricted our analysis for two scalar fields - dilaton and Higgsotic fields minimally coupled with Einstein gravity, which can be generalized for any arbitrary number of scalar field contents with generalized non-canonical and non-minimal interactions. We have explicitly used R"2 gravity, from which we have studied the attractor and non-attractor phases by exactly computing two point, three point and four point correlation functions from scalar fluctuations using the In-In (Schwinger-Keldysh) and the δN formalisms. We have also presented theoretical bounds on the amplitude, tilt and running of the primordial power spectrum, various shapes (equilateral, squeezed, folded kite or counter-collinear) of the amplitude as obtained from three and four point scalar functions, which are consistent with observed data. Also the results from two point tensor fluctuations and the field excursion formula are explicitly presented for the attractor and non-attractor phase. Further, reheating constraints, scale dependent behavior of the couplings and the dynamical solution for the dilaton and Higgsotic fields are also presented. New sets of consistency relations between two, three and four point observables are also presented, which shows significant deviation from canonical slow-roll models. Additionally, three possible theoretical proposals have presented to overcome the tachyonic instability at the time of late time acceleration. Finally, we have also provided the bulk interpretation from the three and four point scalar correlation functions for completeness. (orig.)

  4. Multiple attractors and critical parameters and how to find them numerically: the right, the wrong and the gambling way

    True, Hans

    2013-03-01

    In recent years, several authors have proposed 'easier numerical methods' to find the critical speed in railway dynamical problems. Actually, the methods do function in some cases, but in most cases it is really a gamble. In this article, the methods are discussed and the pros and contras are commented upon. I also address the questions when a linearisation is allowed and the curious fact that the hunting motion is more robust than the ideal stationary-state motion on the track. Concepts such as 'multiple attractors', 'subcritical and supercritical bifurcations', 'permitted linearisation', 'the danger of running at supercritical speeds' and 'chaotic motion' are addressed.

  5. Effect of Parametric Dichotomic Markov Noise on the Properties of Chaotic Transitions in Dynamical Systems

    Gac, J. M.; Żebrowski, J. J.

    A chaotic transition occurs when a continuous change of one of the parameters of the system causes a discontinuous change in the properties of the chaotic attractor of the system. Such phenomena are present in many dynamical systems, in which a chaotic behavior occurs. The best known of these transitions are: the period-doubling bifurcation cascade, intermittency and crises. The effect of dichotomous Markov noise (DMN) on the properties of systems with chaotic transitions is discussed. DMN is a very simple two-valued stochastic process, with constant transition rates between the two states. In spite of its simplicity, this kind of noise is a very powerful tool to describe various phenomena present in many physical, chemical or biological systems. Many interesting phenomena induced by DMN are known. However, there is no research on the effect of this kind of noise on intermittency or crises. We present the change of the mean laminar phase length and of laminar phase length distribution caused by DMN modulating the parameters of a system with intermittency and the modification of the mean life time on the pre-crisis attractor in the case of a boundary crisis. The results obtained analytically are compared with numerical simulations for several simple dynamical systems.

  6. A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form

    Kingni, Sifeu Takougang; Pham, Viet-Thanh; Jafari, Sajad; Woafo, Paul

    2017-01-01

    A three-dimensional autonomous chaotic system with an infinite number of equilibrium points located on a line and a hyperbola is proposed in this paper. To analyze the dynamical behaviors of the proposed system, mathematical tools such as Routh-Hurwitz criteria, Lyapunov exponents and bifurcation diagram are exploited. For a suitable choice of the parameters, the proposed system can generate periodic oscillations and chaotic attractors of different shapes such as bistable and monostable chaotic attractors. In addition, an electronic circuit is designed and implemented to verify the feasibility of the proposed system. A good qualitative agreement is shown between the numerical simulations and the Orcard-PSpice results. Moreover, the fractional-order form of the proposed system is studied using analog and numerical simulations. It is found that chaos, periodic oscillations and periodic spiking exist in this proposed system with order less than three. Then an electronic circuit is designed for the commensurate fractional order α = 0.98, from which we can observe that a chaotic attractor exists in the fractional-order form of the proposed system. Finally, the problem of drive-response generalized projective synchronization of the fractional-order form of the chaotic proposed autonomous system is considered.

  7. Chaotic phenomena in plasmas

    Kawai, Y.

    1991-08-01

    It has recently been recognized that the research on various aspects of chaotic dynamics grows rapidly as one of some areas in nonlinear science. On the other hands, the plasma has long been called a treasure-house of nonlinear phenomena, so it is easy to imagine that the plasma is abundant in chaotic phenomena. In fact, the research on plasma chaos is going on, such as the research on the stochastic magnetic field and the chaotic orbit in the toroidal helical system, as well as the research in other experiments. To review the present status of the research on plasma chaos and to make clear the basic common physics, a working group was organized in 1990 as a collaboration research of National Institute for Fusion Science. This is the report on its activity in 1990, with a stress on experimental data obtained in basic plasma experiments and RFP, and on the relaxed theories and computer simulations. (author)

  8. Cryptography with chaotic mixing

    Oliveira, Luiz P.L. de; Sobottka, Marcelo

    2008-01-01

    We propose a cryptosystem based on one-dimensional chaotic maps of the form H p (x)=r p -1 0G0r p (x) defined in the interval [0, 10 p ) for a positive integer parameter p, where G(x)=10x(mod10) and r p (x)= p √(x), which is a topological conjugacy between G and the shift map σ on the space Σ of the sequences with 10 symbols. There are three advantages in comparison with the recently proposed cryptosystem based on chaotic logistic maps F μ (x)=μx(1-x) with 3 p is always chaotic for all parameters p, (b) the knowledge of an ergodic measure allows assignments of the alphabetic symbols to equiprobable sites of H p 's domain and (c) for each p, the security of the cryptosystem is manageable against brute force attacks

  9. An Improved Memristive Diode Bridge-Based Band Pass Filter Chaotic Circuit

    Quan Xu

    2017-01-01

    Full Text Available By replacing a series resistor in active band pass filter (BPF with an improved memristive diode bridge emulator, a third-order memristive BPF chaotic circuit is presented. The improved memristive diode bridge emulator without grounded limitation is equivalently achieved by a diode bridge cascaded with only one inductor, whose fingerprints of pinched hysteresis loop are examined by numerical simulations and hardware experiments. The memristive BPF chaotic circuit has only one zero unstable saddle point but causes complex dynamical behaviors including period, chaos, period doubling bifurcation, and coexisting bifurcation modes. Specially, it should be highly significant that two kinds of bifurcation routes are displayed under different initial conditions and the coexistence of three different topological attractors is found in a narrow parameter range. Moreover, hardware circuit using discrete components is fabricated and experimental measurements are performed, upon which the numerical simulations are validated. Notably, the proposed memristive BPF chaotic circuit is only third-order and has simple topological structure.

  10. Analysis of bus width and delay on a fully digital signum nonlinearity chaotic oscillator

    Mansingka, Abhinav S.

    2012-07-29

    This paper introduces the first fully digital implementation of a 3rd order ODE-based chaotic oscillator with signum nonlinearity. A threshold bus width of 12-bits for reliable chaotic behavior is observed, below which the system output becomes periodic. Beyond this threshold, the maximum Lyapunov exponent (MLE) is shown to improve up to a peak value at 16-bits and subsequently decrease with increasing bus width. The MLE is also shown to gradually increase with number of introduced internal delay cycles until a peak value at 14 cycles, after which the system loses chaotic properties. Introduced external delay cycles are shown to rotate the attractors in 3-D phase space. Bus width and delay elements can be independently modulated to optimize the system to suit specifications. The experimental results of the system show low area and high performance on a Xilinx Virtex 4 FPGA with throughput of 13.35 Gbits/s for a 32-bit implementation.

  11. Analysis of bus width and delay on a fully digital signum nonlinearity chaotic oscillator

    Mansingka, Abhinav S.; Radwan, Ahmed G.; Salama, Khaled N.; Zidan, Mohammed A.

    2012-01-01

    This paper introduces the first fully digital implementation of a 3rd order ODE-based chaotic oscillator with signum nonlinearity. A threshold bus width of 12-bits for reliable chaotic behavior is observed, below which the system output becomes periodic. Beyond this threshold, the maximum Lyapunov exponent (MLE) is shown to improve up to a peak value at 16-bits and subsequently decrease with increasing bus width. The MLE is also shown to gradually increase with number of introduced internal delay cycles until a peak value at 14 cycles, after which the system loses chaotic properties. Introduced external delay cycles are shown to rotate the attractors in 3-D phase space. Bus width and delay elements can be independently modulated to optimize the system to suit specifications. The experimental results of the system show low area and high performance on a Xilinx Virtex 4 FPGA with throughput of 13.35 Gbits/s for a 32-bit implementation.

  12. Black hole entropy functions and attractor equations

    Lopes Cardoso, Gabriel; Wit, Bernard de; Mahapatra, Swapna

    2007-01-01

    The entropy and the attractor equations for static extremal black hole solutions follow from a variational principle based on an entropy function. In the general case such an entropy function can be derived from the reduced action evaluated in a near-horizon geometry. BPS black holes constitute special solutions of this variational principle, but they can also be derived directly from a different entropy function based on supersymmetry enhancement at the horizon. Both functions are consistent with electric/magnetic duality and for BPS black holes their corresponding OSV-type integrals give identical results at the semi-classical level. We clarify the relation between the two entropy functions and the corresponding attractor equations for N = 2 supergravity theories with higher-derivative couplings in four space-time dimensions. We discuss how non-holomorphic corrections will modify these entropy functions

  13. D0-branes in black hole attractors

    Gaiotto, Davide; Simons, Aaron; Strominger, Andrew; Yin Xi

    2006-01-01

    Configurations of N probe D0-branes in a Calabi-Yau black hole are studied. A large degeneracy of near-horizon bound states are found which can be described as lowest Landau levels tiling the horizon of the black hole. These states preserve some of the enhanced supersymmetry of the near-horizon AdS 2 x S 2 x CY 3 attractor geometry, but not of the full asymptotically flat solution. Supersymmetric non-abelian configurations are constructed which, via the Myers effect, develop charges associated with higher-dimensional branes wrapping CY 3 cycles. An SU(1,1/2) superconformal quantum mechanics describing D0-branes in the attractor geometry is explicitly constructed

  14. Sneutrino Inflation with $\\alpha$-attractors

    Kallosh, Renata; Roest, Diederik; Wrase, Timm

    2016-11-22

    Sneutrino inflation employs the fermionic partners of the inflaton and stabilizer field as right-handed neutrinos to realize the seesaw mechanism for light neutrino masses. A crucial ingredient in existing constructions for sneutrino (multi-)natural inflation is an unbroken discrete shift symmetry. We demonstrate that a similar construction applies to $\\alpha$-attractor models. In this case the hyperbolic geometry protects the neutrino Yukawa couplings to the inflaton field, and the masses of leptons and Higgs fields, from blowing up when the inflaton is super-Planckian. We find that the predictions for $n_s$ and $r$ for $\\alpha$-attractor cosmological models, compatible with the current cosmological data, are preserved in the presence of the neutrino sector.

  15. Attractor dynamics in local neuronal networks

    Jean-Philippe eThivierge

    2014-03-01

    Full Text Available Patterns of synaptic connectivity in various regions of the brain are characterized by the presence of synaptic motifs, defined as unidirectional and bidirectional synaptic contacts that follow a particular configuration and link together small groups of neurons. Recent computational work proposes that a relay network (two populations communicating via a third, relay population of neurons can generate precise patterns of neural synchronization. Here, we employ two distinct models of neuronal dynamics and show that simulated neural circuits designed in this way are caught in a global attractor of activity that prevents neurons from modulating their response on the basis of incoming stimuli. To circumvent the emergence of a fixed global attractor, we propose a mechanism of selective gain inhibition that promotes flexible responses to external stimuli. We suggest that local neuronal circuits may employ this mechanism to generate precise patterns of neural synchronization whose transient nature delimits the occurrence of a brief stimulus.

  16. The power spectrum of inflationary attractors

    Broy, Benedict J.; Westphal, Alexander; Roest, Diederik

    2014-08-01

    Inflationary attractors predict the spectral index and tensor-to-scalar ratio to take specific values that are consistent with Planck. An example is the universal attractor for models with a generalised non-minimal coupling, leading to Starobinsky inflation. In this letter we demonstrate that it also predicts a specific relation between the amplitude of the power spectrum and the number of e-folds. The length and height of the inflationary plateau are related via the non-minimal coupling: in a wide variety of examples, the observed power normalisation leads to at least 55 flat e-foldings. Prior to this phase, the inflationary predictions vary and can account for the observational indications of power loss at large angular scales.

  17. Contractive function systems, their attractors and metrization

    Banakh, T.; Kubiś, Wieslaw; Novosad, N.; Nowak, M.; Strobin, F.

    2015-01-01

    Roč. 46, č. 2 (2015), s. 1029-1066 ISSN 1230-3429 R&D Projects: GA ČR(CZ) GA14-07880S Institutional support: RVO:67985840 Keywords : fractal * attractor * iterated function system * contracting function system Subject RIV: BA - General Mathematics Impact factor: 0.717, year: 2015 http://www.apcz.pl/czasopisma/index.php/TMNA/article/view/TMNA.2015.076

  18. Internal wave attractors: different scenarios of instability

    Brouzet, Christophe; Ermanyuk, E. V.; Joubaud, Sylvain; Pillet, Grimaud; Dauxois, Thierry

    2017-01-01

    International audience; This paper presents an experimental study of different instability scenarios in a parallelogram-shaped internal wave attractor in a trapezoidal domain filled with a uniformly stratified fluid.Energy is injected into the system via the oscillatory motion of a vertical wall of the trapezoidal domain. Whole-field velocity measurements are performed with the conventional PIV technique. In the linear regime, the total kinetic energyof the fluid system is used to quantify th...

  19. A tangent subsolar merging line

    Crooker, N.U.; Siscoe, G.L.; Toffoletto, F.R.

    1990-01-01

    The authors describe a global magnetospheric model with a single subsolar merging line whose position is determined neither locally by the relative orientations and strengths of the merging fields nor globally by the orientation of a separator line--the governing parameters of most previous models--but by the condition of tangential contact between the external field and the magnetopause. As in previous models, the tilt of the merging line varies with IMF orientation, but here it also depends upon the ratio of Earth's magnetic flux that leaks out of the magnetopause to IMF flux that penetrates in. In the limiting case treated by Alekseyev and Belen'kaya, with no leakage of Earth's field and total IMF penetration, the merging line forms a great circle around a spherical magnetosphere where undeviated IMF lines lie tangent to its surface. This tangent merging line lies perpendicular to the IMF. They extend their work to the case of finite leakage and partial penetration, which distort the IMF into a draped pattern, thus changing the locus of tangency to the sphere. In the special case where the penetrating IMF flux is balanced by an equal amount of Earth flux leakage, the tangent merging line bisects the angle between the IMF and Earth's northward subsolar field. This result is identical to the local merging line model result for merging fields with equal magnitude. Here a global flux balance condition replaces the local equal magnitude condition

  20. Dayside merging and cusp geometry

    Crooker, N.U.

    1979-01-01

    Geometrical considerations are presented to show that dayside magnetic merging when constrained to act only where the fields are antiparallel results in lines of merging that converge at the polar cusps. An important consequence of this geometry is that no accelerated flows are predicted across the dayside magnetopause. Acceleration owing to merging acts in opposition to the magnetosheath flow at the merging point and produces the variably directed, slower-than-magnetosheath flows observed in the entry layer. Another consequence of the merging geometry is that much of the time closed field lines constitute the subsolar region of the magnetopause. The manner in which the polar cap convection patterns predicted by the proposed geometry change as the interplanetary field is rotated through 360 0 provides a unifying description of how the observed single circular vortex and the crescent-shaped double vortex patterns mutually evolve under the influence of a single operating principle

  1. On the short-term predictability of fully digital chaotic oscillators for pseudo-random number generation

    Radwan, Ahmed Gomaa

    2014-06-18

    This paper presents a digital implementation of a 3rd order chaotic system using the Euler approximation. Short-term predictability is studied in relation to system precision, Euler step size and attractor size and optimal parameters for maximum performance are derived. Defective bits from the native chaotic output are neglected and the remaining pass the NIST SP. 800-22 tests without post-processing. The resulting optimized pseudorandom number generator has throughput up to 17.60 Gbits/s for a 64-bit design experimentally verified on a Xilinx Virtex 4 FPGA with logic utilization less than 1.85%.

  2. On the short-term predictability of fully digital chaotic oscillators for pseudo-random number generation

    Radwan, Ahmed Gomaa; Mansingka, Abhinav S.; Salama, Khaled N.; Zidan, Mohammed A.

    2014-01-01

    This paper presents a digital implementation of a 3rd order chaotic system using the Euler approximation. Short-term predictability is studied in relation to system precision, Euler step size and attractor size and optimal parameters for maximum performance are derived. Defective bits from the native chaotic output are neglected and the remaining pass the NIST SP. 800-22 tests without post-processing. The resulting optimized pseudorandom number generator has throughput up to 17.60 Gbits/s for a 64-bit design experimentally verified on a Xilinx Virtex 4 FPGA with logic utilization less than 1.85%.

  3. ESO's Two Observatories Merge

    2005-02-01

    On February 1, 2005, the European Southern Observatory (ESO) has merged its two observatories, La Silla and Paranal, into one. This move will help Europe's prime organisation for astronomy to better manage its many and diverse projects by deploying available resources more efficiently where and when they are needed. The merged observatory will be known as the La Silla Paranal Observatory. Catherine Cesarsky, ESO's Director General, comments the new development: "The merging, which was planned during the past year with the deep involvement of all the staff, has created unified maintenance and engineering (including software, mechanics, electronics and optics) departments across the two sites, further increasing the already very high efficiency of our telescopes. It is my great pleasure to commend the excellent work of Jorge Melnick, former director of the La Silla Observatory, and of Roberto Gilmozzi, the director of Paranal." ESO's headquarters are located in Garching, in the vicinity of Munich (Bavaria, Germany), and this intergovernmental organisation has established itself as a world-leader in astronomy. Created in 1962, ESO is now supported by eleven member states (Belgium, Denmark, Finland, France, Germany, Italy, The Netherlands, Portugal, Sweden, Switzerland, and the United Kingdom). It operates major telescopes on two remote sites, all located in Chile: La Silla, about 600 km north of Santiago and at an altitude of 2400m; Paranal, a 2600m high mountain in the Atacama Desert 120 km south of the coastal city of Antofagasta. Most recently, ESO has started the construction of an observatory at Chajnantor, a 5000m high site, also in the Atacama Desert. La Silla, north of the town of La Serena, has been the bastion of the organization's facilities since 1964. It is the site of two of the most productive 4-m class telescopes in the world, the New Technology Telescope (NTT) - the first major telescope equipped with active optics - and the 3.6-m, which hosts HARPS

  4. Distributed Merge Trees

    Morozov, Dmitriy; Weber, Gunther

    2013-01-08

    Improved simulations and sensors are producing datasets whose increasing complexity exhausts our ability to visualize and comprehend them directly. To cope with this problem, we can detect and extract significant features in the data and use them as the basis for subsequent analysis. Topological methods are valuable in this context because they provide robust and general feature definitions. As the growth of serial computational power has stalled, data analysis is becoming increasingly dependent on massively parallel machines. To satisfy the computational demand created by complex datasets, algorithms need to effectively utilize these computer architectures. The main strength of topological methods, their emphasis on global information, turns into an obstacle during parallelization. We present two approaches to alleviate this problem. We develop a distributed representation of the merge tree that avoids computing the global tree on a single processor and lets us parallelize subsequent queries. To account for the increasing number of cores per processor, we develop a new data structure that lets us take advantage of multiple shared-memory cores to parallelize the work on a single node. Finally, we present experiments that illustrate the strengths of our approach as well as help identify future challenges.

  5. Eternal chaotic inflation

    Linde, A.D.

    1986-05-01

    It is shown that the universe evolution in the chaotic inflation scenario has no end and may have no beginning. According to this scenario, the universe consists of exponentially large number of different mini-universes inside which all possible metastable vacuum states and all possible types of compactification are realized. (author)

  6. Chaotic characteristics of corona discharges in atmospheric air

    Tan Xiangyu; Zhang Qiaogen; Wang Xiuhuan; Sun Fu; Zha Wei; Jia Zhijie

    2008-01-01

    A point-plane electrode system in atmospheric air is established to investigate the mechanism of the corona discharge. By using this system, the current pulses of the corona discharges under the 50 Hz ac voltage are measured using partial discharge (PD) measurement instrument and constitute the point-plane voltage-current (V-I) characteristic equation together with the voltage. Then, this paper constructs the nonlinear circuit model and differential equations of the system in an attempt to give the underlying dynamic mechanism based on the nonlinear V-I characteristics of the point-plane corona discharges. The results show that the chaotic phenomenon is found in the corona circuit by the experimental study and nonlinear dynamic analysis. The basic dynamic characteristics, including the Lyapunov exponent, the existence of the strange attractors, and the equilibrium points, are also found and analyzed in the development process of the corona circuit. Moreover, the time series of the corona current pulses obtained in the experiment is used to demonstrate the chaotic characteristics of the corona current based on the nonlinear dynamic circuit theory and the experimental basis. It is pointed out that the corona phenomenon is not a purely stochastic phenomenon but a short term deterministic chaotic activity

  7. The geometry of chaotic dynamics — a complex network perspective

    Donner, R. V.; Heitzig, J.; Donges, J. F.; Zou, Y.; Marwan, N.; Kurths, J.

    2011-12-01

    Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently ɛ-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using ɛ-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that ɛ-recurrence networks exhibit an important link between dynamical systems and graph theory.

  8. Black-Hole Attractors in N=1 Supergravity

    Andrianopoli, L; Ferrara, Sergio; Trigiante, M; Andrianopoli, Laura; Auria, Riccardo D'; Ferrara, Sergio; Trigiante, Mario

    2007-01-01

    We study the attractor mechanism for N=1 supergravity coupled to vector and chiral multiplets and compute the attractor equations of these theories. These equations may have solutions depending on the choice of the holomorphic symmetric matrix f_{\\Lambda\\Sigma} which appears in the kinetic lagrangian of the vector sector. Models with non trivial electric-magnetic duality group which have or have not attractor behavior are exhibited. For a particular class of models, based on an N=1 reduction of homogeneous special geometries, the attractor equations are related to the theory of pure spinors.

  9. Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bidge-based Jerk circuit

    Njitacke, Z.T.; Kengne, J.; Fotsin, H.B.; Negou, A. Nguomkam; Tchiotsop, D.

    2016-01-01

    In the present paper, a new memristor based oscillator is obtained from the autonomous Jerk circuit [Kengne et al., Nonlinear Dynamics (2016) 83: 751̶765] by substituting the nonlinear element of the original circuit with a first order memristive diode bridge. The model is described by a continuous time four-dimensional autonomous system with smooth nonlinearities. Various nonlinear analysis tools such as phase portraits, time series, bifurcation diagrams, Poincaré section and the spectrum of Lyapunov exponents are exploited to characterize different scenarios to chaos in the novel circuit. It is found that the system experiences period doubling and crisis routes to chaos. One of the major results of this work is the finding of a window in the parameters’ space in which the circuit develops hysteretic behaviors characterized by the coexistence of four different (periodic and chaotic) attractors for the same values of the system parameters. Basins of attractions of various coexisting attractors are plotted showing complex basin boundaries. As far as the authors’ knowledge goes, the novel memristive jerk circuit represents one of the simplest electrical circuits (no analog multiplier chip is involved) capable of four disconnected coexisting attractors reported to date. Both PSpice simulations of the nonlinear dynamics of the oscillator and laboratory experimental measurements are carried out to validate the theoretical analysis.

  10. A solution for two-dimensional mazes with use of chaotic dynamics in a recurrent neural network model.

    Suemitsu, Yoshikazu; Nara, Shigetoshi

    2004-09-01

    Chaotic dynamics introduced into a neural network model is applied to solving two-dimensional mazes, which are ill-posed problems. A moving object moves from the position at t to t + 1 by simply defined motion function calculated from firing patterns of the neural network model at each time step t. We have embedded several prototype attractors that correspond to the simple motion of the object orienting toward several directions in two-dimensional space in our neural network model. Introducing chaotic dynamics into the network gives outputs sampled from intermediate state points between embedded attractors in a state space, and these dynamics enable the object to move in various directions. System parameter switching between a chaotic and an attractor regime in the state space of the neural network enables the object to move to a set target in a two-dimensional maze. Results of computer simulations show that the success rate for this method over 300 trials is higher than that of random walk. To investigate why the proposed method gives better performance, we calculate and discuss statistical data with respect to dynamical structure.

  11. Dynamics of chaotic strings

    Schaefer, Mirko

    2011-01-01

    The main topic of this thesis is the investigation of dynamical properties of coupled Tchebycheff map networks. The results give insights into the chaotic string model and its network generalization from a dynamical point of view. As a first approach, discrete symmetry transformations of the model are studied. These transformations are formulated in a general way in order to be also applicable to similar dynamics on bipartite network structures. The dynamics is studied numerically via Lyapunov measures, spatial correlations, and ergodic properties. It is shown that the zeros of the interaction energy are distinguished only with respect to this specific observable, but not by a more general dynamical principle. The original chaotic string model is defined on a one-dimensional lattice (ring-network) as the underlying network topology. This thesis studies a modification of the model based on the introduction of tunable disorder. The effects of inhomogeneous coupling weights as well as small-world perturbations of the ring-network structure on the interaction energy are discussed. Synchronization properties of the chaotic string model and its network generalization are studied in later chapters of this thesis. The analysis is based on the master stability formalism, which relates the stability of the synchronized state to the spectral properties of the network. Apart from complete synchronization, where the dynamics at all nodes of the network coincide, also two-cluster synchronization on bipartite networks is studied. For both types of synchronization it is shown that depending on the type of coupling the synchronized dynamics can display chaotic as well as periodic or quasi-periodic behaviour. The semi-analytical calculations reveal that the respective synchronized states are often stable for a wide range of coupling values even for the ring-network, although the respective basins of attraction may inhabit only a small fraction of the phase space. To provide

  12. Chaos, strange attractors, and fractal basin boundaries

    Grebogi, C.

    1989-01-01

    Even simple mathematical models of physical systems are often observed to exhibit rather complex time evolution. Upon observation, one often has the feeling that such complex time evolutions could, for most practical purposes, be best characterized by statistical properties rather than by detailed knowledge of the exact process. In such situations, the time evolution is often labeled chaotic or turbulent. The study of chaotic dynamics has recently undergone explosive growth. Motivation for this comes partly from the fact that chaotic dynamics is being found to be of fundamental importance in many branches of science and engineering. Examples illustrating the wide-ranging applications of chaotic dynamics to scientific and engineering problems are the following: fluid dynamics, biology, ecology, meteorology, optics, electronics, mechanical engineerings, physiology, economics, chemistry, accelerator technology, thermonuclear fusion, celestial mechanics, and oceanography. The common element in all of the above topics is that they involve nonlinearity in some way. Indeed chaos is expected to be common whenever nonlinearity plays a role. Since nonlinearity is inherent in so much of science and engineering, an understanding of chaos is essential. Given the varied nature of applications where chaos is important, it is natural that researchers in a broad range of fields have become interested in and have contributed to recent developments in chaos

  13. 3rd School on Attractor Mechanism

    SAM 2007; The Attractor Mechanism: Proceedings of the INFN-Laboratori Nazionali di Frascati School 2007

    2010-01-01

    This book is based upon lectures presented in June 2007 at the INFN-Laboratori Nazionali di Frascati School on Attractor Mechanism, directed by Stefano Bellucci. The symposium included such prestigious lecturers as S. Ferrara, M. Gunaydin, P. Levay, and T. Mohaupt. All lectures were given at a pedagogical, introductory level, which is reflected in the specific "flavor" of this volume. The book also benefits from extensive discussions about, and related reworking of, the various contributions. In addition, this volume contains contributions originating from short presentations of rece

  14. The chaotic dynamical aperture

    Lee, S.Y.; Tepikian, S.

    1985-01-01

    Nonlinear magnetic forces become more important for particles in the modern large accelerators. These nonlinear elements are introduced either intentionally to control beam dynamics or by uncontrollable random errors. Equations of motion in the nonlinear Hamiltonian are usually non-integrable. Because of the nonlinear part of the Hamiltonian, the tune diagram of accelerators is a jungle. Nonlinear magnet multipoles are important in keeping the accelerator operation point in the safe quarter of the hostile jungle of resonant tunes. Indeed, all the modern accelerator design have taken advantages of nonlinear mechanics. On the other hand, the effect of the uncontrollable random multipoles should be evaluated carefully. A powerful method of studying the effect of these nonlinear multipoles is using a particle tracking calculation, where a group of test particles are tracing through these magnetic multipoles in the accelerator hundreds to millions of turns in order to test the dynamical aperture of the machine. These methods are extremely useful in the design of a large accelerator such as SSC, LEP, HERA and RHIC. These calculations unfortunately take tremendous amount of computing time. In this paper, we try to apply the existing method in the nonlinear dynamics to study the possible alternative solution. When the Hamiltonian motion becomes chaotic, the tune of the machine becomes undefined. The aperture related to the chaotic orbit can be identified as chaotic dynamical aperture. We review the method of determining chaotic orbit and apply the method to nonlinear problems in accelerator physics. We then discuss the scaling properties and effect of random sextupoles

  15. Chaotic solar oscillations

    Blacher, S; Perdang, J [Institut d' Astrophysique, B-4200 Cointe-Ougree (Belgium)

    1981-09-01

    A numerical experiment on Hamiltonian oscillations demonstrates the existence of chaotic motions which satisfy the property of phase coherence. It is observed that the low-frequency end of the power spectrum of such motions is remarkably similar in structure to the low-frequency SCLERA spectra. Since the smallness of the observed solar amplitudes is not a sufficient mathematical ground for inefficiency of non-linear effects the possibility of chaos among solar oscillations cannot be discarded a priori.

  16. Triadic split-merge sampler

    van Rossum, Anne C.; Lin, Hai Xiang; Dubbeldam, Johan; van der Herik, H. Jaap

    2018-04-01

    In machine vision typical heuristic methods to extract parameterized objects out of raw data points are the Hough transform and RANSAC. Bayesian models carry the promise to optimally extract such parameterized objects given a correct definition of the model and the type of noise at hand. A category of solvers for Bayesian models are Markov chain Monte Carlo methods. Naive implementations of MCMC methods suffer from slow convergence in machine vision due to the complexity of the parameter space. Towards this blocked Gibbs and split-merge samplers have been developed that assign multiple data points to clusters at once. In this paper we introduce a new split-merge sampler, the triadic split-merge sampler, that perform steps between two and three randomly chosen clusters. This has two advantages. First, it reduces the asymmetry between the split and merge steps. Second, it is able to propose a new cluster that is composed out of data points from two different clusters. Both advantages speed up convergence which we demonstrate on a line extraction problem. We show that the triadic split-merge sampler outperforms the conventional split-merge sampler. Although this new MCMC sampler is demonstrated in this machine vision context, its application extend to the very general domain of statistical inference.

  17. Digital chaotic sequence generator based on coupled chaotic systems

    Shu-Bo, Liu; Jing, Sun; Jin-Shuo, Liu; Zheng-Quan, Xu

    2009-01-01

    Chaotic systems perform well as a new rich source of cryptography and pseudo-random coding. Unfortunately their digital dynamical properties would degrade due to the finite computing precision. Proposed in this paper is a modified digital chaotic sequence generator based on chaotic logistic systems with a coupling structure where one chaotic subsystem generates perturbation signals to disturb the control parameter of the other one. The numerical simulations show that the length of chaotic orbits, the output distribution of chaotic system, and the security of chaotic sequences have been greatly improved. Moreover the chaotic sequence period can be extended at least by one order of magnitude longer than that of the uncoupled logistic system and the difficulty in decrypting increases 2 128 *2 128 times indicating that the dynamical degradation of digital chaos is effectively improved. A field programmable gate array (FPGA) implementation of an algorithm is given and the corresponding experiment shows that the output speed of the generated chaotic sequences can reach 571.4 Mbps indicating that the designed generator can be applied to the real-time video image encryption. (general)

  18. Existence of global attractor for the Trojan Y Chromosome model

    Xiaopeng Zhao

    2012-04-01

    Full Text Available This paper is concerned with the long time behavior of solution for the equation derived by the Trojan Y Chromosome (TYC model with spatial spread. Based on the regularity estimates for the semigroups and the classical existence theorem of global attractors, we prove that this equations possesses a global attractor in $H^k(\\Omega^4$ $(k\\geq 0$ space.

  19. Attractors for a class of doubly nonlinear parabolic systems

    Hamid El Ouardi

    2006-03-01

    Full Text Available In this paper, we establish the existence and boundedness of solutions of a doubly nonlinear parabolic system. We also obtain the existence of a global attractor and the regularity property for this attractor in $\\left[ L^{\\infty }(\\Omega \\right] ^{2}$ and ${\\prod_{i=1}^{2}}{B_{\\infty }^{1+\\sigma_{i},p_{i}}( \\Omega } $.

  20. Existence and attractors of solutions for nonlinear parabolic systems

    Hamid El Ouardi

    2001-01-01

    Full Text Available We prove existence and asymptotic behaviour results for weak solutions of a mixed problem (S. We also obtain the existence of the global attractor and the regularity for this attractor in $\\left[H^{2}(\\Omega \\right] ^{2}$ and we derive estimates of its Haussdorf and fractal dimensions.

  1. Synchronization in Coupled Oscillators with Two Coexisting Attractors

    Han-Han, Zhu; Jun-Zhong, Yang

    2008-01-01

    Dynamics in coupled Duffing oscillators with two coexisting symmetrical attractors is investigated. For a pair of Duffing oscillators coupled linearly, the transition to the synchronization generally consists of two steps: Firstly, the two oscillators have to jump onto a same attractor, then they reach synchronization similarly to coupled monostable oscillators. The transition scenarios to the synchronization observed are strongly dependent on initial conditions. (general)

  2. Intermittent and sustained periodic windows in networked chaotic Rössler oscillators

    He, Zhiwei; Sun, Yong; Zhan, Meng

    2013-01-01

    Route to chaos (or periodicity) in dynamical systems is one of fundamental problems. Here, dynamical behaviors of coupled chaotic Rössler oscillators on complex networks are investigated and two different types of periodic windows with the variation of coupling strength are found. Under a moderate coupling, the periodic window is intermittent, and the attractors within the window extremely sensitively depend on the initial conditions, coupling parameter, and topology of the network. Therefore, after adding or removing one edge of network, the periodic attractor can be destroyed and substituted by a chaotic one, or vice versa. In contrast, under an extremely weak coupling, another type of periodic window appears, which insensitively depends on the initial conditions, coupling parameter, and network. It is sustained and unchanged for different types of network structure. It is also found that the phase differences of the oscillators are almost discrete and randomly distributed except that directly linked oscillators more likely have different phases. These dynamical behaviors have also been generally observed in other networked chaotic oscillators

  3. Ulam's scheme revisited digital modeling of chaotic attractors via micro-perturbations

    Domokos, Gabor K

    2002-01-01

    We consider discretizations $f_N$ of expanding maps $f:I \\to I$ in the strict sense: i.e. we assume that the only information available on the map is a finite set of integers. Using this definition for computability, we show that by adding a random perturbation of order $1/N$, the invariant measure corresponding to $f$ can be approximated and we can also give estimates of the error term. We prove that the randomized discrete scheme is equivalent to Ulam's scheme applied to the polygonal approximation of $f$, thus providing a new interpretation of Ulam's scheme. We also compare the efficiency of the randomized iterative scheme to the direct solution of the $N \\times N$ linear system.

  4. Construction of the Lyapunov Spectrum in a Chaotic System Displaying Phase Synchronization

    Carlo, Leonardo De, E-mail: neoleodeo@gmail.com [Gran Sasso Science Institute (GSSI) (Italy); Gentile, Guido, E-mail: gentile@mat.uniroma3.it; Giuliani, Alessandro, E-mail: giuliani@mat.uniroma3.it [Università degli Studi Roma Tre, Dipartimento di Matematica e Fisica (Italy)

    2016-06-15

    We consider a three-dimensional chaotic system consisting of the suspension of Arnold’s cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relative phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.

  5. Chaotic behavior of light-assisted physical aging in arsenoselenide glasses

    Shpotyuk, O., E-mail: shpotyuk@novas.lviv.ua [Lviv Scientific Research Institute of Materials of SRC “Carat,” 202, Stryjska Str., Lviv 79031 (Ukraine); Institute of Physics of Jan Dlugosz University, 13/15, al. Armii Krajowej, Czestochowa 42201 (Poland); Balitska, V. [Lviv Scientific Research Institute of Materials of SRC “Carat,” 202, Stryjska Str., Lviv 79031 (Ukraine); Lviv State University of Vital Activity Safety, 35, Kleparivska str., Lviv 79007 (Ukraine); Kozdras, A. [Opole University of Technology, 75, Ozimska str., Opole 45370 (Poland); Hacinliyan, A. S. [Department of Physics, Yeditepe University, Atasehir 34755, Istanbul (Turkey); Department of Physics, Bogazici University, Bebek, Istanbul (Turkey); Department of Information Systems and Technologies, Yeditepe University, Atasehir 34755, Istanbul (Turkey); Skarlatos, Y. [Department of Physics, Bogazici University, Bebek, Istanbul (Turkey); Kusbeyzi Aybar, I. [Department of Computer Education and Instructional Technology, Yeditepe University, Atasehir 34755, Istanbul (Turkey); Aybar, O. O. [Department of Mathematics, Faculty of Science and Letters, Piri Reis University, Tuzla 34940, Istanbul (Turkey)

    2014-12-15

    The theory of strange attractors is shown to be adequately applicable for analyzing the kinetics of light-assisted physical aging revealed in structural relaxation of Se-rich As-Se glasses below glass transition. Kinetics of enthalpy losses is used to determine the phase space reconstruction parameters. Observed chaotic behaviour (involving chaos and fractal consideration such as detrended fluctuation analysis, attractor identification using phase space representation, delay coordinates, mutual information, false nearest neighbours, etc.) reconstructed via the TISEAN program package is treated within a microstructure model describing multistage aging behaviour in arsenoselenide glasses. This simulation testifies that photoexposure acts as an initiating factor only at the beginning stage of physical aging, thus facilitating further atomic shrinkage of a glassy backbone.

  6. Chaotic behavior of light-assisted physical aging in arsenoselenide glasses

    Shpotyuk, O.; Balitska, V.; Kozdras, A.; Hacinliyan, A. S.; Skarlatos, Y.; Kusbeyzi Aybar, I.; Aybar, O. O.

    2014-01-01

    The theory of strange attractors is shown to be adequately applicable for analyzing the kinetics of light-assisted physical aging revealed in structural relaxation of Se-rich As-Se glasses below glass transition. Kinetics of enthalpy losses is used to determine the phase space reconstruction parameters. Observed chaotic behaviour (involving chaos and fractal consideration such as detrended fluctuation analysis, attractor identification using phase space representation, delay coordinates, mutual information, false nearest neighbours, etc.) reconstructed via the TISEAN program package is treated within a microstructure model describing multistage aging behaviour in arsenoselenide glasses. This simulation testifies that photoexposure acts as an initiating factor only at the beginning stage of physical aging, thus facilitating further atomic shrinkage of a glassy backbone

  7. Dynamic control of chaotic resonators

    Di Falco, A.; Bruck, R.; Liu, C.; Muskens, O.; Fratalocchi, Andrea

    2016-01-01

    We report on the all-optical control of chaotic optical resonators based on silicon on insulator (SOI) platform. We show that simple non-chaotic cavities can be tuned to exhibit chaotic behavior via intense optical pump- ing, inducing a local change of refractive index. To this extent we have fabricated a number of devices and demonstrated experimentally and theoretically that chaos can be triggered on demand on an optical chip. © 2016 SPIE.

  8. Dynamic control of chaotic resonators

    Di Falco, A.

    2016-02-16

    We report on the all-optical control of chaotic optical resonators based on silicon on insulator (SOI) platform. We show that simple non-chaotic cavities can be tuned to exhibit chaotic behavior via intense optical pump- ing, inducing a local change of refractive index. To this extent we have fabricated a number of devices and demonstrated experimentally and theoretically that chaos can be triggered on demand on an optical chip. © 2016 SPIE.

  9. Google matrix, dynamical attractors, and Ulam networks.

    Shepelyansky, D L; Zhirov, O V

    2010-03-01

    We study the properties of the Google matrix generated by a coarse-grained Perron-Frobenius operator of the Chirikov typical map with dissipation. The finite-size matrix approximant of this operator is constructed by the Ulam method. This method applied to the simple dynamical model generates directed Ulam networks with approximate scale-free scaling and characteristics being in certain features similar to those of the world wide web with approximate scale-free degree distributions as well as two characteristics similar to the web: a power-law decay in PageRank that mirrors the decay of PageRank on the world wide web and a sensitivity to the value alpha in PageRank. The simple dynamical attractors play here the role of popular websites with a strong concentration of PageRank. A variation in the Google parameter alpha or other parameters of the dynamical map can drive the PageRank of the Google matrix to a delocalized phase with a strange attractor where the Google search becomes inefficient.

  10. Attractor mechanism as a distillation procedure

    Levay, Peter; Szalay, Szilard

    2010-01-01

    In a recent paper it was shown that for double extremal static spherical symmetric BPS black hole solutions in the STU model the well-known process of moduli stabilization at the horizon can be recast in a form of a distillation procedure of a three-qubit entangled state of a Greenberger-Horne-Zeilinger type. By studying the full flow in moduli space in this paper we investigate this distillation procedure in more detail. We introduce a three-qubit state with amplitudes depending on the conserved charges, the warp factor, and the moduli. We show that for the recently discovered non-BPS solutions it is possible to see how the distillation procedure unfolds itself as we approach the horizon. For the non-BPS seed solutions at the asymptotically Minkowski region we are starting with a three-qubit state having seven nonequal nonvanishing amplitudes and finally at the horizon we get a Greenberger-Horne-Zeilinger state with merely four nonvanishing ones with equal magnitudes. The magnitude of the surviving nonvanishing amplitudes is proportional to the macroscopic black hole entropy. A systematic study of such attractor states shows that their properties reflect the structure of the fake superpotential. We also demonstrate that when starting with the very special values for the moduli corresponding to flat directions the uniform structure at the horizon deteriorates due to errors generalizing the usual bit flips acting on the qubits of the attractor states.

  11. Using Chaotic System in Encryption

    Findik, Oğuz; Kahramanli, Şirzat

    In this paper chaotic systems and RSA encryption algorithm are combined in order to develop an encryption algorithm which accomplishes the modern standards. E.Lorenz's weather forecast' equations which are used to simulate non-linear systems are utilized to create chaotic map. This equation can be used to generate random numbers. In order to achieve up-to-date standards and use online and offline status, a new encryption technique that combines chaotic systems and RSA encryption algorithm has been developed. The combination of RSA algorithm and chaotic systems makes encryption system.

  12. A simple chaotic delay differential equation

    Sprott, J.C.

    2007-01-01

    The simplest chaotic delay differential equation with a sinusoidal nonlinearity is described, including the route to chaos, Lyapunov exponent spectrum, and chaotic diffusion. It is prototypical of many other high-dimensional chaotic systems

  13. Predicting chaotic time series

    Farmer, J.D.; Sidorowich, J.J.

    1987-01-01

    We present a forecasting technique for chaotic data. After embedding a time series in a state space using delay coordinates, we ''learn'' the induced nonlinear mapping using local approximation. This allows us to make short-term predictions of the future behavior of a time series, using information based only on past values. We present an error estimate for this technique, and demonstrate its effectiveness by applying it to several examples, including data from the Mackey-Glass delay differential equation, Rayleigh-Benard convection, and Taylor-Couette flow

  14. Counting and classifying attractors in high dimensional dynamical systems.

    Bagley, R J; Glass, L

    1996-12-07

    Randomly connected Boolean networks have been used as mathematical models of neural, genetic, and immune systems. A key quantity of such networks is the number of basins of attraction in the state space. The number of basins of attraction changes as a function of the size of the network, its connectivity and its transition rules. In discrete networks, a simple count of the number of attractors does not reveal the combinatorial structure of the attractors. These points are illustrated in a reexamination of dynamics in a class of random Boolean networks considered previously by Kauffman. We also consider comparisons between dynamics in discrete networks and continuous analogues. A continuous analogue of a discrete network may have a different number of attractors for many different reasons. Some attractors in discrete networks may be associated with unstable dynamics, and several different attractors in a discrete network may be associated with a single attractor in the continuous case. Special problems in determining attractors in continuous systems arise when there is aperiodic dynamics associated with quasiperiodicity of deterministic chaos.

  15. Stochastic and Chaotic Relaxation Oscillations

    Grasman, J.; Roerdink, J.B.T.M.

    1988-01-01

    For relaxation oscillators stochastic and chaotic dynamics are investigated. The effect of random perturbations upon the period is computed. For an extended system with additional state variables chaotic behavior can be expected. As an example, the Van der Pol oscillator is changed into a

  16. Analysis, Adaptive Control and Anti-Synchronization of a Six-Term Novel Jerk Chaotic System with two Exponential Nonlinearities and its Circuit Simulation

    S. Vaidyanathan

    2014-11-01

    Full Text Available This research work proposes a six-term novel 3-D jerk chaotic system with two exponential nonlinearities. This work also analyses system’s fundamental properties such as dissipativity, equilibria, Lyapunov exponents and Kaplan-Yorke dimension. The phase portraits of the jerk chaotic system simulated using MATLAB, depict the strange chaotic attractor of the system. For the parameter values and initial conditions chosen in this work, the Lyapunov exponents of the novel jerk chaotic system are obtained as L1 = 0.24519, L2 = 0 and L3 = −0.84571. Also, the Kaplan-Yorke dimension of the novel jerk chaotic system is obtained as DKY = 2.2899. Next, an adaptive backstepping controller is designed to stabilize the novel jerk chaotic system having two unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve global chaos anti-synchronization of two identical novel jerk chaotic systems with two unknown system parameters. Finally, an electronic circuit realization of the novel jerk chaotic system is presented using SPICE to confirm the feasibility of the theoretical model.

  17. Dynamics of neural networks with continuous attractors

    Fung, C. C. Alan; Wong, K. Y. Michael; Wu, Si

    2008-10-01

    We investigate the dynamics of continuous attractor neural networks (CANNs). Due to the translational invariance of their neuronal interactions, CANNs can hold a continuous family of stationary states. We systematically explore how their neutral stability facilitates the tracking performance of a CANN, which is believed to have wide applications in brain functions. We develop a perturbative approach that utilizes the dominant movement of the network stationary states in the state space. We quantify the distortions of the bump shape during tracking, and study their effects on the tracking performance. Results are obtained on the maximum speed for a moving stimulus to be trackable, and the reaction time to catch up an abrupt change in stimulus.

  18. Black hole microstates and attractor without supersymmetry

    Dabholkar, Atish; Trivedi, Sandip P.; Sen, Ashoke

    2007-01-01

    Due to the attractor mechanism, the entropy of an extremal black hole does not vary continuously as we vary the asymptotic values of various moduli fields. Using this fact we argue that the entropy of an extremal black hole in string theory, calculated for a range of values of the asymptotic moduli for which the microscopic theory is strongly coupled, should match the statistical entropy of the same system calculated for a range of values of the asymptotic moduli for which the microscopic theory is weakly coupled. This argument does not rely on supersymmetry and applies equally well to nonsupersymmetric extremal black holes. We discuss several examples which support this argument and also several caveats which could invalidate this argument

  19. Fibre inflation and α-attractors

    Kallosh, Renata; Linde, Andrei [Stanford Univ., Stanford, CA (United States). Stanford Inst. for Theoretical Physics and Dept. of Physics; Leiden Univ. (Netherlands). Lorentz Inst. for Theoretical Physics; Roest, Diederik [Groningen Univ. (Netherlands). Van Swinderen Inst. for Particle Physics and Gravity; Westphal, Alexander [DESY, Hamburg (Germany). Theory Group; Yamada, Yusuke [Stanford Univ., Stanford, CA (United States). Stanford Inst. for Theoretical Physics and Dept. of Physics

    2017-07-15

    Fibre inflation is a specific string theory construction based on the Large Volume Scenario that produces an inflationary plateau. We outline its relation to α-attractor models for inflation, with the cosmological sector originating from certain string theory corrections leading to α=2 and α=1/2. Above a certain field range, the steepening effect of higher-order corrections leads first to the breakdown of single-field slow-roll and after that to the onset of 2-field dynamics: the overall volume of the extra dimensions starts to participate in the effective dynamics. Finally, we propose effective supergravity models of fibre inflation based on an D3 uplift term with a nilpotent superfield. Specific moduli dependent D3 induced geometries lead to cosmological fibre models but have in addition a de Sitter minimum exit. These supergravity models motivated by fibre inflation are relatively simple, stabilize the axions and disentangle the Hubble parameter from supersymmetry breaking.

  20. Attractor cosmology from nonminimally coupled gravity

    Odintsov, S. D.; Oikonomou, V. K.

    2018-03-01

    By using a bottom-up reconstruction technique for nonminimally coupled scalar-tensor theories, we realize the Einstein frame attractor cosmologies in the Ω (ϕ )-Jordan frame. For our approach, what is needed for the reconstruction method to work is the functional form of the nonminimal coupling Ω (ϕ ) and of the scalar-to-tensor ratio, and also the assumption of the slow-roll inflation in the Ω (ϕ )-Jordan frame. By appropriately choosing the scalar-to-tensor ratio, we demonstrate that the observational indices of the attractor cosmologies can be realized directly in the Ω (ϕ )-Jordan frame. We investigate the special conditions that are required to hold true in for this realization to occur, and we provide the analytic form of the potential in the Ω (ϕ )-Jordan frame. Also, by performing a conformal transformation, we find the corresponding Einstein frame canonical scalar-tensor theory, and we calculate in detail the corresponding observational indices. The result indicates that although the spectral index of the primordial curvature perturbations is the same in the Jordan and Einstein frames, at leading order in the e -foldings number, the scalar-to-tensor ratio differs. We discuss the possible reasons behind this discrepancy, and we argue that the difference is due to some approximation we performed to the functional form of the potential in the Einstein frame, in order to obtain analytical results, and also due to the difference in the definition of the e -foldings number in the two frames, which is also pointed out in the related literature. Finally, we find the F (R ) gravity corresponding to the Einstein frame canonical scalar-tensor theory.

  1. Chaos and multi-scroll attractors in RCL-shunted junction coupled Jerk circuit connected by memristor.

    Jun Ma

    Full Text Available In this paper, a new four-variable dynamical system is proposed to set chaotic circuit composed of memristor and Josephson junction, and the dependence of chaotic behaviors on nonlinearity is investigated. A magnetic flux-controlled memristor is used to couple with the RCL-shunted junction circuit, and the dynamical behaviors can be modulated by changing the coupling intensity between the memristor and the RCL-shunted junction. Bifurcation diagram and Lyapunov exponent are calculated to confirm the emergence of chaos in the improved dynamical system. The outputs and dynamical behaviors can be controlled by the initial setting and external stimulus as well. As a result, chaos can be suppressed and spiking occurs in the sampled outputs under negative feedback, while applying positive feedback type via memristor can be effective to trigger chaos. Furthermore, it is found that the number of multi-attractors in the Jerk circuit can be modulated when memristor coupling is applied on the circuit. These results indicate that memristor coupling can be effective to control chaotic circuits and it is also useful to reproduce dynamical behaviors for neuronal activities.

  2. Chaos and multi-scroll attractors in RCL-shunted junction coupled Jerk circuit connected by memristor

    Zhou, Ping; Ahmad, Bashir; Ren, Guodong; Wang, Chunni

    2018-01-01

    In this paper, a new four-variable dynamical system is proposed to set chaotic circuit composed of memristor and Josephson junction, and the dependence of chaotic behaviors on nonlinearity is investigated. A magnetic flux-controlled memristor is used to couple with the RCL-shunted junction circuit, and the dynamical behaviors can be modulated by changing the coupling intensity between the memristor and the RCL-shunted junction. Bifurcation diagram and Lyapunov exponent are calculated to confirm the emergence of chaos in the improved dynamical system. The outputs and dynamical behaviors can be controlled by the initial setting and external stimulus as well. As a result, chaos can be suppressed and spiking occurs in the sampled outputs under negative feedback, while applying positive feedback type via memristor can be effective to trigger chaos. Furthermore, it is found that the number of multi-attractors in the Jerk circuit can be modulated when memristor coupling is applied on the circuit. These results indicate that memristor coupling can be effective to control chaotic circuits and it is also useful to reproduce dynamical behaviors for neuronal activities. PMID:29342178

  3. Chaos and multi-scroll attractors in RCL-shunted junction coupled Jerk circuit connected by memristor.

    Ma, Jun; Zhou, Ping; Ahmad, Bashir; Ren, Guodong; Wang, Chunni

    2018-01-01

    In this paper, a new four-variable dynamical system is proposed to set chaotic circuit composed of memristor and Josephson junction, and the dependence of chaotic behaviors on nonlinearity is investigated. A magnetic flux-controlled memristor is used to couple with the RCL-shunted junction circuit, and the dynamical behaviors can be modulated by changing the coupling intensity between the memristor and the RCL-shunted junction. Bifurcation diagram and Lyapunov exponent are calculated to confirm the emergence of chaos in the improved dynamical system. The outputs and dynamical behaviors can be controlled by the initial setting and external stimulus as well. As a result, chaos can be suppressed and spiking occurs in the sampled outputs under negative feedback, while applying positive feedback type via memristor can be effective to trigger chaos. Furthermore, it is found that the number of multi-attractors in the Jerk circuit can be modulated when memristor coupling is applied on the circuit. These results indicate that memristor coupling can be effective to control chaotic circuits and it is also useful to reproduce dynamical behaviors for neuronal activities.

  4. Correlation order, merging and diversification

    Dhaene, J.; Denuit, M.; Vanduffel, S.

    2009-01-01

    We investigate the influence of the dependence between random losses on the shortfall and on the diversification benefit that arises from merging these losses. We prove that increasing the dependence between losses, expressed in terms of correlation order, has an increasing effect on the shortfall,

  5. Characterization of chaotic dynamics in the human menstrual cycle

    Derry, Gregory; Derry, Paula

    2010-03-01

    The human menstrual cycle exhibits much unexplained variability, which is typically dismissed as random variation. Given the many delayed nonlinear feedbacks in the reproductive endocrine system, however, the menstrual cycle might well be a nonlinear dynamical system in a chaotic trajectory, and that this instead accounts for the observed variability. Here, we test this hypothesis by performing a time series analysis on data for 7438 menstrual cycles from 38 women in the 20-40 year age range, using the database maintained by the Tremin Research Program on Women's Health. Using phase space reconstruction techniques with a maximum embedding dimension of 6, we find appropriate scaling behavior in the correlation sums for this data, indicating low dimensional deterministic dynamics. A correlation dimension of 2.6 is measured in this scaling regime, and this result is confirmed by recalculation using the Takens estimator. These results may be interpreted as offering an approximation to the fractal dimension of a strange attractor governing the chaotic dynamics of the menstrual cycle.

  6. Numerical Simulation Bidirectional Chaotic Synchronization of Spiegel-Moore Circuit and Its Application for Secure Communication

    Sanjaya, W. S. M.; Anggraeni, D.; Denya, R.; Ismail, N.

    2017-03-01

    Spiegel-Moore is a dynamical chaotic system which shows irregular variability in the luminosity of stars. In this paper present the performed the design and numerical simulation of the synchronization Spiegel-Moore circuit and applied to security system for communication. The initial study in this paper is to analyze the eigenvalue structures, various attractors, Bifurcation diagram, and Lyapunov exponent analysis. We have studied the dynamic behavior of the system in the case of the bidirectional coupling via a linear resistor. Both experimental and simulation results have shown that chaotic synchronization is possible. Finally, the effectiveness of the bidirectional coupling scheme between two identical Spiegel-Moore circuits in a secure communication system is presented in details. Integration of theoretical electronic circuit, the numerical simulation by using MATLAB®, as well as the implementation of circuit simulations by using Multisim® has been performed in this study.

  7. A financial market model with two discontinuities: Bifurcation structures in the chaotic domain

    Panchuk, Anastasiia; Sushko, Iryna; Westerhoff, Frank

    2018-05-01

    We continue the investigation of a one-dimensional piecewise linear map with two discontinuity points. Such a map may arise from a simple asset-pricing model with heterogeneous speculators, which can help us to explain the intricate bull and bear behavior of financial markets. Our focus is on bifurcation structures observed in the chaotic domain of the map's parameter space, which is associated with robust multiband chaotic attractors. Such structures, related to the map with two discontinuities, have been not studied before. We show that besides the standard bandcount adding and bandcount incrementing bifurcation structures, associated with two partitions, there exist peculiar bandcount adding and bandcount incrementing structures involving all three partitions. Moreover, the map's three partitions may generate intriguing bistability phenomena.

  8. Circuit Implementation, Synchronization of Multistability, and Image Encryption of a Four-Wing Memristive Chaotic System

    Guangya Peng

    2018-01-01

    Full Text Available The four-wing memristive chaotic system used in synchronization is applied to secure communication which can increase the difficulty of deciphering effectively and enhance the security of information. In this paper, a novel four-wing memristive chaotic system with an active cubic flux-controlled memristor is proposed based on a Lorenz-like circuit. Dynamical behaviors of the memristive system are illustrated in terms of Lyapunov exponents, bifurcation diagrams, coexistence Poincaré maps, coexistence phase diagrams, and attraction basins. Besides, the modular equivalent circuit of four-wing memristive system is designed and the corresponding results are observed to verify its accuracy and rationality. A nonlinear synchronization controller with exponential function is devised to realize synchronization of the coexistence of multiple attractors, and the synchronization control scheme is applied to image encryption to improve secret key space. More interestingly, considering different influence of multistability on encryption, the appropriate key is achieved to enhance the antideciphering ability.

  9. Chaotic dynamics in optimal monetary policy

    Gomes, O.; Mendes, V. M.; Mendes, D. A.; Sousa Ramos, J.

    2007-05-01

    There is by now a large consensus in modern monetary policy. This consensus has been built upon a dynamic general equilibrium model of optimal monetary policy as developed by, e.g., Goodfriend and King [ NBER Macroeconomics Annual 1997 edited by B. Bernanke and J. Rotemberg (Cambridge, Mass.: MIT Press, 1997), pp. 231 282], Clarida et al. [J. Econ. Lit. 37, 1661 (1999)], Svensson [J. Mon. Econ. 43, 607 (1999)] and Woodford [ Interest and Prices: Foundations of a Theory of Monetary Policy (Princeton, New Jersey, Princeton University Press, 2003)]. In this paper we extend the standard optimal monetary policy model by introducing nonlinearity into the Phillips curve. Under the specific form of nonlinearity proposed in our paper (which allows for convexity and concavity and secures closed form solutions), we show that the introduction of a nonlinear Phillips curve into the structure of the standard model in a discrete time and deterministic framework produces radical changes to the major conclusions regarding stability and the efficiency of monetary policy. We emphasize the following main results: (i) instead of a unique fixed point we end up with multiple equilibria; (ii) instead of saddle-path stability, for different sets of parameter values we may have saddle stability, totally unstable equilibria and chaotic attractors; (iii) for certain degrees of convexity and/or concavity of the Phillips curve, where endogenous fluctuations arise, one is able to encounter various results that seem intuitively correct. Firstly, when the Central Bank pays attention essentially to inflation targeting, the inflation rate has a lower mean and is less volatile; secondly, when the degree of price stickiness is high, the inflation rate displays a larger mean and higher volatility (but this is sensitive to the values given to the parameters of the model); and thirdly, the higher the target value of the output gap chosen by the Central Bank, the higher is the inflation rate and its

  10. Hyperbolic Plykin attractor can exist in neuron models

    Belykh, V.; Belykh, I.; Mosekilde, Erik

    2005-01-01

    Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study...... of the neuron model, we derive a flow-defined Poincare map giving ail accurate account of the system's dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic...... holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary...

  11. Multi-wing hyperchaotic attractors from coupled Lorenz systems

    Grassi, Giuseppe; Severance, Frank L.; Miller, Damon A.

    2009-01-01

    This paper illustrates an approach to generate multi-wing attractors in coupled Lorenz systems. In particular, novel four-wing (eight-wing) hyperchaotic attractors are generated by coupling two (three) identical Lorenz systems. The paper shows that the equilibria of the proposed systems have certain symmetries with respect to specific coordinate planes and the eigenvalues of the associated Jacobian matrices exhibit the property of similarity. In analogy with the original Lorenz system, where the two-wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four-wings (eight-wings) of these attractors are located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.

  12. Attractor neural networks with resource-efficient synaptic connectivity

    Pehlevan, Cengiz; Sengupta, Anirvan

    Memories are thought to be stored in the attractor states of recurrent neural networks. Here we explore how resource constraints interplay with memory storage function to shape synaptic connectivity of attractor networks. We propose that given a set of memories, in the form of population activity patterns, the neural circuit choses a synaptic connectivity configuration that minimizes a resource usage cost. We argue that the total synaptic weight (l1-norm) in the network measures the resource cost because synaptic weight is correlated with synaptic volume, which is a limited resource, and is proportional to neurotransmitter release and post-synaptic current, both of which cost energy. Using numerical simulations and replica theory, we characterize optimal connectivity profiles in resource-efficient attractor networks. Our theory explains several experimental observations on cortical connectivity profiles, 1) connectivity is sparse, because synapses are costly, 2) bidirectional connections are overrepresented and 3) are stronger, because attractor states need strong recurrence.

  13. Correlation control theory of chaotic laser systems

    Li Fuli.

    1986-04-01

    A novel control theory of chaotic systems is studied. The correlation functions are calculated and used as feedback signals of the chaotic lasers. Computer experiments have shown that in this way the chaotic systems can be controlled to have time-independent output when the external control parameters are in chaotic domain. (author)

  14. Attractor of reaction-diffusion equations in Banach spaces

    José Valero

    2001-04-01

    Full Text Available In this paper we prove first some abstract theorems on existence of global attractors for differential inclusions generated by w-dissipative operators. Then these results are applied to reaction-diffusion equations in which the Babach space Lp is used as phase space. Finally, new results concerning the fractal dimension of the global attractor in the space L2 are obtained.

  15. Revisiting non-Gaussianity from non-attractor inflation models

    Cai, Yi-Fu; Chen, Xingang; Namjoo, Mohammad Hossein; Sasaki, Misao; Wang, Dong-Gang; Wang, Ziwei

    2018-05-01

    Non-attractor inflation is known as the only single field inflationary scenario that can violate non-Gaussianity consistency relation with the Bunch-Davies vacuum state and generate large local non-Gaussianity. However, it is also known that the non-attractor inflation by itself is incomplete and should be followed by a phase of slow-roll attractor. Moreover, there is a transition process between these two phases. In the past literature, this transition was approximated as instant and the evolution of non-Gaussianity in this phase was not fully studied. In this paper, we follow the detailed evolution of the non-Gaussianity through the transition phase into the slow-roll attractor phase, considering different types of transition. We find that the transition process has important effect on the size of the local non-Gaussianity. We first compute the net contribution of the non-Gaussianities at the end of inflation in canonical non-attractor models. If the curvature perturbations keep evolving during the transition—such as in the case of smooth transition or some sharp transition scenarios—the Script O(1) local non-Gaussianity generated in the non-attractor phase can be completely erased by the subsequent evolution, although the consistency relation remains violated. In extremal cases of sharp transition where the super-horizon modes freeze immediately right after the end of the non-attractor phase, the original non-attractor result can be recovered. We also study models with non-canonical kinetic terms, and find that the transition can typically contribute a suppression factor in the squeezed bispectrum, but the final local non-Gaussianity can still be made parametrically large.

  16. Internal Waves and Wave Attractors in Enceladus' Subsurface Ocean

    van Oers, A. M.; Maas, L. R.; Vermeersen, B. L. A.

    2016-12-01

    One of the most peculiar features on Saturn moon Enceladus is its so-called tiger stripe pattern at the geologically active South Polar Terrain (SPT), as first observed in detail by the Cassini spacecraft early 2005. It is generally assumed that the four almost parallel surface lines that constitute this pattern are faults in the icy surface overlying a confined salty water reservoir. In 2013, we formulated the original idea [Vermeersen et al., AGU Fall Meeting 2013, abstract #P53B-1848] that the tiger stripe pattern is formed and maintained by induced, tidally and rotationally driven, wave-attractor motions in the ocean underneath the icy surface of the tiger-stripe region. Such wave-attractor motions are observed in water tank experiments in laboratories on Earth and in numerical experiments [Maas et al., Nature, 338, 557-561, 1997; Drijfhout and Maas, J. Phys. Oceanogr., 37, 2740-2763, 2007; Hazewinkel et al., Phys. Fluids, 22, 107102, 2010]. Numerical simulations show the persistence of wave attractors for a range of ocean shapes and stratifications. The intensification of the wave field near the location of the surface reflections of wave attractors has been numerically and experimentally confirmed. We measured the forces a wave attractor exerts on a solid surface, near a reflection point. These reflection points would correspond to the location of the tiger stripes. Combining experiments and numerical simulations we conclude that (1) wave attractors can exist in Enceladus' subsurface sea, (2) their shape can be matched to the tiger stripes, (3) the wave attractors cause a localized force at the water-ice boundaries, (4) this force could have been large enough to contribute to fracturing the ice and (5) the wave attractors localize energy (and particles) and cause dissipation along its path, helping explain Enceladus' enigmatic heat output at the tiger stripes.

  17. Chaotic behavior in nuclei

    Mitchel, G.; Shriner, J.

    2005-01-01

    Although the predictions of Random Matrix Theory (RMT) were available by the early 1960s, data of sufficiently high quality to adequately test the theory were only obtained a decade later by Rainwater. It was another decade later that Bohigas, Haq and Pandey combined the best available nuclear resonance data - the Columbia neutron resonances in heavy nuclei and the TUNL proton resonances in lighter nuclei - to form the Nuclear Data Ensemble. They obtained excellent agreement for the level statistics with the RMT predictions. The expected Porter-Thomas (PT) distribution was considered very early. However, since the widths (amplitudes squared) are measured, the predicted Gaussian distribution for the amplitudes was only qualitatively confirmed. A much more sensitive test was performed by measuring two widths and the relative phase between the two amplitudes. By comparison of the width and amplitude correlations, the Gaussian distribution was confirmed at the 1% level. Following the Bohigas conjecture - that quantum analogs of classically chaotic systems obey RMT - there was an explosion of activity utilizing level statistics in many different quantum systems. In nuclei the focus was verifying the range of applicability of RMT. Of particular interest was the effect of collectivity and of excitation energy on statistical properties. The effect of symmetry breaking on level statistics was examined and early predictions by Dyson were confirmed. The effect of symmetry breaking on the width distribution was also measured for the first time. Although heuristic arguments predicted no change from the PT distribution, experimentally there was a large deviation from the PT prediction. Later theoretical efforts were consistent with this result. The stringent conditions placed on the experiments - for eigenvalue tests the data need to be essentially perfect (few or no missing levels or mis assigned quantum numbers) - has limited the amount of suitable experimental data. The

  18. Chaotic advection in the ocean

    Koshel' , Konstantin V; Prants, Sergei V [V.I. Il' ichev Pacific Oceanological Institute, Far-Eastern Division of the Russian Academy of Sciences, Vladivostok (Russian Federation)

    2006-11-30

    The problem of chaotic advection of passive scalars in the ocean and its topological, dynamical, and fractal properties are considered from the standpoint of the theory of dynamical systems. Analytic and numerical results on Lagrangian transport and mixing in kinematic and dynamic chaotic advection models are described for meandering jet currents, topographical eddies in a barotropic ocean, and a two-layer baroclinic ocean. Laboratory experiments on hydrodynamic flows in rotating tanks as an imitation of geophysical chaotic advection are described. Perspectives of a dynamical system approach in physical oceanography are discussed. (reviews of topical problems)

  19. Modeling of Coupled Chaotic Oscillators

    Lai, Y.; Grebogi, C.

    1999-01-01

    Chaotic dynamics may impose severe limits to deterministic modeling by dynamical equations of natural systems. We give theoretical argument that severe modeling difficulties may occur for high-dimensional chaotic systems in the sense that no model is able to produce reasonably long solutions that are realized by nature. We make these ideas concrete by investigating systems of coupled chaotic oscillators. They arise in many situations of physical and biological interests, and they also arise from discretization of nonlinear partial differential equations. copyright 1999 The American Physical Society

  20. Eigenfunctions in chaotic quantum systems

    Baecker, Arnd

    2007-07-01

    The structure of wavefunctions of quantum systems strongly depends on the underlying classical dynamics. In this text a selection of articles on eigenfunctions in systems with fully chaotic dynamics and systems with a mixed phase space is summarized. Of particular interest are statistical properties like amplitude distribution and spatial autocorrelation function and the implication of eigenfunction structures on transport properties. For systems with a mixed phase space the separation into regular and chaotic states does not always hold away from the semiclassical limit, such that chaotic states may completely penetrate into the region of the regular island. The consequences of this flooding are discussed and universal aspects highlighted. (orig.)

  1. Eigenfunctions in chaotic quantum systems

    Baecker, Arnd

    2007-01-01

    The structure of wavefunctions of quantum systems strongly depends on the underlying classical dynamics. In this text a selection of articles on eigenfunctions in systems with fully chaotic dynamics and systems with a mixed phase space is summarized. Of particular interest are statistical properties like amplitude distribution and spatial autocorrelation function and the implication of eigenfunction structures on transport properties. For systems with a mixed phase space the separation into regular and chaotic states does not always hold away from the semiclassical limit, such that chaotic states may completely penetrate into the region of the regular island. The consequences of this flooding are discussed and universal aspects highlighted. (orig.)

  2. Chaotic dynamics of flexible Euler-Bernoulli beams

    Awrejcewicz, J., E-mail: awrejcew@p.lodz.pl [Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland and Department of Vehicles, Warsaw University of Technology, 84 Narbutta St., 02-524 Warsaw (Poland); Krysko, A. V., E-mail: anton.krysko@gmail.com [Department of Applied Mathematics and Systems Analysis, Saratov State Technical University, Politehnicheskaya 77, 410054 Saratov (Russian Federation); Kutepov, I. E., E-mail: iekutepov@gmail.com; Zagniboroda, N. A., E-mail: tssrat@mail.ru; Dobriyan, V., E-mail: Dobriy88@yandex.ru; Krysko, V. A., E-mail: tak@san.ru [Department of Mathematics and Modeling, Saratov State Technical University, Politehnicheskaya 77, 410054 Saratov (Russian Federation)

    2013-12-15

    Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c{sup 2}) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions is carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q{sub 0} and frequency ω{sub p} of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.

  3. Psychotherapy Is Chaotic-(Not Only) in a Computational World.

    Schiepek, Günter K; Viol, Kathrin; Aichhorn, Wolfgang; Hütt, Marc-Thorsten; Sungler, Katharina; Pincus, David; Schöller, Helmut J

    2017-01-01

    Objective: The aim of this article is to outline the role of chaotic dynamics in psychotherapy. Besides some empirical findings of chaos at different time scales, the focus is on theoretical modeling of change processes explaining and simulating chaotic dynamics. It will be illustrated how some common factors of psychotherapeutic change and psychological hypotheses on motivation, emotion regulation, and information processing of the client's functioning can be integrated into a comprehensive nonlinear model of human change processes. Methods: The model combines 5 variables (intensity of emotions, problem intensity, motivation to change, insight and new perspectives, therapeutic success) and 4 parameters into a set of 5 coupled nonlinear difference equations. The results of these simulations are presented as time series, as phase space embedding of these time series (i.e., attractors), and as bifurcation diagrams. Results: The model creates chaotic dynamics, phase transition-like phenomena, bi- or multi-stability, and sensibility of the dynamic patterns on parameter drift. These features are predicted by chaos theory and by Synergetics and correspond to empirical findings. The spectrum of these behaviors illustrates the complexity of psychotherapeutic processes. Conclusion: The model contributes to the development of an integrative conceptualization of psychotherapy. It is consistent with the state of scientific knowledge of common factors, as well as other psychological topics, such as: motivation, emotion regulation, and cognitive processing. The role of chaos theory is underpinned, not only in the world of computer simulations, but also in practice. In practice, chaos demands technologies capable of real-time monitoring and reporting on the nonlinear features of the ongoing process (e.g., its stability or instability). Based on this monitoring, a client-centered, continuous, and cooperative process of feedback and control becomes possible. By contrast, restricted

  4. Reconfigurable chaotic logic gates based on novel chaotic circuit

    Behnia, S.; Pazhotan, Z.; Ezzati, N.; Akhshani, A.

    2014-01-01

    Highlights: • A novel method for implementing logic gates based on chaotic maps is introduced. • The logic gates can be implemented without any changes in the threshold voltage. • The chaos-based logic gates may serve as basic components of future computing devices. - Abstract: The logical operations are one of the key issues in today’s computer architecture. Nowadays, there is a great interest in developing alternative ways to get the logic operations by chaos computing. In this paper, a novel implementation method of reconfigurable logic gates based on one-parameter families of chaotic maps is introduced. The special behavior of these chaotic maps can be utilized to provide same threshold voltage for all logic gates. However, there is a wide interval for choosing a control parameter for all reconfigurable logic gates. Furthermore, an experimental implementation of this nonlinear system is presented to demonstrate the robustness of computing capability of chaotic circuits

  5. Spheromak Merging Experiments on SSX

    Brown, M. R.; Kornack, T. W.; Sollins, P. K.; Luh, W. J.

    1997-11-01

    Spheromak merging experiments are underway at the Swarthmore Spheromak Experiment (SSX) at Swarthmore College. The spheromaks are formed by identical magnetized plasma guns and equilibrium is established in close fitting 0.5 m diameter copper flux conservers. Partial merging is achieved through openings in the back wall. We observe the formation of a reconnection boundary layer at the interface of the two spheromaks using a linear probe array. The characteristic scale of the flux reversal is about 1 cm (consistent with the diffusion scale δ_diff, the ion Larmor radius ρi and the ion inertial length c/ω_pi). Movies of the formation and evolution of the layer will be presented. Correlations between reconnection events and pulses of soft x-rays and energetic particles will be presented if available. Plans for 2D and 3D imaging of the layer will also be discussed.

  6. Universality for the parameter-mismatching effect on weak synchronization in coupled chaotic systems

    Lim, Woochang; Kim, Sang-Yoon

    2004-01-01

    To examine the universality for the parameter-mismatching effect on weak chaotic synchronization, we study coupled multidimensional invertible systems such as the coupled Henon maps and coupled pendula. By generalizing the method proposed in coupled one-dimensional (1D) noninvertible maps, we introduce the parameter sensitivity exponent δ to measure the degree of the parameter sensitivity of a weakly stable synchronous chaotic attractor. In terms of the parameter sensitivity exponents, we characterize the effect of the parameter mismatch on the intermittent bursting and the basin riddling occurring in the regime of weak synchronization. It is thus found that the scaling exponent μ for the average characteristic time (i.e., the average interburst time and the average chaotic transient lifetime) for both the bubbling and riddling cases is given by the reciprocal of the parameter sensitivity exponent, as in the simple system of coupled 1D maps. Hence, the reciprocal relation (i.e., μ = 1/δ) seems to be 'universal', in the sense that it holds in typical coupled chaotic systems of different nature

  7. Applications of modularized circuit designs in a new hyper-chaotic system circuit implementation

    Wang, Rui; Sun, Hui; Wang, Jie-Zhi; Wang, Lu; Wang, Yan-Chao

    2015-02-01

    Modularized circuit designs for chaotic systems are introduced in this paper. Especially, a typical improved modularized design strategy is proposed and applied to a new hyper-chaotic system circuit implementation. In this paper, the detailed design procedures are described. Multisim simulations and physical experiments are conducted, and the simulation results are compared with Matlab simulation results for different system parameter pairs. These results are consistent with each other and they verify the existence of the hyper-chaotic attractor for this new hyper-chaotic system. Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 61403395), the Natural Science Foundation of Tianjin, China (Grant No. 13JCYBJC39000), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China, the Fund from the Tianjin Key Laboratory of Civil Aircraft Airworthiness and Maintenance in Civil Aviation of China (Grant No. 104003020106), the National Basic Research Program of China (Grant No. 2014CB744904), and the Fund for the Scholars of Civil Aviation University of China (Grant No. 2012QD21x).

  8. Chaotic behavior in a hydrodynamic model of a fluidized bed reactor

    Schouten, J.C.; van den Bleek, C.M.

    1991-01-01

    Recent preliminary experimental studies using time-series analysis have demonstrated that the multi-phase flow in fluidized bed reactors can be characterized as chaotic. In the present paper, it is therefore argued that the chaotic time-dependence of fluidization is a characteristic feature which should be included in scaling rules for fluidized bed reactors. For example, the similarity groups applied in dimensionless fluidized bed scaling should be improved by extending them with functions of the relevant numbers from chaos theory, such as the correlation and embedding dimension or the maximum Lyapunov exponent. This requires that the dependence of these numbers on fluidization parameters must be theoretically and experimentally investigated. The concept of chaos in fluidization also requires that the classical, empirically developed, hydrodynamic models that are applied in fluidized bed scaling are amended to include time-dependence, non-linearity as well as a sufficient level of complexity before they can predict any chaotic behavior. An example is given of chaotic behavior generated in the classical counter-current flow model according to Van Deemter by writing the upwards solids velocity as a harmonic oscillating function of time. A low-dimensional strange attractor is found, embedded in two-dimensional phase space, of which the correlation dimension depends on the solids exchange coefficient

  9. Predicting the bounds of large chaotic systems using low-dimensional manifolds.

    Asger M Haugaard

    Full Text Available Predicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to discretised manifolds of low dimension (low-D embedded in high-dimensional (high-D phase space. As a central feature, the method exploits that strange attractor dimension is generally much smaller than parent system phase space dimension. This is important, since the computational cost associated with discretised manifolds depends exponentially on their dimension. Thus, systems that would otherwise be associated with tremendous computational challenges, can be tackled on a laptop. As a test, bounding manifolds are calculated for high-D modifications of the canonical Duffing system. Parameters can be set such that the bounding manifold displays harmonic behaviour even if the underlying system is chaotic. Thus, solving for one post-transient forcing cycle of the bounding manifold predicts the extrema of the underlying chaotic problem indefinitely.

  10. Modeling cooperative driving behavior in freeway merges.

    2011-11-01

    Merging locations are major sources of freeway bottlenecks and are therefore important for freeway operations analysis. Microscopic simulation tools have been successfully used to analyze merging bottlenecks and to design optimum geometric configurat...

  11. Automatic generation of data merging program codes.

    Hyensook, Kim; Oussena, Samia; Zhang, Ying; Clark, Tony

    2010-01-01

    Data merging is an essential part of ETL (Extract-Transform-Load) processes to build a data warehouse system. To avoid rewheeling merging techniques, we propose a Data Merging Meta-model (DMM) and its transformation into executable program codes in the manner of model driven engineering. DMM allows defining relationships of different model entities and their merging types in conceptual level. Our formalized transformation described using ATL (ATLAS Transformation Language) enables automatic g...

  12. Nuclear friction and chaotic motion

    Srokowski, T.; Szczurek, A.; Drozdz, S.

    1990-01-01

    The concept of nuclear friction is considered from the point of view of regular versus chaotic motion in an atomic nucleus. Using a realistic nuclear Hamiltonian it is explicitly shown that the frictional description of the gross features of nuclear collisions is adequate if the system behaves chaotically. Because of the core in the Hamiltonian, the three-body nuclear system already reveals a structure of the phase space rich enough for this concept to be applicable

  13. Chaotic diagonal recurrent neural network

    Wang Xing-Yuan; Zhang Yi

    2012-01-01

    We propose a novel neural network based on a diagonal recurrent neural network and chaos, and its structure and learning algorithm are designed. The multilayer feedforward neural network, diagonal recurrent neural network, and chaotic diagonal recurrent neural network are used to approach the cubic symmetry map. The simulation results show that the approximation capability of the chaotic diagonal recurrent neural network is better than the other two neural networks. (interdisciplinary physics and related areas of science and technology)

  14. Oscillatory attractors: a new cosmological phase

    Bains, Jasdeep S. [Center for the Fundamental Laws of Nature, Harvard University, 17 Oxford St, Cambridge, MA 02138 (United States); Hertzberg, Mark P. [Institute of Cosmology, Department of Physics and Astronomy, Tufts University, 574 Boston Ave, Medford, MA 02155 (United States); Wilczek, Frank, E-mail: bains@physics.harvard.edu, E-mail: mark.hertzberg@tufts.edu, E-mail: wilczek@mit.edu [Center for Theoretical Physics, Department of Physics, MIT, 77 Massachusetts Ave, Cambridge, MA 02139 (United States)

    2017-05-01

    In expanding FRW spacetimes, it is usually the case that homogeneous scalar fields redshift and their amplitudes approach limiting values: Hubble friction usually ensures that the field relaxes to its minimum energy configuration, which is usually a static configuration. Here we discover a class of relativistic scalar field models in which the attractor behavior is the field oscillating indefinitely, with finite amplitude, in an expanding FRW spacetime, despite the presence of Hubble friction. This is an example of spontaneous breaking of time translation symmetry. We find that the effective equation of state of the field has average value ( w )=−1, implying that the field itself could drive an inflationary or dark energy dominated phase. This behavior is reminiscent of ghost condensate models, but in the new models, unlike in the ghost condensate models, the energy-momentum tensor is time dependent, so that these new models embody a more definitive breaking of time translation symmetry. We explore (quantum) fluctuations around the homogeneous background solution, and find that low k -modes can be stable, while high k -modes are typically unstable. We discuss possible interpretations and implications of that instability.

  15. Quintessential inflation with α-attractors

    Dimopoulos, Konstantinos; Owen, Charlotte, E-mail: k.dimopoulos1@lancaster.ac.uk, E-mail: c.owen@lancaster.ac.uk [Consortium for Fundamental Physics, Physics Department, Lancaster University, Lancaster LA1 4YB (United Kingdom)

    2017-06-01

    A novel approach to quintessential inflation model building is studied, within the framework of α-attractors, motivated by supergravity theories. Inflationary observables are in excellent agreement with the latest CMB observations, while quintessence explains the dark energy observations without any fine-tuning. The model is kept intentionally minimal, avoiding the introduction of many degrees of freedom, couplings and mass scales. In stark contrast to ΛCDM, for natural values of the parameters, the model attains transient accelerated expansion, which avoids the future horizon problem, while it maintains the field displacement mildly sub-Planckian such that the flatness of the quintessential tail is not lifted by radiative corrections and violations of the equivalence principle (fifth force) are under control. In particular, the required value of the cosmological constant is near the eletroweak scale. Attention is paid to the reheating of the Universe, which avoids gravitino overproduction and respects nucleosynthesis constraints. Kination is treated in a model independent way. A spike in gravitational waves, due to kination, is found not to disturb nucleosynthesis as well.

  16. Fibre inflation and α-attractors

    Kallosh, Renata; Linde, Andrei; Roest, Diederik; Westphal, Alexander; Yamada, Yusuke

    2018-02-01

    Fibre inflation is a specific string theory construction based on the Large Volume Scenario that produces an inflationary plateau. We outline its relation to α-attractor models for inflation, with the cosmological sector originating from certain string theory corrections leading to α = 2 and α = 1/2. Above a certain field range, the steepening effect of higher-order corrections leads first to the breakdown of single-field slow-roll and after that to the onset of 2-field dynamics: the overall volume of the extra dimensions starts to participate in the effective dynamics. Finally, we propose effective supergravity models of fibre inflation based on an \\overline{D3} uplift term with a nilpotent superfield. Specific moduli dependent \\overline{D3} induced geometries lead to cosmological fibre models but have in addition a de Sitter minimum exit. These supergravity models motivated by fibre inflation are relatively simple, stabilize the axions and disentangle the Hubble parameter from supersymmetry breaking.

  17. Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems

    Lucarini, Valerio; Faranda, Davide; Wouters, Jeroen; Kuna, Tobias

    2014-02-01

    In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

  18. Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems.

    Lucarini, Valerio; Faranda, Davide; Wouters, Jeroen; Kuna, Tobias

    2014-01-01

    In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

  19. A Gaussian mixture model based cost function for parameter estimation of chaotic biological systems

    Shekofteh, Yasser; Jafari, Sajad; Sprott, Julien Clinton; Hashemi Golpayegani, S. Mohammad Reza; Almasganj, Farshad

    2015-02-01

    As we know, many biological systems such as neurons or the heart can exhibit chaotic behavior. Conventional methods for parameter estimation in models of these systems have some limitations caused by sensitivity to initial conditions. In this paper, a novel cost function is proposed to overcome those limitations by building a statistical model on the distribution of the real system attractor in state space. This cost function is defined by the use of a likelihood score in a Gaussian mixture model (GMM) which is fitted to the observed attractor generated by the real system. Using that learned GMM, a similarity score can be defined by the computed likelihood score of the model time series. We have applied the proposed method to the parameter estimation of two important biological systems, a neuron and a cardiac pacemaker, which show chaotic behavior. Some simulated experiments are given to verify the usefulness of the proposed approach in clean and noisy conditions. The results show the adequacy of the proposed cost function.

  20. Generalized correlation integral vectors: A distance concept for chaotic dynamical systems

    Haario, Heikki, E-mail: heikki.haario@lut.fi [School of Engineering Science, Lappeenranta University of Technology, Lappeenranta (Finland); Kalachev, Leonid, E-mail: KalachevL@mso.umt.edu [Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812-0864 (United States); Hakkarainen, Janne [Earth Observation Unit, Finnish Meteorological Institute, Helsinki (Finland)

    2015-06-15

    Several concepts of fractal dimension have been developed to characterise properties of attractors of chaotic dynamical systems. Numerical approximations of them must be calculated by finite samples of simulated trajectories. In principle, the quantities should not depend on the choice of the trajectory, as long as it provides properly distributed samples of the underlying attractor. In practice, however, the trajectories are sensitive with respect to varying initial values, small changes of the model parameters, to the choice of a solver, numeric tolerances, etc. The purpose of this paper is to present a statistically sound approach to quantify this variability. We modify the concept of correlation integral to produce a vector that summarises the variability at all selected scales. The distribution of this stochastic vector can be estimated, and it provides a statistical distance concept between trajectories. Here, we demonstrate the use of the distance for the purpose of estimating model parameters of a chaotic dynamic model. The methodology is illustrated using computational examples for the Lorenz 63 and Lorenz 95 systems, together with a framework for Markov chain Monte Carlo sampling to produce posterior distributions of model parameters.

  1. Output-Feedback Control of a Chaotic MEMS Resonator for Oscillation Amplitude Enhancement

    Alexander Jimenez-Triana

    2014-01-01

    Full Text Available The present work addresses the problem of chaos control in an electrostatic MEMS resonator by using an output-feedback control scheme. One of the unstable orbits immersed in the chaotic attractor is stabilized in order to produce a sustained oscillation of the movable plate composing the microstructure. The orbit is carefully chosen so as to produce a high amplitude oscillation. This approach allows the enhancement of oscillation amplitude of the resonator at a reduced control effort, since the unstable orbit already exists in the system and it is not necessary to spend energy to create it. Realistic operational conditions of the MEMS are considered including parametric uncertainties in the model and constraints due to the difficulty in measuring the speed of the plates of the microstructure. A control law is constructed recursively by using the technique of backstepping. Finally, numerical simulations are carried out to confirm the validity of the developed control scheme and to demonstrate the effect of controlling orbits immersed in the chaotic attractor.

  2. Non-linguistic Conditions for Causativization as a Linguistic Attractor.

    Nichols, Johanna

    2017-01-01

    An attractor, in complex systems theory, is any state that is more easily or more often entered or acquired than departed or lost; attractor states therefore accumulate more members than non-attractors, other things being equal. In the context of language evolution, linguistic attractors include sounds, forms, and grammatical structures that are prone to be selected when sociolinguistics and language contact make it possible for speakers to choose between competing forms. The reasons why an element is an attractor are linguistic (auditory salience, ease of processing, paradigm structure, etc.), but the factors that make selection possible and propagate selected items through the speech community are non-linguistic. This paper uses the consonants in personal pronouns to show what makes for an attractor and how selection and diffusion work, then presents a survey of several language families and areas showing that the derivational morphology of pairs of verbs like fear and frighten , or Turkish korkmak 'fear, be afraid' and korkutmak 'frighten, scare', or Finnish istua 'sit' and istutta 'seat (someone)', or Spanish sentarse 'sit down' and sentar 'seat (someone)' is susceptible to selection. Specifically, the Turkish and Finnish pattern, where 'seat' is derived from 'sit' by addition of a suffix-is an attractor and a favored target of selection. This selection occurs chiefly in sociolinguistic contexts of what is defined here as linguistic symbiosis, where languages mingle in speech, which in turn is favored by certain demographic, sociocultural, and environmental factors here termed frontier conditions. Evidence is surveyed from northern Eurasia, the Caucasus, North and Central America, and the Pacific and from both modern and ancient languages to raise the hypothesis that frontier conditions and symbiosis favor causativization.

  3. Non-linguistic Conditions for Causativization as a Linguistic Attractor

    Johanna Nichols

    2018-01-01

    Full Text Available An attractor, in complex systems theory, is any state that is more easily or more often entered or acquired than departed or lost; attractor states therefore accumulate more members than non-attractors, other things being equal. In the context of language evolution, linguistic attractors include sounds, forms, and grammatical structures that are prone to be selected when sociolinguistics and language contact make it possible for speakers to choose between competing forms. The reasons why an element is an attractor are linguistic (auditory salience, ease of processing, paradigm structure, etc., but the factors that make selection possible and propagate selected items through the speech community are non-linguistic. This paper uses the consonants in personal pronouns to show what makes for an attractor and how selection and diffusion work, then presents a survey of several language families and areas showing that the derivational morphology of pairs of verbs like fear and frighten, or Turkish korkmak ‘fear, be afraid’ and korkutmak ‘frighten, scare’, or Finnish istua ‘sit’ and istutta ‘seat (someone’, or Spanish sentarse ‘sit down’ and sentar ‘seat (someone’ is susceptible to selection. Specifically, the Turkish and Finnish pattern, where ‘seat’ is derived from ‘sit’ by addition of a suffix—is an attractor and a favored target of selection. This selection occurs chiefly in sociolinguistic contexts of what is defined here as linguistic symbiosis, where languages mingle in speech, which in turn is favored by certain demographic, sociocultural, and environmental factors here termed frontier conditions. Evidence is surveyed from northern Eurasia, the Caucasus, North and Central America, and the Pacific and from both modern and ancient languages to raise the hypothesis that frontier conditions and symbiosis favor causativization.

  4. A Novel Image Encryption Scheme Based on Clifford Attractor and Noisy Logistic Map for Secure Transferring Images in Navy

    Mohadeseh Kanafchian

    2017-04-01

    In this paper, we first give a brief introduction into chaotic image encryption and then we investigate some important properties and behaviour of the logistic map. The logistic map, aperiodic trajectory, or random-like fluctuation, could not be obtained with some choice of initial condition. Therefore, a noisy logistic map with an additive system noise is introduced. The proposed scheme is based on the extended map of the Clifford strange attractor, where each dimension has a specific role in the encryption process. Two dimensions are used for pixel permutation and the third dimension is used for pixel diffusion. In order to optimize the Clifford encryption system we increase the space key by using the noisy logistic map and a novel encryption scheme based on the Clifford attractor and the noisy logistic map for secure transfer images is proposed. This algorithm consists of two parts: the noisy logistic map shuffle of the pixel position and the pixel value. We use times for shuffling the pixel position and value then we generate the new pixel position and value by the Clifford system. To illustrate the efficiency of the proposed scheme, various types of security analysis are tested. It can be concluded that the proposed image encryption system is a suitable choice for practical applications.

  5. NUMERICAL ANALYSIS OF A CHAOTIC SYSTEM

    任志坚

    2001-01-01

    This paper further proves that a single spiral strange attractor can be observed in an extremely simple autonomous electrical circuit by computer simulation. It is of third order and has only one nonlinear element: a three-segment piecewise linear resistor. The digital analyses show that the strange attractor has peculiar features compared with other third-order differential systems.

  6. Studies in Chaotic adiabatic dynamics

    Jarzynski, C.

    1994-01-01

    Chaotic adiabatic dynamics refers to the study of systems exhibiting chaotic evolution under slowly time-dependent equations of motion. In this dissertation the author restricts his attention to Hamiltonian chaotic adiabatic systems. The results presented are organized around a central theme, namely, that the energies of such systems evolve diffusively. He begins with a general analysis, in which he motivates and derives a Fokker-Planck equation governing this process of energy diffusion. He applies this equation to study the open-quotes goodnessclose quotes of an adiabatic invariant associated with chaotic motion. This formalism is then applied to two specific examples. The first is that of a gas of noninteracting point particles inside a hard container that deforms slowly with time. Both the two- and three-dimensional cases are considered. The results are discussed in the context of the Wall Formula for one-body dissipation in nuclear physics, and it is shown that such a gas approaches, asymptotically with time, an exponential velocity distribution. The second example involves the Fermi mechanism for the acceleration of cosmic rays. Explicit evolution equations are obtained for the distribution of cosmic ray energies within this model, and the steady-state energy distribution that arises when this equation is modified to account for the injection and removal of cosmic rays is discussed. Finally, the author re-examines the multiple-time-scale approach as applied to the study of phase space evolution under a chaotic adiabatic Hamiltonian. This leads to a more rigorous derivation of the above-mentioned Fokker-Planck equation, and also to a new term which has relevance to the problem of chaotic adiabatic reaction forces (the forces acting on slow, heavy degrees of freedom due to their coupling to light, fast chaotic degrees)

  7. Chaotic scattering and quantum dynamics

    Doron, Eyal.

    1992-11-01

    The main concern of this thesis is the application of the semiclassical approximation to quantum chaotic scattering systems. We deal with two separate, although interconnected, subjects. The first subject dealt with is the semiclassical characterization of the fluctuations of the S matrix. A particular important parameter is the magnetic field B, and we show how the correlation length and line shape of S matrix elements under a change of B may be derived. An effect which is present in many physical wave systems is absorption of energy flux. We show how absorption affects both the reflectivity and the scattering phase and time delay of a scattering system. In the second part of the thesis, we show how the formalism and results obtained from chaotic scattering can be applied to the investigation of closed chaotic systems, and in particular to chaotic billiards. The semiclassical expansion for billiards is presented. In the last part of the thesis we deal with the statistics of S matrices of chaotic scattering systems. The main message of this work is that scattering matrix, and its classical counterpart the Poincare Scattering Map can be used to yield a powerful formulation of the quantum mechanical dynamics of bounded systems. (author)

  8. The dimension of attractors underlying periodic turbulent Poiseuille flow

    Keefe, Laurence; Moin, Parviz; Kim, John

    1992-01-01

    A lower bound on the Liapunov dimenison, D-lambda, of the attractor underlying turbulent, periodic Poiseuille flow at a pressure-gradient Reynolds number of 3200 is calculated, on the basis of a coarse-grained (16x33x8) numerical solution, to be approximately 352. Comparison of Liapunov exponent spectra from this and a higher-resolution (16x33x16) simulation on the same spatial domain shows these spectra to have a universal shape when properly scaled. On the basis of these scaling properties, and a partial exponent spectrum from a still higher-resolution (32x33x32) simulation, it is argued that the actual dimension of the attractor underlying motion of the given computational domain is approximately 780. It is suggested that this periodic turbulent shear flow is deterministic chaos, and that a strange attractor does underly solutions to the Navier-Stokes equations in such flows.

  9. Separation of attractors in 1-modulus quantum corrected special geometry

    Bellucci, S; Marrani, A; Shcherbakov, A

    2008-01-01

    We study the solutions to the N=2, d=4 Attractor Equations in a dyonic, extremal, static, spherically symmetric and asymptotically flat black hole background, in the simplest case of perturbative quantum corrected cubic Special Kahler geometry consistent with continuous axion-shift symmetry, namely in the 1-modulus Special Kahler geometry described (in a suitable special symplectic coordinate) by the holomorphic Kahler gauge-invariant prepotential F=t^3+i*lambda, with lambda real. By performing computations in the ``magnetic'' charge configuration, we find evidence for interesting phenomena (absent in the classical limit of vanishing lambda). Namely, for a certain range of the quantum parameter lambda we find a ``splitting'' of attractors, i.e. the existence of multiple solutions to the Attractor Equations for fixed supporting charge configuration. This corresponds to the existence of ``area codes'' in the radial evolution of the scalar t, determined by the various disconnected regions of the moduli space, wh...

  10. Strange attractors in weakly turbulent Couette-Taylor flow

    Brandstater, A.; Swinney, Harry L.

    1987-01-01

    An experiment is conducted on the transition from quasi-periodic to weakly turbulent flow of a fluid contained between concentric cylinders with the inner cylinder rotating and the outer cylinder at rest. Power spectra, phase-space portraits, and circle maps obtained from velocity time-series data indicate that the nonperiodic behavior observed is deterministic, that is, it is described by strange attractors. Various problems that arise in computing the dimension of strange attractors constructed from experimental data are discussed and it is shown that these problems impose severe requirements on the quantity and accuracy of data necessary for determining dimensions greater than about 5. In the present experiment the attractor dimension increases from 2 at the onset of turbulence to about 4 at a Reynolds number 50-percent above the onset of turbulence.

  11. Some Convex Functions Based Measures of Independence and Their Application to Strange Attractor Reconstruction

    Kazuyuki Aihara

    2011-04-01

    Full Text Available The classical information-theoretic measures such as the entropy and the mutual information (MI are widely applicable to many areas in science and engineering. Csiszar generalized the entropy and the MI by using the convex functions. Recently, we proposed the grid occupancy (GO and the quasientropy (QE as measures of independence. The QE explicitly includes a convex function in its definition, while the expectation of GO is a subclass of QE. In this paper, we study the effect of different convex functions on GO, QE, and Csiszar’s generalized mutual information (GMI. A quality factor (QF is proposed to quantify the sharpness of their minima. Using the QF, it is shown that these measures can have sharper minima than the classical MI. Besides, a recursive algorithm for computing GMI, which is a generalization of Fraser and Swinney’s algorithm for computing MI, is proposed. Moreover, we apply GO, QE, and GMI to chaotic time series analysis. It is shown that these measures are good criteria for determining the optimum delay in strange attractor reconstruction.

  12. Attractors of equations of non-Newtonian fluid dynamics

    Zvyagin, V G; Kondrat'ev, S K

    2014-01-01

    This survey describes a version of the trajectory-attractor method, which is applied to study the limit asymptotic behaviour of solutions of equations of non-Newtonian fluid dynamics. The trajectory-attractor method emerged in papers of the Russian mathematicians Vishik and Chepyzhov and the American mathematician Sell under the condition that the corresponding trajectory spaces be invariant under the translation semigroup. The need for such an approach was caused by the fact that for many equations of mathematical physics for which the Cauchy initial-value problem has a global (weak) solution with respect to the time, the uniqueness of such a solution has either not been established or does not hold. In particular, this is the case for equations of fluid dynamics. At the same time, trajectory spaces invariant under the translation semigroup could not be constructed for many equations of non-Newtonian fluid dynamics. In this connection, a different approach to the construction of trajectory attractors for dissipative systems was proposed in papers of Zvyagin and Vorotnikov without using invariance of trajectory spaces under the translation semigroup and is based on the topological lemma of Shura-Bura. This paper presents examples of equations of non-Newtonian fluid dynamics (the Jeffreys system describing movement of the Earth's crust, the model of motion of weak aqueous solutions of polymers, a system with memory) for which the aforementioned construction is used to prove the existence of attractors in both the autonomous and the non-autonomous cases. At the beginning of the paper there is also a brief exposition of the results of Ladyzhenskaya on the existence of attractors of the two-dimensional Navier-Stokes system and the result of Vishik and Chepyzhov for the case of attractors of the three-dimensional Navier-Stokes system. Bibliography: 34 titles

  13. Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

    Kuznetsov, N. V.; Leonov, G. A.; Mokaev, T. N.; Prasad, A.; Shrimali, M. D.

    2015-01-01

    The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a hidden attractor in the case of multistability as well as a classical self-excited attractor. The hidden attractor in this system can be localized by analytical/numerical methods based on the continuation and perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of attractors. A con...

  14. On synchronization of three chaotic systems

    Yan Jianping; Li Changpin

    2005-01-01

    In this paper, a simple but efficient method is applied to the synchronization of three chaotic systems, i.e., the chaotic Lorenz, Chua, and Chen systems. Numerical simulations show this method works very well

  15. Intermittent chaotic chimeras for coupled rotators

    Olmi, Simona; Martens, Erik Andreas; Thutupalli, Shashi

    2015-01-01

    Two symmetrically coupled populations of N oscillators with inertia m display chaotic solutions with broken symmetry similar to experimental observations with mechanical pendulums. In particular, we report evidence of intermittent chaotic chimeras, where one population is synchronized and the other...

  16. On the Design of Chaotic Oscillators

    Lindberg, Erik; Tamasevicius, A; Cenys, A.

    1998-01-01

    A discussion of the chaotic oscillator concept from a design methodology pointof view. The attributes of some chaoticoscillators are discussed and a systematicdesign method based on eigenvalue investigation is proposed. The method isillustrated with a chaotic Wien-bridgeoscillator design....

  17. Driven-dissipative Euler close-quote s equations for a rigid body: A chaotic system relevant to fluid dynamics

    Turner, L.

    1996-01-01

    Adhering to the lore that vorticity is a critical ingredient of fluid turbulence, a triad of coupled helicity (vorticity) states of the incompressible Navier-Stokes fluid are followed. Effects of the remaining states of the fluid on the triad are then modeled as a simple driving term. Numerical solution of the equations yield attractors that seem strange and chaotic. This suggests that the unpredictability of nonlinear fluid dynamics (i.e., turbulence) may be traced back to the most primordial structure of the Navier-Stokes equation; namely, the driven triadic interaction. copyright 1996 The American Physical Society

  18. Analysis of chaos attractors of MCG-recordings.

    Jiang, Shiqin; Yang, Fan; Yi, Panke; Chen, Bo; Luo, Ming; Wang, Lemin

    2006-01-01

    By studying the chaos attractor of cardiac magnetic induction strength B(z) generated by the electrical activity of the heart, we found that its projection in the reconstructed phase space has a similar shape with the map of the total current dipole vector. It is worth noting that the map of the total current dipole vector is computed with MCG recordings measured at 36 locations, whereas the chaos attractor of B(z) is generated by only one cardiac magnetic field recordings on the measured plan. We discuss only two subjects of different ages in this paper.

  19. Strange attractor in the Potts spin glass on hierarchical lattices

    Lima, Washington de [Universidade Federal de Pernambuco, Centro Acadêmico do Agreste, Pernambuco (Brazil); Camelo-Neto, G. [Universidade Federal de Alagoas, Núcleo de Ciências Exatas, Laboratório de Física Teórica e Computacional, CEP 57309-005 Arapiraca, Alagoas (Brazil); Coutinho, S., E-mail: sergio@ufpe.br [Universidade Federal de Pernambuco, Departamento de Física, Laboratório de Física Teórica e Computacional, Cidade Universitária, CEP 50670-901 Recife, Pernambuco (Brazil)

    2013-11-29

    The spin-glass q-state Potts model on d-dimensional diamond hierarchical lattices is investigated by an exact real space renormalization group scheme. Above a critical dimension d{sub l}(q) for q>2, the coupling constants probability distribution flows to a low-temperature strange attractor or to the high-temperature paramagnetic fixed point, according to the temperature is below or above the critical temperature T{sub c}(q,d). The strange attractor was investigated considering four initial different distributions for q=3 and d=5 presenting strong robustness in shape and temperature interval suggesting a condensed phase with algebraic decay.

  20. STRANGE ATTRACTORS ON PSEUDOSPECTRAL SOLUTIONS FOR DISSIPATIVE ZAKHAROV EQUATIONS

    马书清; 常谦顺

    2004-01-01

    In this paper, the pseudospcctral method to solve the dissipative Zakharov equations is used. Its convergence is proved by priori estinates. The existence of the global attractors and the estimates of dimension are presented. A class of steady state solutions is also disscussed. The numerical results show that if the steady state solutions satisfy some special conditions, they become unstable and limit cycles and strange attractors will occur for very small perturbations.The largest Lyapunov exponent and analysis of the lincarized system are applied to explain these phenomena.

  1. Simplified Chua's attractor via bridging a diode pair

    Quan Xu

    2015-04-01

    Full Text Available In this paper, a simplified Chua's circuit is realised by bridging a diode pair between a passive LC (inductance and capacitance in parallel connection - LC oscillator and an active RC (resistance and capacitance in parallel connection - RC filter. The dynamical behaviours of the circuit are investigated by numerical simulations and verified by experimental measurements. It is found that the simplified Chua's circuit generates Chua's attractors similarly and demonstrates complex non-linear phenomena including coexisting bifurcation modes and coexisting attractors in particular.

  2. A new chaotic secure communication scheme

    Hua Changchun; Yang Bo; Ouyang Gaoxiang; Guan Xinping

    2005-01-01

    A new chaotic secure communication scheme is constructed. Unified chaotic system is used to encrypt the emitted signal. Different from the existing chaotic secure communication methods, the useful information is embodied in the parameter of chaotic systems in this Letter. The receiver is designed which can succeed in recovering the former signal. Finally computer simulations are done to verify the proposed methods, and the results show that the obtained theoretic results are feasible and efficient

  3. Chaotic structure of oil prices

    Bildirici, Melike; Sonustun, Fulya Ozaksoy

    2018-01-01

    The fluctuations in oil prices are very complicated and therefore, it is unable to predict its effects on economies. For modelling complex system of oil prices, linear economic models are not sufficient and efficient tools. Thus, in recent years, economists attached great attention to non-linear structure of oil prices. For analyzing this relationship, GARCH types of models were used in some papers. Distinctively from the other papers, in this study, we aimed to analyze chaotic pattern of oil prices. Thus, it was used the Lyapunov Exponents and Hennon Map to determine chaotic behavior of oil prices for the selected time period.

  4. Chaotic signals in digital communications

    Eisencraft, Marcio; Suyama, Ricardo

    2013-01-01

    Chaotic Signals in Digital Communications combines fundamental background knowledge with state-of-the-art methods for using chaotic signals and systems in digital communications. The book builds a bridge between theoretical works and practical implementation to help researchers attain consistent performance in realistic environments. It shows the possible shortcomings of the chaos-based communication systems proposed in the literature, particularly when they are subjected to non-ideal conditions. It also presents a toolbox of techniques for researchers working to actually implement such system

  5. Is merging and acquisition profitable?

    Skjeret, Frode; Soergard, Lars

    2002-01-01

    This report deals with mergers and acquisitions in the electricity sector in Norway. The background is the fact that the profitability of these activities proves to be low. In buying, it is typically the selling shareholder who profits from the transaction, while the buying company does not really earn much. This result appears to be a robust result both in different countries, between sectors and independent of methodology. The report provides theoretical justification for merging and buying up and empirical evaluations of the effects of company integration. It is asserted that what can be learned in general from the literature may also occur in the European power sector. Furthermore, the report discusses the challenges faced by the companies if they want to expand through mergers and acquisitions

  6. Morphology of magnetic merging at the magnetopause

    Crooker, N.U.

    1990-01-01

    To illustrate the basic features of magnetospheric topology, the development of a global model is traced from the superposition of dipole and uniform fields to the effects of adding, in turn, diffusion regions, surface currents, and a magnetic field component normal to the magnetopause. The subsolar, antiparallel, tearing, and patchy merging geometries proposed in the past all emerge under various conditions, but models tht deduce merging geometry from global boundary conditions are lacking. An exception is a model in which the external field merges wherever it falls tangent to the magnetopause. The result is a subsolar merging line that has all the characteristics of early sketches based on local arguments. Magnetosheath plasma beta affects magnetospheric topology and, consequently, merging geometry. Low, high, and variable beta favor subsolar, tearing, and patchy merging, respectively. Proposed flux transfer event models of burst reconnection from a single merging line, flux ropes from multiple merging lines, and flux tube elbows from patches can also be categorized by plasma beta in the same respective order. The topological modeling reviewed here may prove to be most useful for interpreting merging results from MHD simulations. (author)

  7. Merging By Decentralized Eventual Consistency Algorithms

    Ahmed-Nacer Mehdi

    2015-12-01

    Full Text Available Merging mechanism is an essential operation for version control systems. When each member of collaborative development works on an individual copy of the project, software merging allows to reconcile modifications made concurrently as well as managing software change through branching. The collaborative system is in charge to propose a merge result that includes user’s modifications. Theusers now have to check and adapt this result. The adaptation should be as effort-less as possible, otherwise, the users may get frustrated and will quit the collaboration. This paper aims to reduce the conflicts during the collaboration and im prove the productivity. It has three objectives: study the users’ behavior during the collaboration, evaluate the quality of textual merging results produced by specific algorithms and propose a solution to improve the r esult quality produced by the default merge tool of distributed version control systems. Through a study of eight open-source repositories totaling more than 3 million lines of code, we observe the behavior of the concurrent modifications during t he merge p rocedure. We i dentified when th e ex isting merge techniques under-perform, and we propose solutions to improve the quality of the merge. We finally compare with the traditional merge tool through a large corpus of collaborative editing.

  8. Chaotic oscillations in a low pressure two-phase natural circulation loop under low power and high inlet subcooling conditions

    Wu, C.Y.; Wang, S.B.; Pan, C.

    1996-01-01

    The oscillation characteristics of a low pressure two-phase natural circulation loop have been investigated experimentally in this study. Experimental results indicate that the characteristics of the thermal hydraulic oscillations can be periodic, with 2-5 fundamental frequencies, or chaotic, depending on the heating power and inlet subcooling. The number of fundamental frequencies of oscillation increases if the inlet subcooling is increased at a given heating power or the heating power is decreased at a given inlet subcooling; chaotic oscillations appear if the inlet subcooling is further increased and/or the heating power is further decreased. A map of the oscillation characteristics is thus established. The change in oscillation characteristics is evident from the time evolution and power spectrum of a thermal hydraulic parameter and the phase portraits of two thermal hydraulic parameters. These results reveal that a strange attractor exists in a low pressure two-phase natural circulation loop with low power and very high inlet subcooling. (orig.)

  9. Aging in a Chaotic System

    Barkai, E.

    2002-01-01

    We demonstrate aging behavior in a simple non-linear system. Our model is a chaotic map which generates deterministically sub-diffusion. Asymptotic behaviors of the diffusion process are described using aging continuous time random walks, introduced previously to model diffusion in glasses.

  10. Chaotic dynamics from interspike intervals

    Pavlov, A N; Sosnovtseva, Olga; Mosekilde, Erik

    2001-01-01

    Considering two different mathematical models describing chaotic spiking phenomena, namely, an integrate-and-fire and a threshold-crossing model, we discuss the problem of extracting dynamics from interspike intervals (ISIs) and show that the possibilities of computing the largest Lyapunov expone...

  11. MAXIMUM-LIKELIHOOD-ESTIMATION OF THE ENTROPY OF AN ATTRACTOR

    SCHOUTEN, JC; TAKENS, F; VANDENBLEEK, CM

    In this paper, a maximum-likelihood estimate of the (Kolmogorov) entropy of an attractor is proposed that can be obtained directly from a time series. Also, the relative standard deviation of the entropy estimate is derived; it is dependent on the entropy and on the number of samples used in the

  12. The Geometric Structure of Strange Attractors in the Lozi Map

    YongluoCAO; ZengrongLIU

    1998-01-01

    In this paper,the structure of the strange attractors in the Lozi map is investigated on basis of the results gotten by the authors in 1991-1993,The new results of the strange atrtractors of the Lozi map show that our viewpoint is correct.

  13. Global attractors for the coupled suspension bridge system with temperature

    Dell'Oro, Filippo; Giorgi, C.

    2016-01-01

    Roč. 39, č. 4 (2016), s. 864-875 ISSN 0170-4214 Institutional support: RVO:67985840 Keywords : absorbing set * coupled bridge system * global attractor Subject RIV: BA - General Mathematics Impact factor: 1.017, year: 2016 http://onlinelibrary.wiley.com/doi/10.1002/mma.3526/abstract

  14. Sourcing dark matter and dark energy from α-attractors

    Mishra, Swagat S.; Sahni, Varun; Shtanov, Yuri

    2017-01-01

    In [1], Kallosh and Linde drew attention to a new family of superconformal inflationary potentials, subsequently called α-attractors [2]. The α-attractor family can interpolate between a large class of inflationary models. It also has an important theoretical underpinning within the framework of supergravity. We demonstrate that the α-attractors have an even wider appeal since they may describe dark matter and perhaps even dark energy. The dark matter associated with the α-attractors, which we call α-dark matter (αDM), shares many of the attractive features of fuzzy dark matter, with V (φ) = ½ m 2 φ 2 , while having none of its drawbacks. Like fuzzy dark matter, αDM can have a large Jeans length which could resolve the cusp-core and substructure problems faced by standard cold dark matter. αDM also has an appealing tracker property which enables it to converge to the late-time dark matter asymptote, ( w ) ≅ 0, from a wide range of initial conditions. It thus avoids the enormous fine-tuning problems faced by the m 2 φ 2 potential in describing dark matter.

  15. Probability Density Function Method for Observing Reconstructed Attractor Structure

    陆宏伟; 陈亚珠; 卫青

    2004-01-01

    Probability density function (PDF) method is proposed for analysing the structure of the reconstructed attractor in computing the correlation dimensions of RR intervals of ten normal old men. PDF contains important information about the spatial distribution of the phase points in the reconstructed attractor. To the best of our knowledge, it is the first time that the PDF method is put forward for the analysis of the reconstructed attractor structure. Numerical simulations demonstrate that the cardiac systems of healthy old men are about 6 - 6.5 dimensional complex dynamical systems. It is found that PDF is not symmetrically distributed when time delay is small, while PDF satisfies Gaussian distribution when time delay is big enough. A cluster effect mechanism is presented to explain this phenomenon. By studying the shape of PDFs, that the roles played by time delay are more important than embedding dimension in the reconstruction is clearly indicated. Results have demonstrated that the PDF method represents a promising numerical approach for the observation of the reconstructed attractor structure and may provide more information and new diagnostic potential of the analyzed cardiac system.

  16. Sourcing dark matter and dark energy from α-attractors

    Mishra, Swagat S.; Sahni, Varun [Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007 (India); Shtanov, Yuri, E-mail: swagat@iucaa.in, E-mail: varun@iucaa.in, E-mail: shtanov@bitp.kiev.ua [Bogolyubov Institute for Theoretical Physics, Kiev 03680 (Ukraine)

    2017-06-01

    In [1], Kallosh and Linde drew attention to a new family of superconformal inflationary potentials, subsequently called α-attractors [2]. The α-attractor family can interpolate between a large class of inflationary models. It also has an important theoretical underpinning within the framework of supergravity. We demonstrate that the α-attractors have an even wider appeal since they may describe dark matter and perhaps even dark energy. The dark matter associated with the α-attractors, which we call α-dark matter (αDM), shares many of the attractive features of fuzzy dark matter, with V (φ) = ½ m {sup 2}φ{sup 2}, while having none of its drawbacks. Like fuzzy dark matter, αDM can have a large Jeans length which could resolve the cusp-core and substructure problems faced by standard cold dark matter. αDM also has an appealing tracker property which enables it to converge to the late-time dark matter asymptote, ( w ) ≅ 0, from a wide range of initial conditions. It thus avoids the enormous fine-tuning problems faced by the m {sup 2}φ{sup 2} potential in describing dark matter.

  17. Our universe as an attractor in a superstring model

    Maeda, Keiichi.

    1986-11-01

    One preferential scenario of the evolution of the universe is discussed in a superstring model. The universe can reach the present state as an attractor in the dynamical system. The kinetic terms of the ''axions'' play an important role so that our present universe is realized almost uniquely. (author)

  18. Attractor horizons in six-dimensional type IIB supergravity

    Astefanesei, Dumitru, E-mail: dumitru.astefanesei@ucv.cl [Instituto de Fisica, Pontificia Universidad Catolica de Valparaiso, Casilla 4059, Valparaiso (Chile); Miskovic, Olivera, E-mail: olivera.miskovic@ucv.cl [Instituto de Fisica, Pontificia Universidad Catolica de Valparaiso, Casilla 4059, Valparaiso (Chile); Olea, Rodrigo, E-mail: rodrigo.olea@unab.cl [Universidad Andres Bello, Departamento de Ciencias Fisicas, Republica 220, Santiago (Chile)

    2012-08-14

    We consider near horizon geometries of extremal black holes in six-dimensional type IIB supergravity. In particular, we use the entropy function formalism to compute the charges and thermodynamic entropy of these solutions. We also comment on the role of attractor mechanism in understanding the entropy of the Hopf T-dual solutions in type IIA supergravity.

  19. COMPARISON OF CHAOTIC AND FRACTAL PROPERTIES OF POLAR FACULAE WITH SUNSPOT ACTIVITY

    Deng, L. H.; Xiang, Y. Y.; Dun, G. T. [Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216 (China); Li, B., E-mail: wooden@escience.cn [Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and Physics, Shandong University at Weihai, Weihai 264209 (China)

    2016-01-15

    The solar magnetic activity is governed by a complex dynamo mechanism and exhibits a nonlinear dissipation behavior in nature. The chaotic and fractal properties of solar time series are of great importance to understanding the solar dynamo actions, especially with regard to the nonlinear dynamo theories. In the present work, several nonlinear analysis approaches are proposed to investigate the nonlinear dynamical behavior of the polar faculae and sunspot activity for the time interval from 1951 August to 1998 December. The following prominent results are found: (1) both the high- and the low-latitude solar activity are governed by a three-dimensional chaotic attractor, and the chaotic behavior of polar faculae is the most complex, followed by that of the sunspot areas, and then the sunspot numbers; (2) both the high- and low-latitude solar activity exhibit a high degree of persistent behavior, and their fractal nature is due to such long-range correlation; (3) the solar magnetic activity cycle is predictable in nature, but the high-accuracy prediction should only be done for short- to mid-term due to its intrinsically dynamical complexity. With the help of the Babcock–Leighton dynamo model, we suggest that the nonlinear coupling of the polar magnetic fields with strong active-region fields exhibits a complex manner, causing the statistical similarities and differences between the polar faculae and the sunspot-related indicators.

  20. Power Forecasting of Combined Heating and Cooling Systems Based on Chaotic Time Series

    Liu Hai

    2015-01-01

    Full Text Available Theoretic analysis shows that the output power of the distributed generation system is nonlinear and chaotic. And it is coupled with the microenvironment meteorological data. Chaos is an inherent property of nonlinear dynamic system. A predicator of the output power of the distributed generation system is to establish a nonlinear model of the dynamic system based on real time series in the reconstructed phase space. Firstly, chaos should be detected and quantified for the intensive studies of nonlinear systems. If the largest Lyapunov exponent is positive, the dynamical system must be chaotic. Then, the embedding dimension and the delay time are chosen based on the improved C-C method. The attractor of chaotic power time series can be reconstructed based on the embedding dimension and delay time in the phase space. By now, the neural network can be trained based on the training samples, which are observed from the distributed generation system. The neural network model will approximate the curve of output power adequately. Experimental results show that the maximum power point of the distributed generation system will be predicted based on the meteorological data. The system can be controlled effectively based on the prediction.

  1. Applications of Chaotic Dynamics in Robotics

    Xizhe Zang

    2016-03-01

    Full Text Available This article presents a summary of applications of chaos and fractals in robotics. Firstly, basic concepts of deterministic chaos and fractals are discussed. Then, fundamental tools of chaos theory used for identifying and quantifying chaotic dynamics will be shared. Principal applications of chaos and fractal structures in robotics research, such as chaotic mobile robots, chaotic behaviour exhibited by mobile robots interacting with the environment, chaotic optimization algorithms, chaotic dynamics in bipedal locomotion and fractal mechanisms in modular robots will be presented. A brief survey is reported and an analysis of the reviewed publications is also presented.

  2. ?Strange Attractors (chaos) in the hydro-climatology of Colombia?

    Poveda Jaramillo, German

    1997-01-01

    Inter annual hydro-climatology of Colombia is strongly influenced by extreme phases of ENSO, a phenomenon exhibiting many features of chaotic non-linear system. The possible chaotic nature of Colombian hydrology is examined by using time series of monthly precipitation at Bogota (1866-1992) and Medellin (1908-1995), and average stream flows of the Magdalena River at Puerto Berrio. The power spectrum, the Haussdorf-Besikovich (fractal) dimension, the correlation dimension, and the largest Lyapunov exponent are estimated for the time series. Ideas of hydrologic forecasting and predictability are discussed in the context of nonlinear dynamical systems exhibit chaotic behavior

  3. Dynamic Parameter-Control Chaotic System.

    Hua, Zhongyun; Zhou, Yicong

    2016-12-01

    This paper proposes a general framework of 1-D chaotic maps called the dynamic parameter-control chaotic system (DPCCS). It has a simple but effective structure that uses the outputs of a chaotic map (control map) to dynamically control the parameter of another chaotic map (seed map). Using any existing 1-D chaotic map as the control/seed map (or both), DPCCS is able to produce a huge number of new chaotic maps. Evaluations and comparisons show that chaotic maps generated by DPCCS are very sensitive to their initial states, and have wider chaotic ranges, better unpredictability and more complex chaotic behaviors than their seed maps. Using a chaotic map of DPCCS as an example, we provide a field-programmable gate array design of this chaotic map to show the simplicity of DPCCS in hardware implementation, and introduce a new pseudo-random number generator (PRNG) to investigate the applications of DPCCS. Analysis and testing results demonstrate the excellent randomness of the proposed PRNG.

  4. Visibility graphlet approach to chaotic time series

    Mutua, Stephen [Business School, University of Shanghai for Science and Technology, Shanghai 200093 (China); Computer Science Department, Masinde Muliro University of Science and Technology, P.O. Box 190-50100, Kakamega (Kenya); Gu, Changgui, E-mail: gu-changgui@163.com, E-mail: hjyang@ustc.edu.cn; Yang, Huijie, E-mail: gu-changgui@163.com, E-mail: hjyang@ustc.edu.cn [Business School, University of Shanghai for Science and Technology, Shanghai 200093 (China)

    2016-05-15

    Many novel methods have been proposed for mapping time series into complex networks. Although some dynamical behaviors can be effectively captured by existing approaches, the preservation and tracking of the temporal behaviors of a chaotic system remains an open problem. In this work, we extended the visibility graphlet approach to investigate both discrete and continuous chaotic time series. We applied visibility graphlets to capture the reconstructed local states, so that each is treated as a node and tracked downstream to create a temporal chain link. Our empirical findings show that the approach accurately captures the dynamical properties of chaotic systems. Networks constructed from periodic dynamic phases all converge to regular networks and to unique network structures for each model in the chaotic zones. Furthermore, our results show that the characterization of chaotic and non-chaotic zones in the Lorenz system corresponds to the maximal Lyapunov exponent, thus providing a simple and straightforward way to analyze chaotic systems.

  5. Advances and applications in chaotic systems

    Volos, Christos

    2016-01-01

    This book reports on the latest advances and applications of chaotic systems. It consists of 25 contributed chapters by experts who are specialized in the various topics addressed in this book. The chapters cover a broad range of topics of chaotic systems such as chaos, hyperchaos, jerk systems, hyperjerk systems, conservative and dissipative systems, circulant chaotic systems, multi-scroll chaotic systems, finance chaotic system, highly chaotic systems, chaos control, chaos synchronization, circuit realization and applications of chaos theory in secure communications, mobile robot, memristors, cellular neural networks, etc. Special importance was given to chapters offering practical solutions, modeling and novel control methods for the recent research problems in chaos theory. This book will serve as a reference book for graduate students and researchers with a basic knowledge of chaos theory and control systems. The resulting design procedures on the chaotic systems are emphasized using MATLAB software.

  6. COSMOS-e{sup '}-soft Higgsotic attractors

    Choudhury, Sayantan [Tata Institute of Fundamental Research, Department of Theoretical Physics, Mumbai (India)

    2017-07-15

    In this work, we have developed an elegant algorithm to study the cosmological consequences from a huge class of quantum field theories (i.e. superstring theory, supergravity, extra dimensional theory, modified gravity, etc.), which are equivalently described by soft attractors in the effective field theory framework. In this description we have restricted our analysis for two scalar fields - dilaton and Higgsotic fields minimally coupled with Einstein gravity, which can be generalized for any arbitrary number of scalar field contents with generalized non-canonical and non-minimal interactions. We have explicitly used R{sup 2} gravity, from which we have studied the attractor and non-attractor phases by exactly computing two point, three point and four point correlation functions from scalar fluctuations using the In-In (Schwinger-Keldysh) and the δN formalisms. We have also presented theoretical bounds on the amplitude, tilt and running of the primordial power spectrum, various shapes (equilateral, squeezed, folded kite or counter-collinear) of the amplitude as obtained from three and four point scalar functions, which are consistent with observed data. Also the results from two point tensor fluctuations and the field excursion formula are explicitly presented for the attractor and non-attractor phase. Further, reheating constraints, scale dependent behavior of the couplings and the dynamical solution for the dilaton and Higgsotic fields are also presented. New sets of consistency relations between two, three and four point observables are also presented, which shows significant deviation from canonical slow-roll models. Additionally, three possible theoretical proposals have presented to overcome the tachyonic instability at the time of late time acceleration. Finally, we have also provided the bulk interpretation from the three and four point scalar correlation functions for completeness. (orig.)

  7. The 2003 Merged Model for Vietnam

    Jensen, Henning Tarp; Tarp, Finn

    This monograph documents the 2003 Merged Model for Vietnam. The initialization and calibration of the model is based on a financial 2003 SAM framework and an auxiliary 2002-3 data set. The recursive nature of the solution of the Merged Model is discussed with reference to the four main sectors...

  8. Neural network modeling of chaotic dynamics in nuclear reactor flows

    Welstead, S.T.

    1992-01-01

    Neural networks have many scientific applications in areas such as pattern classification and time series prediction. The universal approximation property of these networks, however, can also be exploited to provide researchers with tool for modeling observed nonlinear phenomena. It has been shown that multilayer feed forward networks can capture important global nonlinear properties, such as chaotic dynamics, merely by training the network on a finite set of observed data. The network itself then provides a model of the process that generated the data. Characterizations such as the existence and general shape of a strange attractor and the sign of the largest Lyapunov exponent can then be extracted from the neural network model. In this paper, the author applies this idea to data generated from a nonlinear process that is representative of convective flows that can arise in nuclear reactor applications. Such flows play a role in forced convection heat removal from pressurized water reactors and boiling water reactors, and decay heat removal from liquid-metal-cooled reactors, either by natural convection or by thermosyphons

  9. BNFL/Siemens to merge?

    Anon.

    1997-01-01

    Negotiations are being conducted on the creation of a joint venture between British Nuclear Fuels Limited (BNFL) and the German company Siemens. The venture would merge Siemens' nuclear fuel, engineering and construction services businesses with BNFL's Magnox and AGR fuel fabrication business which is based at Springfields in Lancashire. It would incorporate Siemens' share of Nuclear Power International, a joint venture with the French company Framatone which is developing the European Pressurised Water Reactor project, and its US subsidiary Siemens' Power Corp. BNFL's mixed oxide (MOX) fuel fabrication business, its stake in the Dutch-German-UK uranium enrichment company Urenco and its nuclear waste reprocessing business would not be involved. There is already speculation that Siemens' greater input will lead to it taking a majority stake in what will be the world's second largest nuclear fuel manufacturer. Reaction to the news is reported. This has been muted in the United Kingdom, mixed in Germany and adverse in France because of the implications for the Siemens' Framatone collaboration. (UK)

  10. Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems

    Hu Manfeng; Xu Zhenyuan; Zhang Rong; Hu Aihua

    2007-01-01

    Based on the active control idea and the invariance principle of differential equations, a general scheme of adaptive full state hybrid projective synchronization (FSHPS) and parameters identification of a class of chaotic (hyper-chaotic) systems with linearly dependent uncertain parameters is proposed in this Letter. With this effective scheme parameters identification and FSHPS of chaotic and hyper-chaotic systems can be realized simultaneously. Numerical simulations on the chaotic Chen system and the hyper-chaotic Chen system are presented to verify the effectiveness of the proposed scheme

  11. Time-varying mixed logit model for vehicle merging behavior in work zone merging areas.

    Weng, Jinxian; Du, Gang; Li, Dan; Yu, Yao

    2018-08-01

    This study aims to develop a time-varying mixed logit model for the vehicle merging behavior in work zone merging areas during the merging implementation period from the time of starting a merging maneuver to that of completing the maneuver. From the safety perspective, vehicle crash probability and severity between the merging vehicle and its surrounding vehicles are regarded as major factors influencing vehicle merging decisions. Model results show that the model with the use of vehicle crash risk probability and severity could provide higher prediction accuracy than previous models with the use of vehicle speeds and gap sizes. It is found that lead vehicle type, through lead vehicle type, through lag vehicle type, crash probability of the merging vehicle with respect to the through lag vehicle, crash severities of the merging vehicle with respect to the through lead and lag vehicles could exhibit time-varying effects on the merging behavior. One important finding is that the merging vehicle could become more and more aggressive in order to complete the merging maneuver as quickly as possible over the elapsed time, even if it has high vehicle crash risk with respect to the through lead and lag vehicles. Copyright © 2018 Elsevier Ltd. All rights reserved.

  12. Illusion optics in chaotic light

    Zhang Suheng; Gan Shu; Xiong Jun; Zhang Xiangdong; Wang Kaige

    2010-01-01

    The time-reversal process provides the possibility to counteract the time evolution of a physical system. Recent research has shown that such a process can occur in the first-order field correlation of chaotic light and result in the spatial interference and phase-reversal diffraction in an unbalanced interferometer. Here we report experimental investigations on the invisibility cloak and illusion phenomena in chaotic light. In an unbalanced interferometer illuminated by thermal light, we have observed the cloak effect and the optical transformation of one object into another object. The experimental results can be understood by the phase-reversal diffraction, and they demonstrate the theoretical proposal of similar effects in complementary media.

  13. Chaotic evolution of arms races

    Tomochi, Masaki; Kono, Mitsuo

    1998-12-01

    A new set of model equations is proposed to describe the evolution of the arms race, by extending Richardson's model with special emphases that (1) power dependent defensive reaction or historical enmity could be a motive force to promote armaments, (2) a deterrent would suppress the growth of armaments, and (3) the defense reaction of one nation against the other nation depends nonlinearly on the difference in armaments between two. The set of equations is numerically solved to exhibit stationary, periodic, and chaotic behavior depending on the combinations of parameters involved. The chaotic evolution is realized when the economic situation of each country involved in the arms race is quite different, which is often observed in the real world.

  14. Linear feedback control, adaptive feedback control and their combination for chaos (lag) synchronization of LC chaotic systems

    Yan Zhenya; Yu Pei

    2007-01-01

    In this paper, we study chaos (lag) synchronization of a new LC chaotic system, which can exhibit not only a two-scroll attractor but also two double-scroll attractors for different parameter values, via three types of state feedback controls: (i) linear feedback control; (ii) adaptive feedback control; and (iii) a combination of linear feedback and adaptive feedback controls. As a consequence, ten families of new feedback control laws are designed to obtain global chaos lag synchronization for τ < 0 and global chaos synchronization for τ = 0 of the LC system. Numerical simulations are used to illustrate these theoretical results. Each family of these obtained feedback control laws, including two linear (adaptive) functions or one linear function and one adaptive function, is added to two equations of the LC system. This is simpler than the known synchronization controllers, which apply controllers to all equations of the LC system. Moreover, based on the obtained results of the LC system, we also derive the control laws for chaos (lag) synchronization of another new type of chaotic system

  15. Anomalous diffusion in chaotic scattering

    Srokowski, T.; Ploszajczak, M.

    1994-01-01

    The anomalous diffusion is found for peripheral collision of atomic nuclei described in the framework of the molecular dynamics. Similarly as for chaotic billiards, the long free paths are the source of the long-time correlations and the anomalous diffusion. Consequences of this finding for the energy dissipation in deep-inelastic collisions and the dynamics of fission in hot nuclei are discussed (authors). 30 refs., 2 figs

  16. CHAOTIC CAPTURE OF NEPTUNE TROJANS

    Nesvorny, David; Vokrouhlicky, David

    2009-01-01

    Neptune Trojans (NTs) are swarms of outer solar system objects that lead/trail planet Neptune during its revolutions around the Sun. Observations indicate that NTs form a thick cloud of objects with a population perhaps ∼10 times more numerous than that of Jupiter Trojans and orbital inclinations reaching ∼25 deg. The high inclinations of NTs are indicative of capture instead of in situ formation. Here we study a model in which NTs were captured by Neptune during planetary migration when secondary resonances associated with the mean-motion commensurabilities between Uranus and Neptune swept over Neptune's Lagrangian points. This process, known as chaotic capture, is similar to that previously proposed to explain the origin of Jupiter's Trojans. We show that chaotic capture of planetesimals from an ∼35 Earth-mass planetesimal disk can produce a population of NTs that is at least comparable in number to that inferred from current observations. The large orbital inclinations of NTs are a natural outcome of chaotic capture. To obtain the ∼4:1 ratio between high- and low-inclination populations suggested by observations, planetary migration into a dynamically excited planetesimal disk may be required. The required stirring could have been induced by Pluto-sized and larger objects that have formed in the disk.

  17. Chaotic dynamics of respiratory sounds

    Ahlstrom, C.; Johansson, A.; Hult, P.; Ask, P.

    2006-01-01

    There is a growing interest in nonlinear analysis of respiratory sounds (RS), but little has been done to justify the use of nonlinear tools on such data. The aim of this paper is to investigate the stationarity, linearity and chaotic dynamics of recorded RS. Two independent data sets from 8 + 8 healthy subjects were recorded and investigated. The first set consisted of lung sounds (LS) recorded with an electronic stethoscope and the other of tracheal sounds (TS) recorded with a contact accelerometer. Recurrence plot analysis revealed that both LS and TS are quasistationary, with the parts corresponding to inspiratory and expiratory flow plateaus being stationary. Surrogate data tests could not provide statistically sufficient evidence regarding the nonlinearity of the data. The null hypothesis could not be rejected in 4 out of 32 LS cases and in 15 out of 32 TS cases. However, the Lyapunov spectra, the correlation dimension (D 2 ) and the Kaplan-Yorke dimension (D KY ) all indicate chaotic behavior. The Lyapunov analysis showed that the sum of the exponents was negative in all cases and that the largest exponent was found to be positive. The results are partly ambiguous, but provide some evidence of chaotic dynamics of RS, both concerning LS and TS. The results motivate continuous use of nonlinear tools for analysing RS data

  18. Chaotic dynamics of respiratory sounds

    Ahlstrom, C. [Department of Biomedical Engineering, Linkoepings Universitet, IMT/LIU, Universitetssjukhuset, S-58185 Linkoeping (Sweden) and Biomedical Engineering, Orebro University Hospital, S-70185 Orebro (Sweden)]. E-mail: christer@imt.liu.se; Johansson, A. [Department of Biomedical Engineering, Linkoepings Universitet, IMT/LIU, Universitetssjukhuset, S-58185 Linkoeping (Sweden); Hult, P. [Department of Biomedical Engineering, Linkoepings Universitet, IMT/LIU, Universitetssjukhuset, S-58185 Linkoeping (Sweden); Biomedical Engineering, Orebro University Hospital, S-70185 Orebro (Sweden); Ask, P. [Department of Biomedical Engineering, Linkoepings Universitet, IMT/LIU, Universitetssjukhuset, S-58185 Linkoeping (Sweden); Biomedical Engineering, Orebro University Hospital, S-70185 Orebro (Sweden)

    2006-09-15

    There is a growing interest in nonlinear analysis of respiratory sounds (RS), but little has been done to justify the use of nonlinear tools on such data. The aim of this paper is to investigate the stationarity, linearity and chaotic dynamics of recorded RS. Two independent data sets from 8 + 8 healthy subjects were recorded and investigated. The first set consisted of lung sounds (LS) recorded with an electronic stethoscope and the other of tracheal sounds (TS) recorded with a contact accelerometer. Recurrence plot analysis revealed that both LS and TS are quasistationary, with the parts corresponding to inspiratory and expiratory flow plateaus being stationary. Surrogate data tests could not provide statistically sufficient evidence regarding the nonlinearity of the data. The null hypothesis could not be rejected in 4 out of 32 LS cases and in 15 out of 32 TS cases. However, the Lyapunov spectra, the correlation dimension (D {sub 2}) and the Kaplan-Yorke dimension (D {sub KY}) all indicate chaotic behavior. The Lyapunov analysis showed that the sum of the exponents was negative in all cases and that the largest exponent was found to be positive. The results are partly ambiguous, but provide some evidence of chaotic dynamics of RS, both concerning LS and TS. The results motivate continuous use of nonlinear tools for analysing RS data.

  19. Reconstruction of the El Nino attractor with neural networks

    Grieger, B.; Latif, M.

    1993-01-01

    Based on a combined data set of sea surface temperature, zonal surface wind stress and upper ocean heat content the dynamics of the El Nino phenomenon is investigated. In a reduced phase space spanned by the first four EOFs two different stochastic models are estimated from the data. A nonlinear model represented by a simulated neural network is compared with a linear model obtained with the Principal Oscillation Pattern (POP) analysis. While the linear model is limited to damped oscillations onto a fix point attractor, the nonlinear model recovers a limit cycle attractor. This indicates that the real system is located above the bifurcation point in parameter space supporting self-sustained oscillations. The results are discussed with respect to consistency with current theory. (orig.)

  20. Non-Abelian magnetized blackholes and unstable attractors

    Mosaffa, A.E.; Randjbar-Daemi, S.; Sheikh-Jabbari, M.M.

    2006-12-01

    Fluctuations of non-Abelian gauge fields in a background magnetic flux contain tachyonic modes and hence the background is unstable. We extend these results to the cases where the background flux is coupled to Einstein gravity and show that the corresponding spherically symmetric geometries, which in the absence of a cosmological constant are of the form of Reissner-Nordstroem blackholes or the AdS 2 x S 2 , are also unstable. We discuss the relevance of these instabilities to several places in string theory including various string compactifications and the attractor mechanism. Our results for the latter imply that the attractor mechanism shown to work for the extremal Abelian charged blackholes, cannot be applied in a straightforward way to the extremal non-Abelian colored blackholes. (author)

  1. Attractors of dissipative structure in three dissipative fluids

    Kondoh, Yoshiomi

    1993-10-01

    A general theory with use of auto-correlations for distributions is presented to derive that realization of coherent structures in general dissipative dynamic systems is equivalent to that of self-organized states with the minimum dissipation rate for instantaneously contained energy. Attractors of dissipative structure are shown to be given by eigenfunctions for dissipative dynamic operators of the dynamic system and to constitute the self-organized and self-similar decay phase. Three typical examples applied to incompressible viscous fluids, to incompressible viscous and resistive magnetohydrodynamic (MHD) fluids and to compressible resistive MHD plasmas are presented to lead to attractors in the three dissipative fluids and to describe a common physical picture of self-organization and bifurcation of the dissipative structure. (author)

  2. Torus-doubling process via strange nonchaotic attractors

    Mitsui, Takahito; Uenohara, Seiji; Morie, Takashi; Horio, Yoshihiko; Aihara, Kazuyuki

    2012-01-01

    Torus-doubling bifurcations typically occur only a finite number of times. It has been assumed that torus-doubling bifurcations in quasiperiodically forced systems are interrupted by the appearance of strange nonchaotic attractors (SNAs). In the present Letter, we study a quasiperiodically forced noninvertible map and report the occurrence of a torus-doubling process via SNAs. The mechanism of this process is numerically clarified. Furthermore, this process is experimentally demonstrated in a switched-capacitor integrated circuit. -- Highlights: ► We report the occurrence of a torus-doubling process via strange nonchaotic attractors (SNAs). ► The process consists of the gradual fractalization of a torus and the Heagy–Hammel transition. ► The torus-doubling process via SNAs is also experimentally demonstrated in an electronic circuit.

  3. Logical Attractors: a Boolean Approach to the Dynamics of Psychosis

    Kupper, Z.; Hoffmann, H.

    A Boolean modeling approach to attractors in the dynamics of psychosis is presented: Kinetic Logic, originating from R. Thomas, describes systems on an intermediate level between a purely verbal, qualitative description and a description using nonlinear differential equations. With this method we may model impact, feedback and temporal evolution, as well as analyze the resulting attractors. In our previous research the method has been applied to general and more specific questions in the dynamics of psychotic disorders. In this paper a model is introduced that describes different dynamical patterns of chronic psychosis in the context of vocational rehabilitation. It also shows to be useful in formulating and exploring possible treatment strategies. Finally, some of the limitations and benefits of Kinetic Logic as a modeling tool for psychology and psychiatry are discussed.

  4. A Hyperchaotic Attractor with Multiple Positive Lyapunov Exponents

    Guo-Si, Hu

    2009-01-01

    There are many hyperchaotic systems, but few systems can generate hyperchaotic attractors with more than three PLEs (positive Lyapunov exponents). A new hyperchaotic system, constructed by adding an approximate time-delay state feedback to a five-dimensional hyperchaotic system, is presented. With the increasing number of phase-shift units used in this system, the number of PLEs also steadily increases. Hyperchaotic attractors with 25 PLEs can be generated by this system with 32 phase-shift units. The sum of the PLEs will reach the maximum value when 23 phase-shift units are used. A simple electronic circuit, consisting of 16 operational amplifiers and two analogy multipliers, is presented for confirming hyperchaos of order 5, i.e., with 5 PLEs

  5. Attractor reconstruction for non-linear systems: a methodological note

    Nichols, J.M.; Nichols, J.D.

    2001-01-01

    Attractor reconstruction is an important step in the process of making predictions for non-linear time-series and in the computation of certain invariant quantities used to characterize the dynamics of such series. The utility of computed predictions and invariant quantities is dependent on the accuracy of attractor reconstruction, which in turn is determined by the methods used in the reconstruction process. This paper suggests methods by which the delay and embedding dimension may be selected for a typical delay coordinate reconstruction. A comparison is drawn between the use of the autocorrelation function and mutual information in quantifying the delay. In addition, a false nearest neighbor (FNN) approach is used in minimizing the number of delay vectors needed. Results highlight the need for an accurate reconstruction in the computation of the Lyapunov spectrum and in prediction algorithms.

  6. Sequences by Metastable Attractors: Interweaving Dynamical Systems and Experimental Data

    Axel Hutt

    2017-05-01

    Full Text Available Metastable attractors and heteroclinic orbits are present in the dynamics of various complex systems. Although their occurrence is well-known, their identification and modeling is a challenging task. The present work reviews briefly the literature and proposes a novel combination of their identification in experimental data and their modeling by dynamical systems. This combination applies recurrence structure analysis permitting the derivation of an optimal symbolic representation of metastable states and their dynamical transitions. To derive heteroclinic sequences of metastable attractors in various experimental conditions, the work introduces a Hausdorff clustering algorithm for symbolic dynamics. The application to brain signals (event-related potentials utilizing neural field models illustrates the methodology.

  7. Hierarchical-control-based output synchronization of coexisting attractor networks

    Yun-Zhong, Song; Yi-Fa, Tang

    2010-01-01

    This paper introduces the concept of hierarchical-control-based output synchronization of coexisting attractor networks. Within the new framework, each dynamic node is made passive at first utilizing intra-control around its own arena. Then each dynamic node is viewed as one agent, and on account of that, the solution of output synchronization of coexisting attractor networks is transformed into a multi-agent consensus problem, which is made possible by virtue of local interaction between individual neighbours; this distributed working way of coordination is coined as inter-control, which is only specified by the topological structure of the network. Provided that the network is connected and balanced, the output synchronization would come true naturally via synergy between intra and inter-control actions, where the Tightness is proved theoretically via convex composite Lyapunov functions. For completeness, several illustrative examples are presented to further elucidate the novelty and efficacy of the proposed scheme. (general)

  8. Inflationary α -attractor cosmology: A global dynamical systems perspective

    Alho, Artur; Uggla, Claes

    2017-04-01

    We study flat Friedmann-Lemaître-Robertson-Walker α -attractor E- and T-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimensional compact state spaces. This results in both illustrative figures and a complete description of the entire solution spaces of these models, including asymptotics. In particular, it is shown that observational viability, which requires a sufficient number of e -folds, is associated with a particular solution given by a one-dimensional center manifold of a past asymptotic de Sitter state, where the center manifold structure also explains why nearby solutions are attracted to this "inflationary attractor solution." A center manifold expansion yields a description of the inflationary regime with arbitrary analytic accuracy, where the slow-roll approximation asymptotically describes the tangency condition of the center manifold at the asymptotic de Sitter state.

  9. Evidence of deterministic components in the apparent randomness of GRBs: clues of a chaotic dynamic.

    Greco, G; Rosa, R; Beskin, G; Karpov, S; Romano, L; Guarnieri, A; Bartolini, C; Bedogni, R

    2011-01-01

    Prompt γ-ray emissions from gamma-ray bursts (GRBs) exhibit a vast range of extremely complex temporal structures with a typical variability time-scale significantly short - as fast as milliseconds. This work aims to investigate the apparent randomness of the GRB time profiles making extensive use of nonlinear techniques combining the advanced spectral method of the Singular Spectrum Analysis (SSA) with the classical tools provided by the Chaos Theory. Despite their morphological complexity, we detect evidence of a non stochastic short-term variability during the overall burst duration - seemingly consistent with a chaotic behavior. The phase space portrait of such variability shows the existence of a well-defined strange attractor underlying the erratic prompt emission structures. This scenario can shed new light on the ultra-relativistic processes believed to take place in GRB explosions and usually associated with the birth of a fast-spinning magnetar or accretion of matter onto a newly formed black hole.

  10. Parametric, nonparametric and parametric modelling of a chaotic circuit time series

    Timmer, J.; Rust, H.; Horbelt, W.; Voss, H. U.

    2000-09-01

    The determination of a differential equation underlying a measured time series is a frequently arising task in nonlinear time series analysis. In the validation of a proposed model one often faces the dilemma that it is hard to decide whether possible discrepancies between the time series and model output are caused by an inappropriate model or by bad estimates of parameters in a correct type of model, or both. We propose a combination of parametric modelling based on Bock's multiple shooting algorithm and nonparametric modelling based on optimal transformations as a strategy to test proposed models and if rejected suggest and test new ones. We exemplify this strategy on an experimental time series from a chaotic circuit where we obtain an extremely accurate reconstruction of the observed attractor.

  11. Tunable power law in the desynchronization events of coupled chaotic electronic circuits

    Oliveira, Gilson F. de; Lorenzo, Orlando di; Chevrollier, Martine; Passerat de Silans, Thierry; Oriá, Marcos; Souza Cavalcante, Hugo L. D. de

    2014-01-01

    We study the statistics of the amplitude of the synchronization error in chaotic electronic circuits coupled through linear feedback. Depending on the coupling strength, our system exhibits three qualitatively different regimes of synchronization: weak coupling yields independent oscillations; moderate to strong coupling produces a regime of intermittent synchronization known as attractor bubbling; and stronger coupling produces complete synchronization. In the regime of moderate coupling, the probability distribution for the sizes of desynchronization events follows a power law, with an exponent that can be adjusted by changing the coupling strength. Such power-law distributions are interesting, as they appear in many complex systems. However, most of the systems with such a behavior have a fixed value for the exponent of the power law, while here we present an example of a system where the exponent of the power law is easily tuned in real time

  12. Phase synchronisation in mutually coupled chaotic Josephson junctions: Effect of asymmetry and incommensurate frequencies

    Al-Khawaja, S.

    2011-01-01

    In this paper, synchronising two coupled ratchet Josephson junctions subjected to a quasiperiodic field is achieved. In the limit of weak perturbation of irrational frequencies equal to the square root of the transcendental number π and for small damping parameters, phase locking occurs as the coupling between both junctions is increased. It turns out that the transition from non-synchronous to synchronous chaotic state does not involve attractors appearing and disappearing. The undertaken symmetry analysis of the system demonstrates the suppression of the massive phase fluctuations as the coupling rises, allowing chaos synchronisation between both junctions to take place. The calculations also reveal the persistence of the synchronous state for high coupling strengths, taking into consideration the symmetry particularity of the external drive and potential. (author)

  13. Phase Synchronisation in Mutually Coupled Chaotic Josephson Junctions: Effect of Asymmetry and Incommensurate Frequencies

    Sameer Al-Khawaja

    2010-01-01

    In this paper, synchronising two coupled ratchet Josephson junctions subjected to a quasiperiodic field is achieved. In the limit of weak perturbation of irrational frequencies equal to the square root of the transcendental number π and for small damping parameters, phase locking occurs as the coupling between both junctions is increased. It turns out that the transition from non-synchronous to synchronous chaotic state does not involve attractors appearing and disappearing. The undertaken symmetry analysis of the system demonstrates the suppression of the massive phase fluctuations as the coupling rises, allowing chaos synchronisation between both junctions to take place. The calculations also reveal the persistence of the synchronous state for high coupling strengths, taking into consideration the symmetry particularity of the external drive and potential. (author)

  14. Long time behavior and attractors for energetically insulated fluid systems

    Feireisl, Eduard

    2010-01-01

    Roč. 27, č. 4 (2010), s. 1587-1609 ISSN 1078-0947 R&D Projects: GA ČR GA201/09/0917 Institutional research plan: CEZ:AV0Z10190503 Keywords : Navier-Stokes-Fourier system * attractor * long time behavior Subject RIV: BA - General Mathematics Impact factor: 0.986, year: 2010 http://www.aimsciences.org/journals/displayArticles.jsp?paperID=5040

  15. Attractors of magnetohydrodynamic flows in an Alfvenic state

    Nunez, Manuel; Sanz, Javier [Departamento de Analisis Matematico, Universidad de Valladolid, Valladolid (Spain)

    1999-08-13

    We present a simplified form of the magnetohydrodynamic system which describes the evolution of a plasma where the small-scale velocity and magnetic field are aligned in the form of Alfven waves, such as happens in several turbulent situations. Bounds on the dimension of the global attractor are found, and are shown to be an improvement of the standard ones for the full magnetohydrodynamic equations. (author)

  16. Robust synchronization of chaotic systems via feedback

    Femat, Ricardo [IPICYT, San Luis Potosi (Mexico). Dept. de Matematicas Aplicadas; Solis-Perales, Gualberto [Universidad de Guadalajara, Centro Univ. de Ciencias Exactas e Ingenierias (Mexico). Div. de Electronica y Computacion

    2008-07-01

    This volume includes the results derived during last ten years about both suppression and synchronization of chaotic -continuous time- systems. Along this time, the concept was to study how the intrinsic properties of dynamical systems can be exploited to suppress and to synchronize the chaotic behaviour and what synchronization phenomena can be found under feedback interconnection. A compilation of these findings is described in this book. This book shows a perspective on synchronization of chaotic systems. (orig.)

  17. Global attractors and extinction dynamics of cyclically competing species.

    Rulands, Steffen; Zielinski, Alejandro; Frey, Erwin

    2013-05-01

    Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases.

  18. On the renormalization group perspective of α-attractors

    Narain, Gaurav, E-mail: gaunarain@itp.ac.cn [Kavli Institute for Theoretical Physics China (KITPC), Key Laboratory of Theoretical Physics, Institute of Theoretical Physics (ITP), Chinese Academy of Sciences -CAS, Beijing 100190 (China)

    2017-10-01

    In this short paper we outline a recipe for the reconstruction of F ( R ) gravity starting from single field inflationary potentials in the Einstein frame. For simple potentials one can compute the explicit form of F ( R ), whilst for more involved examples one gets a parametric form of F ( R ). The F ( R ) reconstruction algorithm is used to study various examples: power-law φ {sup n} , exponential and α -attractors. In each case it is seen that for large R (corresponding to large value of inflaton field), F ( R ) ∼ R {sup 2}. For the case of α -attractors F ( R ) ∼ R {sup 2} for all values of inflaton field (for all values of R ) as α → 0. For generic inflaton potential V (φ), it is seen that if V {sup '}/ V →0 (for some φ) then the corresponding F ( R ) ∼ R {sup 2}. We then study α-attractors in more detail using non-perturbative renormalisation group methods to analyse the reconstructed F ( R ). It is seen that α →0 is an ultraviolet stable fixed point of the renormalisation group trajectories.

  19. Topological and metric properties of Henon-type strange attractors

    Cvitanovic, P.; Gunaratne, G.H.; Procaccia, I.

    1988-01-01

    We use the set of all periodic points of Henon-type mappings to develop a theory of the topological and metric properties of their attractors. The topology of a Henon-type attractor is conveniently represented by a two-dimensional symbol plane, with the allowed and disallowed orbits cleanly separated by the ''pruning front.'' The pruning front is a function discontinuous on every binary rational number, but for maps with finite dissipation chemical bondbchemical bond<1, it is well approximated by a few steps, or, in the symbolic dynamics language, by a finite grammar. Thus equipped with the complete list of allowed periodic points, we reconstruct (to resolution of order b/sup n/) the physical attractor by piecing together the linearized neighborhoods of all periodic points of cycle length n. We use this representation to compute the singularity spectrum f(α). The description in terms of periodic points works very well in the ''hyperbolic phase,'' for α larger than some α/sub c/, where α/sub c/ is the value of α corresponding to the (conjectured) phase transition

  20. Investigating parameters participating in the infant respiratory control system attractor.

    Terrill, Philip I; Wilson, Stephen J; Suresh, Sadasivam; Cooper, David M; Dakin, Carolyn

    2008-01-01

    Theoretically, any participating parameter in a non-linear system represents the dynamics of the whole system. Taken's time delay embedding theory provides the fundamental basis for allowing non-linear analysis to be performed on physiological, time-series data. In practice, only one measurable parameter is required to be measured to convey an accurate representation of the system dynamics. In this paper, the infant respiratory control system is represented using three variables-a digitally sampled respiratory inductive plethysmography waveform, and the derived parameters tidal volume and inter-breath interval time series data. For 14 healthy infants, these data streams were analysed using recurrence plot analysis across one night of sleep. The measured attractor size of these variables followed the same qualitative trends across the nights study. Results suggest that the attractor size measures of the derived IBI and tidal volume are representative surrogates for the raw respiratory waveform. The extent to which the relative attractor sizes of IBI and tidal volume remain constant through changing sleep state could potentially be used to quantify pathology, or maturation of breathing control.

  1. Entropies from Markov Models as Complexity Measures of Embedded Attractors

    Julián D. Arias-Londoño

    2015-06-01

    Full Text Available This paper addresses the problem of measuring complexity from embedded attractors as a way to characterize changes in the dynamical behavior of different types of systems with a quasi-periodic behavior by observing their outputs. With the aim of measuring the stability of the trajectories of the attractor along time, this paper proposes three new estimations of entropy that are derived from a Markov model of the embedded attractor. The proposed estimators are compared with traditional nonparametric entropy measures, such as approximate entropy, sample entropy and fuzzy entropy, which only take into account the spatial dimension of the trajectory. The method proposes the use of an unsupervised algorithm to find the principal curve, which is considered as the “profile trajectory”, that will serve to adjust the Markov model. The new entropy measures are evaluated using three synthetic experiments and three datasets of physiological signals. In terms of consistency and discrimination capabilities, the results show that the proposed measures perform better than the other entropy measures used for comparison purposes.

  2. TOWARDS THRESHOLD FREQUENCY IN CHAOTIC COLPITTS OSCILLATOR

    Lindberg, Erik; Tamasevicius, Arunas; Mykolaitis, Gytis

    2007-01-01

    A novel version of chaotic Colpitts oscillator is described. Instead of a linear loss resistor, it includes an extra inductor and diode in the collector circuit of the transistor. The modified circuit in comparison with the common Colpitts oscillator may generate chaotic oscillations at the funda......A novel version of chaotic Colpitts oscillator is described. Instead of a linear loss resistor, it includes an extra inductor and diode in the collector circuit of the transistor. The modified circuit in comparison with the common Colpitts oscillator may generate chaotic oscillations...

  3. A chaotic model for the epidemic of Ebola virus disease in West Africa (2013-2016)

    Mangiarotti, Sylvain; Peyre, Marisa; Huc, Mireille

    2016-11-01

    An epidemic of Ebola Virus Disease (EVD) broke out in Guinea in December 2013. It was only identified in March 2014 while it had already spread out in Liberia and Sierra Leone. The spill over of the disease became uncontrollable and the epidemic could not be stopped before 2016. The time evolution of this epidemic is revisited here with the global modeling technique which was designed to obtain the deterministic models from single time series. A generalized formulation of this technique for multivariate time series is introduced. It is applied to the epidemic of EVD in West Africa focusing on the period between March 2014 and January 2015, that is, before any detected signs of weakening. Data gathered by the World Health Organization, based on the official publications of the Ministries of Health of the three main countries involved in this epidemic, are considered in our analysis. Two observed time series are used: the daily numbers of infections and deaths. A four-dimensional model producing a very complex dynamical behavior is obtained. The model is tested in order to investigate its skills and drawbacks. Our global analysis clearly helps to distinguish three main stages during the epidemic. A characterization of the obtained attractor is also performed. In particular, the topology of the chaotic attractor is analyzed and a skeleton is obtained for its structure.

  4. Chaos control of Hastings–Powell model by combining chaotic motions

    Danca, Marius-F.; Chattopadhyay, Joydev

    2016-01-01

    In this paper, we propose a Parameter Switching (PS) algorithm as a new chaos control method for the Hastings–Powell (HP) system. The PS algorithm is a convergent scheme that switches the control parameter within a set of values while the controlled system is numerically integrated. The attractor obtained with the PS algorithm matches the attractor obtained by integrating the system with the parameter replaced by the averaged value of the switched parameter values. The switching rule can be applied periodically or randomly over a set of given values. In this way, every stable cycle of the HP system can be approximated if its underlying parameter value equalizes the average value of the switching values. Moreover, the PS algorithm can be viewed as a generalization of Parrondo's game, which is applied for the first time to the HP system, by showing that losing strategy can win: “losing + losing = winning.” If “loosing” is replaced with “chaos” and, “winning” with “order” (as the opposite to “chaos”), then by switching the parameter value in the HP system within two values, which generate chaotic motions, the PS algorithm can approximate a stable cycle so that symbolically one can write “chaos + chaos = regular.” Also, by considering a different parameter control, new complex dynamics of the HP model are revealed.

  5. Chaos control of Hastings–Powell model by combining chaotic motions

    Danca, Marius-F., E-mail: danca@rist.ro [Romanian Institute of Science and Technology, 400487 Cluj-Napoca (Romania); Chattopadhyay, Joydev, E-mail: joydev@isical.ac.in [Agricultural and Ecological Research Unit Indian Statistical Institute, 203, B. T. Road, Kolkata 700 108 (India)

    2016-04-15

    In this paper, we propose a Parameter Switching (PS) algorithm as a new chaos control method for the Hastings–Powell (HP) system. The PS algorithm is a convergent scheme that switches the control parameter within a set of values while the controlled system is numerically integrated. The attractor obtained with the PS algorithm matches the attractor obtained by integrating the system with the parameter replaced by the averaged value of the switched parameter values. The switching rule can be applied periodically or randomly over a set of given values. In this way, every stable cycle of the HP system can be approximated if its underlying parameter value equalizes the average value of the switching values. Moreover, the PS algorithm can be viewed as a generalization of Parrondo's game, which is applied for the first time to the HP system, by showing that losing strategy can win: “losing + losing = winning.” If “loosing” is replaced with “chaos” and, “winning” with “order” (as the opposite to “chaos”), then by switching the parameter value in the HP system within two values, which generate chaotic motions, the PS algorithm can approximate a stable cycle so that symbolically one can write “chaos + chaos = regular.” Also, by considering a different parameter control, new complex dynamics of the HP model are revealed.

  6. Chaos control of Hastings-Powell model by combining chaotic motions.

    Danca, Marius-F; Chattopadhyay, Joydev

    2016-04-01

    In this paper, we propose a Parameter Switching (PS) algorithm as a new chaos control method for the Hastings-Powell (HP) system. The PS algorithm is a convergent scheme that switches the control parameter within a set of values while the controlled system is numerically integrated. The attractor obtained with the PS algorithm matches the attractor obtained by integrating the system with the parameter replaced by the averaged value of the switched parameter values. The switching rule can be applied periodically or randomly over a set of given values. In this way, every stable cycle of the HP system can be approximated if its underlying parameter value equalizes the average value of the switching values. Moreover, the PS algorithm can be viewed as a generalization of Parrondo's game, which is applied for the first time to the HP system, by showing that losing strategy can win: "losing + losing = winning." If "loosing" is replaced with "chaos" and, "winning" with "order" (as the opposite to "chaos"), then by switching the parameter value in the HP system within two values, which generate chaotic motions, the PS algorithm can approximate a stable cycle so that symbolically one can write "chaos + chaos = regular." Also, by considering a different parameter control, new complex dynamics of the HP model are revealed.

  7. Bouncing and Merging of Liquid Jets

    Saha, Abhishek; Li, Minglei; Law, Chung K.

    2014-11-01

    Collision of two fluid jets is a technique that is utilized in many industrial applications, such as in rocket engines, to achieve controlled mixing, atomization and sometimes liquid phase reactions. Thus, the dynamics of colliding jets have direct impact on the performance, efficiency and reliability of such applications. In analogy with the dynamics of droplet-droplet collision, in this work we have experimentally demonstrated, for n-alkane hydrocarbons as well as water, that with increasing impact inertia obliquely colliding jets also exhibit the same nonmonotonic responses of merging, bouncing, merging again, and merging followed by disintegration; and that the continuous entrainment of the boundary layer air over the jet surface into the colliding interfacial region leads to two distinguishing features of jet collision, namely: there exists a maximum impact angle beyond which merging is always possible, and that merging is inhibited and then promoted with increasing pressure. These distinct response regimes were mapped and explained on the bases of impact inertia, deformation of the jet surface, viscous loss within the jet interior, and the thickness and pressure build-up within the interfacial region in order to activate the attractive surface van der Waals force to effect merging.

  8. ENHANCED OFF-CENTER STELLAR TIDAL DISRUPTIONS BY SUPERMASSIVE BLACK HOLES IN MERGING GALAXIES

    Liu, F. K.; Chen, Xian

    2013-01-01

    Off-center stellar tidal disruption flares have been suggested to be a powerful probe of recoiling supermassive black holes (SMBHs) out of galactic centers due to anisotropic gravitational wave radiations. However, off-center tidal flares can also be produced by SMBHs in merging galaxies. In this paper, we computed the tidal flare rates by dual SMBHs in two merging galaxies before the SMBHs become self-gravitationally bounded. We employ an analytical model to calculate the tidal loss-cone feeding rates for both SMBHs, taking into account two-body relaxation of stars, tidal perturbations by the companion galaxy, and chaotic stellar orbits in triaxial gravitational potential. We show that for typical SMBHs with masses 10 7 M ☉ , the loss-cone feeding rates are enhanced by mergers up to Γ ∼ 10 –2 yr –1 , about two orders of magnitude higher than those by single SMBHs in isolated galaxies and about four orders of magnitude higher than those by recoiling SMBHs. The enhancements are mainly due to tidal perturbations by the companion galaxy. We suggest that off-center tidal flares are overwhelmed by those from merging galaxies, making the identification of recoiling SMBHs challenging. Based on the calculated rates, we estimate the relative contributions of tidal flare events by single, binary, and dual SMBH systems during cosmic time. Our calculations show that the off-center tidal disruption flares by un-bound SMBHs in merging galaxies contribute a fraction comparable to that by single SMBHs in isolated galaxies. We conclude that off-center tidal disruptions are powerful tracers of the merging history of galaxies and SMBHs.

  9. Noise-induced attractor annihilation in the delayed feedback logistic map

    Pisarchik, A.N.; Martínez-Zérega, B.E.

    2013-01-01

    We study dynamics of the bistable logistic map with delayed feedback, under the influence of white Gaussian noise and periodic modulation applied to the variable. This system may serve as a model to describe population dynamics under finite resources in noisy environment with seasonal fluctuations. While a very small amount of noise has no effect on the global structure of the coexisting attractors in phase space, an intermediate noise totally eliminates one of the attractors. Slow periodic modulation enhances the attractor annihilation.

  10. Merged ozone profiles from four MIPAS processors

    Laeng, Alexandra; von Clarmann, Thomas; Stiller, Gabriele; Dinelli, Bianca Maria; Dudhia, Anu; Raspollini, Piera; Glatthor, Norbert; Grabowski, Udo; Sofieva, Viktoria; Froidevaux, Lucien; Walker, Kaley A.; Zehner, Claus

    2017-04-01

    The Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) was an infrared (IR) limb emission spectrometer on the Envisat platform. Currently, there are four MIPAS ozone data products, including the operational Level-2 ozone product processed at ESA, with the scientific prototype processor being operated at IFAC Florence, and three independent research products developed by the Istituto di Fisica Applicata Nello Carrara (ISAC-CNR)/University of Bologna, Oxford University, and the Karlsruhe Institute of Technology-Institute of Meteorology and Climate Research/Instituto de Astrofísica de Andalucía (KIT-IMK/IAA). Here we present a dataset of ozone vertical profiles obtained by merging ozone retrievals from four independent Level-2 MIPAS processors. We also discuss the advantages and the shortcomings of this merged product. As the four processors retrieve ozone in different parts of the spectra (microwindows), the source measurements can be considered as nearly independent with respect to measurement noise. Hence, the information content of the merged product is greater and the precision is better than those of any parent (source) dataset. The merging is performed on a profile per profile basis. Parent ozone profiles are weighted based on the corresponding error covariance matrices; the error correlations between different profile levels are taken into account. The intercorrelations between the processors' errors are evaluated statistically and are used in the merging. The height range of the merged product is 20-55 km, and error covariance matrices are provided as diagnostics. Validation of the merged dataset is performed by comparison with ozone profiles from ACE-FTS (Atmospheric Chemistry Experiment-Fourier Transform Spectrometer) and MLS (Microwave Limb Sounder). Even though the merging is not supposed to remove the biases of the parent datasets, around the ozone volume mixing ratio peak the merged product is found to have a smaller (up to 0.1 ppmv

  11. Complicated basins and the phenomenon of amplitude death in coupled hidden attractors

    Chaudhuri, Ushnish [Department of Physics, Sri Venkateswara College, University of Delhi, New Delhi 110021 (India); Department of Physics, National University of Singapore, Singapore 117551 (Singapore); Prasad, Awadhesh, E-mail: awadhesh@physics.du.ac.in [Department of Physics and Astrophysics, University of Delhi, Delhi 110007 (India)

    2014-02-07

    Understanding hidden attractors, whose basins of attraction do not contain the neighborhood of equilibrium of the system, are important in many physical applications. We observe riddled-like complicated basins of coexisting hidden attractors both in coupled and uncoupled systems. Amplitude death is observed in coupled hidden attractors with no fixed point using nonlinear interaction. A new route to amplitude death is observed in time-delay coupled hidden attractors. Numerical results are presented for systems with no or one stable fixed point. The applications are highlighted.

  12. An Efficient Algorithm for Computing Attractors of Synchronous And Asynchronous Boolean Networks

    Zheng, Desheng; Yang, Guowu; Li, Xiaoyu; Wang, Zhicai; Liu, Feng; He, Lei

    2013-01-01

    Biological networks, such as genetic regulatory networks, often contain positive and negative feedback loops that settle down to dynamically stable patterns. Identifying these patterns, the so-called attractors, can provide important insights for biologists to understand the molecular mechanisms underlying many coordinated cellular processes such as cellular division, differentiation, and homeostasis. Both synchronous and asynchronous Boolean networks have been used to simulate genetic regulatory networks and identify their attractors. The common methods of computing attractors are that start with a randomly selected initial state and finish with exhaustive search of the state space of a network. However, the time complexity of these methods grows exponentially with respect to the number and length of attractors. Here, we build two algorithms to achieve the computation of attractors in synchronous and asynchronous Boolean networks. For the synchronous scenario, combing with iterative methods and reduced order binary decision diagrams (ROBDD), we propose an improved algorithm to compute attractors. For another algorithm, the attractors of synchronous Boolean networks are utilized in asynchronous Boolean translation functions to derive attractors of asynchronous scenario. The proposed algorithms are implemented in a procedure called geneFAtt. Compared to existing tools such as genYsis, geneFAtt is significantly faster in computing attractors for empirical experimental systems. Availability The software package is available at https://sites.google.com/site/desheng619/download. PMID:23585840

  13. Chaotic bursting in semiconductor lasers

    Ruschel, Stefan; Yanchuk, Serhiy

    2017-11-01

    We investigate the dynamic mechanisms for low frequency fluctuations in semiconductor lasers subjected to delayed optical feedback, using the Lang-Kobayashi model. This system of delay differential equations displays pronounced envelope dynamics, ranging from erratic, so called low frequency fluctuations to regular pulse packages, if the time scales of fast oscillations and envelope dynamics are well separated. We investigate the parameter regions where low frequency fluctuations occur and compute their Lyapunov spectra. Using the geometric singular perturbation theory, we study this intermittent chaotic behavior and characterize these solutions as bursting slow-fast oscillations.

  14. Chaotic fluctuations in mathematical economics

    Yoshida, Hiroyuki, E-mail: yoshida.hiroyuki@nihon-u.ac.jp [College of Economics, Nihon University, Chiyoda-ku, Tokyo 101-8360 (Japan)

    2011-03-01

    In this paper we examine a Cournot duopoly model, which expresses the strategic interaction between two firms. We formulate the dynamic adjustment process and investigate the dynamic properties of the stationary point. By introducing a memory mechanism characterized by distributed lag functions, we presuppose that each firm makes production decisions in a cautious manner. This implies that we have to deal with the system of integro-differential equations. By means of numerical simulations we show the occurrence of chaotic fluctuations in the case of fixed delays.

  15. Design of secure digital communication systems using chaotic modulation, cryptography and chaotic synchronization

    Chien, T.-I.; Liao, T.-L.

    2005-01-01

    This paper presents a secure digital communication system based on chaotic modulation, cryptography, and chaotic synchronization techniques. The proposed system consists of a Chaotic Modulator (CM), a Chaotic Secure Transmitter (CST), a Chaotic Secure Receiver (CSR) and a Chaotic Demodulator (CDM). The CM module incorporates a chaotic system and a novel Chaotic Differential Peaks Keying (CDPK) modulation scheme to generate analog patterns corresponding to the input digital bits. The CST and CSR modules are designed such that a single scalar signal is transmitted in the public channel. Furthermore, by giving certain structural conditions of a particular class of chaotic system, the CST and the nonlinear observer-based CSR with an appropriate observer gain are constructed to synchronize with each other. These two slave systems are driven simultaneously by the transmitted signal and are designed to synchronize and generate appropriate cryptography keys for encryption and decryption purposes. In the CDM module, a nonlinear observer is designed to estimate the chaotic modulating system in the CM. A demodulation mechanism is then applied to decode the transmitted input digital bits. The effectiveness of the proposed scheme is demonstrated through the numerical simulation of an illustrative communication system. Synchronization between the chaotic circuits of the transmitter and receiver modules is guaranteed through the Lyapunov stability theorem. Finally, the security features of the proposed system in the event of attack by an intruder in either the time domain or the frequency domain are discussed

  16. Theory and practice of chaotic cryptography

    Amigo, J.M.; Kocarev, L.; Szczepanski, J.

    2007-01-01

    In this Letter we address some basic questions about chaotic cryptography, not least the very definition of chaos in discrete systems. We propose a conceptual framework and illustrate it with different examples from private and public key cryptography. We elaborate also on possible limits of chaotic cryptography

  17. Synchronization of Time-Continuous Chaotic Oscillators

    Yanchuk, S.; Maistrenko, Yuri; Mosekilde, Erik

    2003-01-01

    Considering a system of two coupled identical chaotic oscillators, the paper first establishes the conditions of transverse stability for the fully synchronized chaotic state. Periodic orbit threshold theory is applied to determine the bifurcations through which low-periodic orbits embedded...

  18. Synthesizing chaotic maps with prescribed invariant densities

    Rogers, Alan; Shorten, Robert; Heffernan, Daniel M.

    2004-01-01

    The Inverse Frobenius-Perron Problem (IFPP) concerns the creation of discrete chaotic mappings with arbitrary invariant densities. In this Letter, we present a new and elegant solution to the IFPP, based on positive matrix theory. Our method allows chaotic maps with arbitrary piecewise-constant invariant densities, and with arbitrary mixing properties, to be synthesized

  19. A Hybrid Chaotic Quantum Evolutionary Algorithm

    Cai, Y.; Zhang, M.; Cai, H.

    2010-01-01

    A hybrid chaotic quantum evolutionary algorithm is proposed to reduce amount of computation, speed up convergence and restrain premature phenomena of quantum evolutionary algorithm. The proposed algorithm adopts the chaotic initialization method to generate initial population which will form a pe...... tests. The presented algorithm is applied to urban traffic signal timing optimization and the effect is satisfied....

  20. Approximating chaotic saddles for delay differential equations.

    Taylor, S Richard; Campbell, Sue Ann

    2007-04-01

    Chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos. They play a key role in mediating transport processes involving scattering and chaotic transients. Here we present evidence (long chaotic transients and fractal basins of attraction) of transient chaos in a "logistic" delay differential equation. We adapt an existing method (stagger-and-step) to numerically construct the chaotic saddle for this system. This is the first such analysis of transient chaos in an infinite-dimensional dynamical system, and in delay differential equations in particular. Using Poincaré section techniques we illustrate approaches to visualizing the saddle set, and confirm that the saddle has the Cantor-like fractal structure consistent with a chaotic saddle generated by horseshoe-type dynamics.

  1. Approximating chaotic saddles for delay differential equations

    Taylor, S. Richard; Campbell, Sue Ann

    2007-04-01

    Chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos. They play a key role in mediating transport processes involving scattering and chaotic transients. Here we present evidence (long chaotic transients and fractal basins of attraction) of transient chaos in a “logistic” delay differential equation. We adapt an existing method (stagger-and-step) to numerically construct the chaotic saddle for this system. This is the first such analysis of transient chaos in an infinite-dimensional dynamical system, and in delay differential equations in particular. Using Poincaré section techniques we illustrate approaches to visualizing the saddle set, and confirm that the saddle has the Cantor-like fractal structure consistent with a chaotic saddle generated by horseshoe-type dynamics.

  2. Investigation of a chaotic thermostat

    Morales, G. J.

    2018-03-01

    A numerical study is presented of a free particle interacting with a deterministic thermostat in which the usual friction force is supplemented with a fluctuating force that depends on the self-consistent damping coefficient associated with coupling to the heat bath. It is found that this addition results in a chaotic environment in which a particle self-heats from rest and moves in positive and negative directions, exhibiting a characteristic diffusive behavior. The frequency power spectrum of the dynamical quantities displays the exponential frequency dependence ubiquitous to chaotic dynamics. The velocity distribution function approximates a Maxwellian distribution, but it does show departures from perfect thermal equilibrium, while the distribution function for the damping coefficient shows a closer fit. The behavior for the classic Nosé-Hoover (NH) thermostat is compared to that of the enlarged Martyna-Klein-Tuckerman (MKT) model. Over a narrow amplitude range, the application of a constant external force results quantitatively in the Einstein relation for the NH thermostat, and for the MKT model it differs by a factor of 2.

  3. Hybrid chaotic ant swarm optimization

    Li Yuying; Wen Qiaoyan; Li Lixiang; Peng Haipeng

    2009-01-01

    Chaotic ant swarm optimization (CASO) is a powerful chaos search algorithm that is used to find the global optimum solution in search space. However, the CASO algorithm has some disadvantages, such as lower solution precision and longer computational time, when solving complex optimization problems. To resolve these problems, an improved CASO, called hybrid chaotic swarm optimization (HCASO), is proposed in this paper. The new algorithm introduces preselection operator and discrete recombination operator into the CASO; meanwhile it replaces the best position found by own and its neighbors' ants with the best position found by preselection operator and discrete recombination operator in evolution equation. Through testing five benchmark functions with large dimensionality, the experimental results show the new method enhances the solution accuracy and stability greatly, as well as reduces the computational time and computer memory significantly when compared to the CASO. In addition, we observe the results can become better with swarm size increasing from the sensitivity study to swarm size. And we gain some relations between problem dimensions and swam size according to scalability study.

  4. Chaotic Transport in Circumterrestrial Orbits

    Rosengren, Aaron Jay

    2018-04-01

    The slow deformation of circumterrestrial orbits in the medium region, subject to lunisolar secular resonances, is well approximated by a Hamiltonian system with 2.5 degrees of freedom. This dynamical model is referred to in the astrophysical and celestial dynamics communities as the quadrupolar, secular, hierarchical three-body problem, and, in the non-autonomous case, gives rise to the classical Kozai-Lidov mechanism. In the time-dependent model, brought about in our case by the Moon's perturbed motion, the action variables of the system may experience chaotic variations and large drifts due to the possible overlap of nearby resonances. Using variational chaos indicators, we compute high-resolution portraits of the action space, revealing the existence of tori and structures filling chaotic regions. Our refined and elaborate calculations allow us to isolate precise initial conditions near specific areas of interest and to study their asymptotic behavior in time. We highlight in particular how the drift in phase space is mediated by the complement of the numerically detected KAM tori. Despite their reputed normality, Earth satellite orbits can possess an extraordinarily rich spectrum of dynamical behaviors, and, like the small body remnants of Solar system formation, they have all the complications that make them very interesting candidates for testing the modern tools of chaos theory.

  5. Cosmic ray modulation and merged interaction regions

    Burlaga, L.F.; Goldstein, M.L.; Mcdonald, F.B.

    1985-01-01

    Beyond several AU, interactions among shocks and streams give rise to merged interaction regions in which the magnetic field is turbulent. The integral intensity of . 75 MeV/Nuc cosmic rays at Voyager is generally observed to decrease when a merged interaction region moves past the spacecraft and to increase during the passage of a rarefaction region. When the separation between interaction regions is relatively large, the cosmic ray intensity tends to increase on a scale of a few months. This was the case at Voyager 1 from July 1, 1983 to May 1, 1984, when the spacecraft moved from 16.7 to 19.6 AU. Changes in cosmic ray intensity were related to the magnetic field strength in a simple way. It is estimated that the diffusion coefficient in merged interaction regions at this distance is similar to 0.6 x 10 to the 22nd power sq cm/s

  6. Application studies of spherical tokamak plasma merging

    Ono, Yasushi; Inomoto, Michiaki

    2012-01-01

    The experiment of plasma merging and heating has long history in compact torus studies since Wells. The study of spherical tokamak (ST), starting from TS-3 plasma merging experiment of Tokyo University in the late 1980s, is followed by START of Culham laboratory in the 1900s, TS-4 and UTST of Tokyo University and MAST of Culham laboratory in the 2000s, and last year by VEST of Soul University. ST has the following advantages: 1) plasma heating by magnetic reconnection at a MW-GW level, 2) rapid start-up of high beta plasma, 3) current drive/flux multiplication and distribution control of ST plasma, 4) fueling and helium-ash exhaust. In the present article, we emphasize that magnetic reconnection and plasma merging phenomena are important in ST plasma study as well as in plasma physics. (author)

  7. Hematopoietic differentiation: a coordinated dynamical process towards attractor stable states

    Rossi Simona

    2010-06-01

    Full Text Available Abstract Background The differentiation process, proceeding from stem cells towards the different committed cell types, can be considered as a trajectory towards an attractor of a dynamical process. This view, taking into consideration the transcriptome and miRNome dynamics considered as a whole, instead of looking at few 'master genes' driving the system, offers a novel perspective on this phenomenon. We investigated the 'differentiation trajectories' of the hematopoietic system considering a genome-wide scenario. Results We developed serum-free liquid suspension unilineage cultures of cord blood (CB CD34+ hematopoietic progenitor cells through erythroid (E, megakaryocytic (MK, granulocytic (G and monocytic (Mo pathways. These cultures recapitulate physiological hematopoiesis, allowing the analysis of almost pure unilineage precursors starting from initial differentiation of HPCs until terminal maturation. By analyzing the expression profile of protein coding genes and microRNAs in unilineage CB E, MK, G and Mo cultures, at sequential stages of differentiation and maturation, we observed a coordinated, fully interconnected and scalable character of cell population behaviour in both transcriptome and miRNome spaces reminiscent of an attractor-like dynamics. MiRNome and transcriptome space differed for a still not terminally committed behaviour of microRNAs. Conclusions Consistent with their roles, the transcriptome system can be considered as the state space of a cell population, while the continuously evolving miRNA space corresponds to the tuning system necessary to reach the attractor. The behaviour of miRNA machinery could be of great relevance not only for the promise of reversing the differentiated state but even for tumor biology.

  8. Is attentional blink a byproduct of neocortical attractors?

    David N Silverstein

    2011-05-01

    Full Text Available This study proposes a computational model for attentional blink or blink of the mind, a phenomenon where a human subject misses perception of a later expected visual pattern as two expected visual patterns are presented less than 500 ms apart. A neocortical patch modeled as an attractor network is stimulated with a sequence of 14 patterns 100 ms apart, two of which are expected targets. Patterns that become active attractors are considered recognized. A neocortical patch is represented as a square matrix of hypercolumns, each containing a set of minicolumns with synaptic connections within and across both minicolumns and hypercolumns. Each minicolumn consists of locally connected layer 2/3 pyramidal cells with interacting basket cells and layer 4 pyramidal cells for input stimulation. All neurons are implemented using the Hodgkin-Huxley multi-compartmental cell formalism and include calcium dynamics, and they interact via saturating and depressing AMPA / NMDA and GABAA synapses. Stored patterns are encoded with global connectivity of minicolumns across hypercolumns and active patterns compete as the result of lateral inhibition in the network. Stored patterns were stimulated over time intervals to create attractor interference measurable with synthetic spike traces. This setup corresponds with item presentations in human visual attentional blink studies. Stored target patterns were depolarized while distractor patterns where hyperpolarized to represent expectation of items in working memory. Additionally, studies on the inhibitory effect of benzodiazopines on attentional blink in human subjects were compared with neocortical simulations where the GABAA receptor conductance and decay time were increased. Simulations showed increases in the attentional blink duration, agreeing with observations in human studies.

  9. Neural attractor network for application in visual field data classification

    Fink, Wolfgang

    2004-01-01

    The purpose was to introduce a novel method for computer-based classification of visual field data derived from perimetric examination, that may act as a ' counsellor', providing an independent 'second opinion' to the diagnosing physician. The classification system consists of a Hopfield-type neural attractor network that obtains its input data from perimetric examination results. An iterative relaxation process determines the states of the neurons dynamically. Therefore, even 'noisy' perimetric output, e.g., early stages of a disease, may eventually be classified correctly according to the predefined idealized visual field defect (scotoma) patterns, stored as attractors of the network, that are found with diseases of the eye, optic nerve and the central nervous system. Preliminary tests of the classification system on real visual field data derived from perimetric examinations have shown a classification success of over 80%. Some of the main advantages of the Hopfield-attractor-network-based approach over feed-forward type neural networks are: (1) network architecture is defined by the classification problem; (2) no training is required to determine the neural coupling strengths; (3) assignment of an auto-diagnosis confidence level is possible by means of an overlap parameter and the Hamming distance. In conclusion, the novel method for computer-based classification of visual field data, presented here, furnishes a valuable first overview and an independent 'second opinion' in judging perimetric examination results, pointing towards a final diagnosis by a physician. It should not be considered a substitute for the diagnosing physician. Thanks to the worldwide accessibility of the Internet, the classification system offers a promising perspective towards modern computer-assisted diagnosis in both medicine and tele-medicine, for example and in particular, with respect to non-ophthalmic clinics or in communities where perimetric expertise is not readily available

  10. Chaotic dynamic characteristics of pressure fluctuation signals in hydro-turbine

    Su, Wen Tao; An, Shi [School of Management, Harbin Institute of Technology, Harbin (China); Li, Xiao Bin; Lan, Chao Feng; Li, Feng Chen [School of Energy Science and Engineering, Harbin Institute of Technology, Harbin (China); Wang, Jian Sheng [Ministry of Education of China, Tianjin (China)

    2016-11-15

    The pressure fluctuation characteristics in a Francis hydro-turbine running at partial flow conditions were studied based on the chaotic dynamic methods. Firstly, the experimental data of pressure fluctuations in the draft tube at various flow conditions was de-noised using lifting wavelet transformation, then, for the de-noised signals, their spectrum distribution on the frequency domain, the energy variation and the energy partition accounting for the total energy was calculated. Hereby, for the flow conditions ranging from no cavitation to severe cavitation, the chaos dynamic features of fluctuation signals were analyzed, including the temporal-frequency distribution, phase trajectory, Lyapunov exponent and Poincaré map etc. It is revealed that, the main energy of pressure fluctuations in the draft tube locates at low-frequency region. As the cavitation grows, the amplitude of power spectrum at frequency domain becomes larger. For all the flow conditions, all the maximal Lyapunov exponents are larger than zero, and they increase with the cavitation level. Therefore, it is believed that there indeed exist the chaotic attractors in the pressure fluctuation signals for a hydro-turbine.

  11. A note on chaotic vs. stochastic behavior of the high-latitude ionospheric plasma density fluctuations

    A. W. Wernik

    1996-01-01

    Full Text Available Four data sets of density fluctuations measured in-situ by the Dynamics Explorer (DE 2 were analyzed in an attempt to study chaotic nature of the high-latitude turbulence and, in this way to complement the conventional spectral analysis. It has been found that the probability distribution function of density differences is far from Gaussian and similar to that observed in the intermittent fluid or MBD turbulence. This indicates that ionospheric density fluctuations are not stochastic but coherent to some extent. Wayland's and surrogate data tests for determinism in a time series of density data allowed us to differentiate between regions of intense shear and moderate shear. We observe that in the region of strong field aligned currents (FAC and intense shear, or along the convection in the collisional regime, ionospheric turbulence behaves like a random noise with non-Gaussian statistics implying that the underlying physical process is nondeterministic. On the other hand, when FACs are weak, and shear is moderate or observations made in the inertial regime the turbulence is chaotic. The attractor dimension is lowest (1.9 for 'old' convected irregularities. The dimension 3.2 is found for turbulence in the inertial regime and considerably smaller (2.4 in the collisional regime. It is suggested that a high dimension in the inertial regime may be caused by a complicated velocity structure in the shear instability region.

  12. Trajectory-probed instability and statistics of desynchronization events in coupled chaotic systems

    Oliveira, Gilson F. de, E-mail: gilson@otica.ufpb.br; Chevrollier, Martine; Oriá, Marcos [Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900 João Pessoa-PB (Brazil); Passerat de Silans, Thierry [Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900 João Pessoa-PB (Brazil); UAF, Universidade Federal de Campina Grande, 58429-900 Campina Grande, PB (Brazil); Souza Cavalcante, Hugo L. D. de [Departamento de Informática, Centro de Informática, Universidade Federal da Paraíba, Av. dos Escoteiros s/n, Mangabeira VII, 58055-000 João Pessoa, PB (Brazil)

    2015-11-15

    Complex systems, such as financial markets, earthquakes, and neurological networks, exhibit extreme events whose mechanisms of formation are not still completely understood. These mechanisms may be identified and better studied in simpler systems with dynamical features similar to the ones encountered in the complex system of interest. For instance, sudden and brief departures from the synchronized state observed in coupled chaotic systems were shown to display non-normal statistical distributions similar to events observed in the complex systems cited above. The current hypothesis accepted is that these desynchronization events are influenced by the presence of unstable object(s) in the phase space of the system. Here, we present further evidence that the occurrence of large events is triggered by the visitation of the system's phase-space trajectory to the vicinity of these unstable objects. In the system studied here, this visitation is controlled by a single parameter, and we exploit this feature to observe the effect of the visitation rate in the overall instability of the synchronized state. We find that the probability of escapes from the synchronized state and the size of those desynchronization events are enhanced in attractors whose shapes permit the chaotic trajectories to approach the region of strong instability. This result shows that the occurrence of large events requires not only a large local instability to amplify noise, or to amplify the effect of parameter mismatch between the coupled subsystems, but also that the trajectories of the system wander close to this local instability.

  13. Giant Suppression of the Activation Rate in Dynamical Systems Exhibiting Chaotic Transitions

    Gac, J. M.; Xafebrowski, J. J.

    2008-05-01

    The phenomenon of giant suppression of activation, when two or more correlated noise signals act on the system, was found a few years ago in dynamical bistable or metastable systems. When the correlation between these noise signals is strong enough and the amplitudes of the noise are chosen correctly --- the life time of the metastable state may be longer than in the case of the application of only a single noise even by many orders of magnitude. In this paper, we investigate similar phenomena in systems exhibiting several chaotic transitions: Pomeau--Manneville intermittency, boundary crisis and interior crisis induced intermittency. Our goal is to show that, in these systems the application of two noise components with the proper choice of the parameters in the case of intermittency can also lengthen the mean laminar phase length or, in the case of boundary crisis, lengthen the time the trajectory spends on the pre-crisis attractor. In systems with crisis induced intermittency, we introduce a new phenomenon we called quasi-deterministic giant suppression of activation in which the lengthening of the laminar phase lengths is caused not by the action of two correlated noise signals but by a single noise term which is correlated with one of the chaotic variables of the system.

  14. Trajectory-probed instability and statistics of desynchronization events in coupled chaotic systems

    Oliveira, Gilson F. de; Chevrollier, Martine; Oriá, Marcos; Passerat de Silans, Thierry; Souza Cavalcante, Hugo L. D. de

    2015-01-01

    Complex systems, such as financial markets, earthquakes, and neurological networks, exhibit extreme events whose mechanisms of formation are not still completely understood. These mechanisms may be identified and better studied in simpler systems with dynamical features similar to the ones encountered in the complex system of interest. For instance, sudden and brief departures from the synchronized state observed in coupled chaotic systems were shown to display non-normal statistical distributions similar to events observed in the complex systems cited above. The current hypothesis accepted is that these desynchronization events are influenced by the presence of unstable object(s) in the phase space of the system. Here, we present further evidence that the occurrence of large events is triggered by the visitation of the system's phase-space trajectory to the vicinity of these unstable objects. In the system studied here, this visitation is controlled by a single parameter, and we exploit this feature to observe the effect of the visitation rate in the overall instability of the synchronized state. We find that the probability of escapes from the synchronized state and the size of those desynchronization events are enhanced in attractors whose shapes permit the chaotic trajectories to approach the region of strong instability. This result shows that the occurrence of large events requires not only a large local instability to amplify noise, or to amplify the effect of parameter mismatch between the coupled subsystems, but also that the trajectories of the system wander close to this local instability

  15. Trajectory-probed instability and statistics of desynchronization events in coupled chaotic systems

    de Oliveira, Gilson F.; Chevrollier, Martine; Passerat de Silans, Thierry; Oriá, Marcos; de Souza Cavalcante, Hugo L. D.

    2015-11-01

    Complex systems, such as financial markets, earthquakes, and neurological networks, exhibit extreme events whose mechanisms of formation are not still completely understood. These mechanisms may be identified and better studied in simpler systems with dynamical features similar to the ones encountered in the complex system of interest. For instance, sudden and brief departures from the synchronized state observed in coupled chaotic systems were shown to display non-normal statistical distributions similar to events observed in the complex systems cited above. The current hypothesis accepted is that these desynchronization events are influenced by the presence of unstable object(s) in the phase space of the system. Here, we present further evidence that the occurrence of large events is triggered by the visitation of the system's phase-space trajectory to the vicinity of these unstable objects. In the system studied here, this visitation is controlled by a single parameter, and we exploit this feature to observe the effect of the visitation rate in the overall instability of the synchronized state. We find that the probability of escapes from the synchronized state and the size of those desynchronization events are enhanced in attractors whose shapes permit the chaotic trajectories to approach the region of strong instability. This result shows that the occurrence of large events requires not only a large local instability to amplify noise, or to amplify the effect of parameter mismatch between the coupled subsystems, but also that the trajectories of the system wander close to this local instability.

  16. Pullback attractors for a singularly nonautonomous plate equation

    Vera Lucia Carbone

    2011-06-01

    Full Text Available We consider the family of singularly nonautonomous plate equations with structural damping $$ u_{tt} + a(t,xu_t - Delta u_t + (-Delta^2 u + lambda u = f(u, $$ in a bounded domain $Omega subset mathbb{R}^n$, with Navier boundary conditions. When the nonlinearity f is dissipative we show that this problem is globally well posed in $H^2_0(Omega imes L^2(Omega$ and has a family of pullback attractors which is upper-semicontinuous under small perturbations of the damping a.

  17. An efficient algorithm for computing fixed length attractors based on bounded model checking in synchronous Boolean networks with biochemical applications.

    Li, X Y; Yang, G W; Zheng, D S; Guo, W S; Hung, W N N

    2015-04-28

    Genetic regulatory networks are the key to understanding biochemical systems. One condition of the genetic regulatory network under different living environments can be modeled as a synchronous Boolean network. The attractors of these Boolean networks will help biologists to identify determinant and stable factors. Existing methods identify attractors based on a random initial state or the entire state simultaneously. They cannot identify the fixed length attractors directly. The complexity of including time increases exponentially with respect to the attractor number and length of attractors. This study used the bounded model checking to quickly locate fixed length attractors. Based on the SAT solver, we propose a new algorithm for efficiently computing the fixed length attractors, which is more suitable for large Boolean networks and numerous attractors' networks. After comparison using the tool BooleNet, empirical experiments involving biochemical systems demonstrated the feasibility and efficiency of our approach.

  18. Interpreting quantum discord through quantum state merging

    Madhok, Vaibhav; Datta, Animesh

    2011-01-01

    We present an operational interpretation of quantum discord based on the quantum state merging protocol. Quantum discord is the markup in the cost of quantum communication in the process of quantum state merging, if one discards relevant prior information. Our interpretation has an intuitive explanation based on the strong subadditivity of von Neumann entropy. We use our result to provide operational interpretations of other quantities like the local purity and quantum deficit. Finally, we discuss in brief some instances where our interpretation is valid in the single-copy scenario.

  19. Merging history of three bimodal clusters

    Maurogordato, S.; Sauvageot, J. L.; Bourdin, H.; Cappi, A.; Benoist, C.; Ferrari, C.; Mars, G.; Houairi, K.

    2011-01-01

    We present a combined X-ray and optical analysis of three bimodal galaxy clusters selected as merging candidates at z ~ 0.1. These targets are part of MUSIC (MUlti-Wavelength Sample of Interacting Clusters), which is a general project designed to study the physics of merging clusters by means of multi-wavelength observations. Observations include spectro-imaging with XMM-Newton EPIC camera, multi-object spectroscopy (260 new redshifts), and wide-field imaging at the ESO 3.6 m and 2.2 m telescopes. We build a global picture of these clusters using X-ray luminosity and temperature maps together with galaxy density and velocity distributions. Idealized numerical simulations were used to constrain the merging scenario for each system. We show that A2933 is very likely an equal-mass advanced pre-merger ~200 Myr before the core collapse, while A2440 and A2384 are post-merger systems (~450 Myr and ~1.5 Gyr after core collapse, respectively). In the case of A2384, we detect a spectacular filament of galaxies and gas spreading over more than 1 h-1 Mpc, which we infer to have been stripped during the previous collision. The analysis of the MUSIC sample allows us to outline some general properties of merging clusters: a strong luminosity segregation of galaxies in recent post-mergers; the existence of preferential axes - corresponding to the merging directions - along which the BCGs and structures on various scales are aligned; the concomitance, in most major merger cases, of secondary merging or accretion events, with groups infalling onto the main cluster, and in some cases the evidence of previous merging episodes in one of the main components. These results are in good agreement with the hierarchical scenario of structure formation, in which clusters are expected to form by successive merging events, and matter is accreted along large-scale filaments. Based on data obtained with the European Southern Observatory, Chile (programs 072.A-0595, 075.A-0264, and 079.A-0425

  20. Development of adaptive control applied to chaotic systems

    Rhode, Martin Andreas

    1997-12-01

    Continuous-time derivative control and adaptive map-based recursive feedback control techniques are used to control chaos in a variety of systems and in situations that are of practical interest. The theoretical part of the research includes the review of fundamental concept of control theory in the context of its applications to deterministic chaotic systems, the development of a new adaptive algorithm to identify the linear system properties necessary for control, and the extension of the recursive proportional feedback control technique, RPF, to high dimensional systems. Chaos control was applied to models of a thermal pulsed combustor, electro-chemical dissolution and the hyperchaotic Rossler system. Important implications for combustion engineering were suggested by successful control of the model of the thermal pulsed combustor. The system was automatically tracked while maintaining control into regions of parameter and state space where no stable attractors exist. In a simulation of the electrochemical dissolution system, application of derivative control to stabilize a steady state, and adaptive RPF to stabilize a period one orbit, was demonstrated. The high dimensional adaptive control algorithm was applied in a simulation using the Rossler hyperchaotic system, where a period-two orbit with two unstable directions was stabilized and tracked over a wide range of a system parameter. In the experimental part, the electrochemical system was studied in parameter space, by scanning the applied potential and the frequency of the rotating copper disk. The automated control algorithm is demonstrated to be effective when applied to stabilize a period-one orbit in the experiment. We show the necessity of small random perturbations applied to the system in order to both learn the dynamics and control the system at the same time. The simultaneous learning and control capability is shown to be an important part of the active feedback control.

  1. Chaotic sedimentation of particle pairs in a vertical channel at low Reynolds number: Multiple states and routes to chaos

    Verjus, Romuald; Guillou, Sylvain; Ezersky, Alexander; Angilella, Jean-Régis

    2016-12-01

    The sedimentation of a pair of rigid circular particles in a two-dimensional vertical channel containing a Newtonian fluid is investigated numerically, for terminal particle Reynolds numbers (ReT) ranging from 1 to 10, and for a confinement ratio equal to 4. While it is widely admitted that sufficiently inertial pairs should sediment by performing a regular DKT oscillation (Drafting-Kissing-Tumbling), the present analysis shows in contrast that a chaotic regime can also exist for such particles, leading to a much slower sedimentation velocity. It consists of a nearly horizontal pair, corresponding to a maximum effective blockage ratio, and performing a quasiperiodic transition to chaos while increasing the particle weight. For less inertial regimes, the classical oblique doublet structure and its complex behavior (multiple stable states and hysteresis, period-doubling cascade and chaotic attractor) are recovered, in agreement with previous work [Aidun, C. K. and Ding, E.-J., "Dynamics of particle sedimentation in a vertical channel: Period-doubling bifurcation and chaotic state," Phys. Fluids 15, 1612 (2003)]. As a consequence of these various behaviors, the link between the terminal Reynolds number and the non-dimensional driving force is complex: it contains several branches displaying hysteresis as well as various bifurcations. For the range of Reynolds number considered here, a global bifurcation diagram is given.

  2. Attractors of multivalued semiflows generated by differential inclusions and their approximations

    Alexei V. Kapustian

    2000-01-01

    Full Text Available We prove the existence of global compact attractors for differential inclusions and obtain some results concerning the continuity and upper semicontinuity of the attractors for approximating and perturbed inclusions. Applications are given to a model of regional economic growth.

  3. Shift of critical points in the parametrically modulated Henon map with coexisting attractors

    Saucedo-Solorio, J.M.; Pisarchik, A.N.; Aboites, V.

    2002-01-01

    We study how the critical point positions change in the parametrically modulated Henon map with coexisting period-1 and period-3 attractors. In particular, a new type of scaling law is found coinciding with that evidenced by laser experiments. We show that resonance phenomena play a crucial role in deformation of attractors and their basins of attraction

  4. Study of chaos in chaotic satellite systems

    Ayub Khan

    2017-12-27

    Dec 27, 2017 ... through strange attractor which is framed of the com- plex patterns. The one ... jectory of the Slave satellite be the measured attitude of the Master satellite [12 ... Bifurcation occurs when a tiny smooth change is made to the parameter .... negative and one of these tends to zero which is the required condition ...

  5. Control of partial synchronization in chaotic oscillators

    2015-02-07

    Feb 7, 2015 ... other real systems such as the brain network or the power grid, where multiple ..... 2D attractors of the driver oscillator (x2 vs. x3 plot) in the left and the response (y2 vs. y3 plot) in the right are given in the uppermost panels.

  6. Chaotic bubbling and nonstagnant foams.

    Tufaile, Alberto; Sartorelli, José Carlos; Jeandet, Philippe; Liger-Belair, Gerard

    2007-06-01

    We present an experimental investigation of the agglomeration of bubbles obtained from a nozzle working in different bubbling regimes. This experiment consists of a continuous production of bubbles from a nozzle at the bottom of a liquid column, and these bubbles create a two-dimensional (2D) foam (or a bubble raft) at the top of this column. The bubbles can assemble in various dynamically stable arrangement, forming different kinds of foams in a liquid mixture of water and glycerol, with the effect that the bubble formation regimes influence the foam obtained from this agglomeration of bubbles. The average number of bubbles in the foam is related to the bubble formation frequency and the bubble mean lifetime. The periodic bubbling can generate regular or irregular foam, while a chaotic bubbling only generates irregular foam.

  7. Chaotic hydrodynamics of fluidized beds

    Van der Stappen, M.L.M. [Unit Process and Systems Engineering, Advanced Manufacturing Technology Group, Unilever Research Laboratorium, Vlaardingen (Netherlands)

    1996-12-31

    The major goals of this thesis are: (1) to develop and evaluate an analysis method based on techniques from non-linear chaos theory to characterize the nonlinear hydrodynamics of gas-solids fluidized beds quantitatively; and (2) to determine the dependence of the chaotic invariants on the operating conditions and investigate how the chaos analysis method can be profitably applied to improve scale-up and design of gas-solids fluidized bed reactors. Chaos theory is introduced in chapter 2 with emphasis on analysis techniques for (experimental) time series, known from literature at the start of this work (1990-1991). In chapter 3, the testing of existing and newly developed techniques on both model and fluidized bed data is described. This leads to the development of the chaos analysis method to analyze measured pressure fluctuations time series of a fluidized bed. Following, in chapter 4, this method is tested and all choices for the parameters are evaluated. The influence of the experimental parameters and external disturbances on the measurements and analysis results is discussed and quantified. The result is a chaos measurement and analysis protocol, which is further used in this work. In chapter 5, the applications to fluidized beds are discussed. It is shown that the entropy is a good measure for the characterization of the dynamical behavior of gas-solids bubbling/slugging fluidized beds. Entropy is applied to characterize the influence of the operating conditions, to assess regime transitions and to analyze dimensionless similar beds of different scale. Quantitative design correlations that relate entropy to the operating parameters (including the bed diameter) are described. Finally, it is discussed how the results of this work might be used in scaling up the chaotic dynamics of fluidized beds. The overall conclusions and outlook from this work are presented in chapter 6. 182 refs.

  8. Regular transport dynamics produce chaotic travel times.

    Villalobos, Jorge; Muñoz, Víctor; Rogan, José; Zarama, Roberto; Johnson, Neil F; Toledo, Benjamín; Valdivia, Juan Alejandro

    2014-06-01

    In the hope of making passenger travel times shorter and more reliable, many cities are introducing dedicated bus lanes (e.g., Bogota, London, Miami). Here we show that chaotic travel times are actually a natural consequence of individual bus function, and hence of public transport systems more generally, i.e., chaotic dynamics emerge even when the route is empty and straight, stops and lights are equidistant and regular, and loading times are negligible. More generally, our findings provide a novel example of chaotic dynamics emerging from a single object following Newton's laws of motion in a regularized one-dimensional system.

  9. Chaotic inflation in models with flat directions

    Graziani, F.; Olive, K.

    1989-01-01

    We consider the chaotic inflationary scenario in models with flat directions. We find that unless the scalars along the flat directions have vacuum expectation values p or 10 14 M p 15 M p depending on the expectation values of the chaotic inflator, Ψ, one or two or more periods of inflation occur but with a resulting energy density perturbation δρ/ρ ≅ 10 -16 , far too small to be of any consequence for galaxy formation. Even with p only limited initial values of ≅ (3-200) M p result in inflation with reasonable density perturbations. Thus chaotic inflation in models with flat directions require rather special initial conditions. (orig.)

  10. Chaotic invasive weed optimization algorithm with application to parameter estimation of chaotic systems

    Ahmadi, Mohamadreza; Mojallali, Hamed

    2012-01-01

    Highlights: ► A new meta-heuristic optimization algorithm. ► Integration of invasive weed optimization and chaotic search methods. ► A novel parameter identification scheme for chaotic systems. - Abstract: This paper introduces a novel hybrid optimization algorithm by taking advantage of the stochastic properties of chaotic search and the invasive weed optimization (IWO) method. In order to deal with the weaknesses associated with the conventional method, the proposed chaotic invasive weed optimization (CIWO) algorithm is presented which incorporates the capabilities of chaotic search methods. The functionality of the proposed optimization algorithm is investigated through several benchmark multi-dimensional functions. Furthermore, an identification technique for chaotic systems based on the CIWO algorithm is outlined and validated by several examples. The results established upon the proposed scheme are also supplemented which demonstrate superior performance with respect to other conventional methods.

  11. Non-Abelian magnetized blackholes and unstable attractors

    Mosaffa, A.E. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of)], E-mail: mosaffa@theory.ipm.ac.ir; Randjbar-Daemi, S. [The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11 34014, Trieste (Italy)], E-mail: seif@ictp.trieste.it; Sheikh-Jabbari, M.M. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of)], E-mail: jabbari@theory.ipm.ac.ir

    2008-01-21

    Fluctuations of non-Abelian gauge fields in a background magnetic charge contain 'tachyonic' modes which as we will show cause an instability of the background. We extend this result to the cases where the background charge (flux) is coupled to four-dimensional Einstein gravity and show that the corresponding spherically symmetric geometries, which in the absence of a cosmological constant are of the form of (colored) Reissner-Nordstroem blackholes or the AdS{sub 2}xS{sup 2}, are also unstable unless the flux assumes its smallest allowed value, in which case the configuration is stable. We discuss the relevance of these instabilities to several places in string theory including various string compactifications and the attractor mechanism. Our results for the latter imply that the attractor mechanism shown to work for the extremal Abelian charged blackholes, cannot be applied in a straightforward way to the extremal non-Abelian colored blackholes, with the exception of the minimally charged stable ones.

  12. Universality of multi-field α-attractors

    Achúcarro, Ana; Kallosh, Renata; Linde, Andrei; Wang, Dong-Gang; Welling, Yvette

    2018-04-01

    We study a particular version of the theory of cosmological α-attractors with α=1/3, in which both the dilaton (inflaton) field and the axion field are light during inflation. The kinetic terms in this theory originate from maximal Script N=4 superconformal symmetry and from maximal Script N=8 supergravity. We show that because of the underlying hyperbolic geometry of the moduli space in this theory, it exhibits double attractor behavior: their cosmological predictions are stable not only with respect to significant modifications of the dilaton potential, but also with respect to significant modifications of the axion potential: nssimeq1‑2/N, rsimeq4/N2. We also show that the universality of predictions extends to other values of α lesssim Script O(1) with general two-field potentials that may or may not have an embedding in supergravity. Our results support the idea that inflation involving multiple, not stabilized, light fields on a hyperbolic manifold may be compatible with current observational constraints for a broad class of potentials.

  13. Split Attractor Flow in N=2 Minimally Coupled Supergravity

    Ferrara, Sergio; Orazi, Emanuele

    2011-01-01

    We classify the stability region, marginal stability walls (MS) and split attractor flows for two-center extremal black holes in four-dimensional N=2 supergravity minimally coupled to n vector multiplets. It is found that two-center (continuous) charge orbits, classified by four duality invariants, either support a stability region ending on a MS wall or on an anti-marginal stability (AMS) wall, but not both. Therefore, the scalar manifold never contains both walls. Moreover, the BPS mass of the black hole composite (in its stability region) never vanishes in the scalar manifold. For these reasons, the "bound state transformation walls" phenomenon does not necessarily occur in these theories. The entropy of the flow trees also satisfies an inequality which forbids "entropy enigma" decays in these models. Finally, the non-BPS case, due to the existence of a "fake" superpotential satisfying a triangle inequality, can be treated as well, and it can be shown to exhibit a split attractor flow dynamics which, at le...

  14. Reconstructing a f ( R ) theory from the α-Attractors

    Miranda, T.; Fabris, J. C.; Piattella, O. F., E-mail: tays.andrade@aluno.ufes.br, E-mail: oliver.piattella@pq.cnpq.br, E-mail: fabris@pq.cnpq.br [Department of Physics, Universidade Federal do Espírito Santo, avenida Fernando Ferrari 514, 29075-910 Vitória, Espírito Santo (Brazil)

    2017-09-01

    We show an analogy at high curvature between a f ( R ) = R + aR {sup n} {sup −} {sup 1} + bR {sup 2} theory and the α-Attractors. We calculate the expressions of the parameters a , b and n as functions of α and the predictions of the model f ( R ) = R + aR {sup n} {sup −} {sup 1} + bR {sup 2} on the scalar spectral index n {sub s} and the tensor-to-scalar ratio r . We find that the power law correction R {sup n} {sup −} {sup 1} allows for a production of gravitational waves enhanced with respect to the one in the Starobinsky model, while maintaining a viable prediction on n {sub s}. We numerically reconstruct the full α-Attractors class of models testing the goodness of our high-energy approximation f ( R ) = R + aR {sup n} {sup −} {sup 1} + bR {sup 2}. Moreover, we also investigate the case of a single power law f ( R ) = γ R {sup 2} {sup −} {sup δ} theory, with γ and δ free parameters. We calculate analytically the predictions of this model on the scalar spectral index n {sub s} and the tensor-to-scalar ratio r and the values of δ which are allowed from the current observational results. We find that −0.015 < δ < 0.016, confirming once again the excellent agreement between the Starobinsky model and observation.

  15. Modeling merging behavior at lane drops.

    2015-02-01

    In work-zone configurations where lane drops are present, merging of traffic at the taper presents an operational concern. In : addition, as flow through the work zone is reduced, the relative traffic safety of the work zone is also reduced. Improvin...

  16. Algorithm 426 : Merge sort algorithm [M1

    Bron, C.

    1972-01-01

    Sorting by means of a two-way merge has a reputation of requiring a clerically complicated and cumbersome program. This ALGOL 60 procedure demonstrates that, using recursion, an elegant and efficient algorithm can be designed, the correctness of which is easily proved [2]. Sorting n objects gives

  17. Cluster Physics with Merging Galaxy Clusters

    Sandor M. Molnar

    2016-02-01

    Full Text Available Collisions between galaxy clusters provide a unique opportunity to study matter in a parameter space which cannot be explored in our laboratories on Earth. In the standard LCDM model, where the total density is dominated by the cosmological constant ($Lambda$ and the matter density by cold dark matter (CDM, structure formation is hierarchical, and clusters grow mostly by merging.Mergers of two massive clusters are the most energetic events in the universe after the Big Bang,hence they provide a unique laboratory to study cluster physics.The two main mass components in clusters behave differently during collisions:the dark matter is nearly collisionless, responding only to gravity, while the gas is subject to pressure forces and dissipation, and shocks and turbulenceare developed during collisions. In the present contribution we review the different methods used to derive the physical properties of merging clusters. Different physical processes leave their signatures on different wavelengths, thusour review is based on a multifrequency analysis. In principle, the best way to analyze multifrequency observations of merging clustersis to model them using N-body/HYDRO numerical simulations. We discuss the results of such detailed analyses.New high spatial and spectral resolution ground and space based telescopeswill come online in the near future. Motivated by these new opportunities,we briefly discuss methods which will be feasible in the near future in studying merging clusters.

  18. Preparation for electron ring - plasma ring merging experiments in RECE-MERGE

    Taggart, D.; Sekiguchi, A.; Fleischmann, H.H.

    1986-01-01

    The formation of a mixed-CT using relativistic electron rings and gun-produced plasma rings by MERGE-ing them axially is simulated. This process is similar to the axial stacking of relativistic electron rings in RECE-Christa. The results of their first plasm production experiment are reported here. After study of the gun-produced plasma's properties is completed, the gun will be mounted at the downstream end of the vacuum tank and the source of relativistic electron rings will be at the upstream end. The two rings, formed at opposite ends of the tank, will be translated axially and merged

  19. A novel sort of adaptive complex synchronizations of two indistinguishable chaotic complex nonlinear models with uncertain parameters and its applications in secure communications

    Mahmoud, Emad E.; Abood, Fatimah S.

    In this paper, we will demonstrate the adaptive complex anti-lag synchronization (CALS) of two indistinguishable complex chaotic nonlinear systems with the parameters which are uncertain. The significance of CALS is not advised well in the literature yet. The CALS contains or consolidate two sorts of synchronizations (anti-lag synchronization ALS and lag synchronization LS). The state variable of the master system synchronizes with an alternate state variable of the slave system. Depending on the function of Lyapunov, a plan is orchestrated to achieve CALS of chaotic attractors of complex systems with unverifiable parameters. CALS of two indistinguishable complexes of Lü systems is viewed as, for example, an occasion for affirming the likelihood of the plan exhibited. In physics, we can see complex chaotic systems in numerous different applications, for example, applied sciences or engineering. With a specific end goal to affirm the proposed synchronization plan viability and demonstrate the hypothetical outcomes, we can compute the numerical simulation. The above outcomes will give the hypothetical establishment to the secure communication applications. CALS of complex chaotic systems in which a state variable of the master system synchronizes with an alternate state variable of the slave system is an encouraging sort of synchronization as it contributes excellent security in secure communication. Amid this secure communication, the synchronization between transmitter and collector is shut and message signals are recouped. The encryption and restoration of the signals are simulated numerically.

  20. Bidirectional communication using delay coupled chaotic directly ...

    Corresponding author. ... 30 September 2009. Abstract. Chaotic synchronization of two directly modulated semiconductor lasers with ... For InGaAsP lasers used in optical communication systems, the nonlinear gain re- duction is very strong and its ...

  1. Bifurcation Control of Chaotic Dynamical Systems

    Wang, Hua O; Abed, Eyad H

    1992-01-01

    A nonlinear system which exhibits bifurcations, transient chaos, and fully developed chaos is considered, with the goal of illustrating the role of two ideas in the control of chaotic dynamical systems...

  2. Prediction and Geometry of Chaotic Time Series

    Leonardi, Mary

    1997-01-01

    This thesis examines the topic of chaotic time series. An overview of chaos, dynamical systems, and traditional approaches to time series analysis is provided, followed by an examination of state space reconstruction...

  3. Encryption in Chaotic Systems with Sinusoidal Excitations

    G. Obregón-Pulido

    2014-01-01

    Full Text Available In this contribution an encryption method using a chaotic oscillator, excited by “n” sinusoidal signals, is presented. The chaotic oscillator is excited by a sum of “n” sinusoidal signals and a message. The objective is to encrypt such a message using the chaotic behavior and transmit it, and, as the chaotic system is perturbed by the sinusoidal signal, the transmission security could be increased due to the effect of such a perturbation. The procedure is based on the regulation theory and consider that the receiver knows the frequencies of the perturbing signal, with this considerations the algorithm estimates the excitation in such a way that the receiver can cancel out the perturbation and all the undesirable dynamics in order to produce only the message. In this way we consider that the security level is increased.

  4. Chaotic Zones around Rotating Small Bodies

    Lages, José; Shevchenko, Ivan I. [Institut UTINAM, Observatoire des Sciences de l’Univers THETA, CNRS, Université de Franche-Comté, Besançon F-25030 (France); Shepelyansky, Dima L., E-mail: jose.lages@utinam.cnrs.fr [Laboratoire de Physique Théorique du CNRS, IRSAMC, Université de Toulouse, UPS, Toulouse F-31062 (France)

    2017-06-01

    Small bodies of the solar system, like asteroids, trans-Neptunian objects, cometary nuclei, and planetary satellites, with diameters smaller than 1000 km usually have irregular shapes, often resembling dumb-bells or contact binaries. The spinning of such a gravitating dumb-bell creates around it a zone of chaotic orbits. We determine its extent analytically and numerically. We find that the chaotic zone swells significantly if the rotation rate is decreased; in particular, the zone swells more than twice if the rotation rate is decreased 10 times with respect to the “centrifugal breakup” threshold. We illustrate the properties of the chaotic orbital zones in examples of the global orbital dynamics about asteroid 243 Ida (which has a moon, Dactyl, orbiting near the edge of the chaotic zone) and asteroid 25143 Itokawa.

  5. Nonlinear chaotic model for predicting storm surges

    M. Siek

    2010-09-01

    Full Text Available This paper addresses the use of the methods of nonlinear dynamics and chaos theory for building a predictive chaotic model from time series. The chaotic model predictions are made by the adaptive local models based on the dynamical neighbors found in the reconstructed phase space of the observables. We implemented the univariate and multivariate chaotic models with direct and multi-steps prediction techniques and optimized these models using an exhaustive search method. The built models were tested for predicting storm surge dynamics for different stormy conditions in the North Sea, and are compared to neural network models. The results show that the chaotic models can generally provide reliable and accurate short-term storm surge predictions.

  6. Indirect adaptive control of discrete chaotic systems

    Salarieh, Hassan; Shahrokhi, Mohammad

    2007-01-01

    In this paper an indirect adaptive control algorithm is proposed to stabilize the fixed points of discrete chaotic systems. It is assumed that the functionality of the chaotic dynamics is known but the system parameters are unknown. This assumption is usually applicable to many chaotic systems, such as the Henon map, logistic and many other nonlinear maps. Using the recursive-least squares technique, the system parameters are identified and based on the feedback linearization method an adaptive controller is designed for stabilizing the fixed points, or unstable periodic orbits of the chaotic maps. The stability of the proposed scheme has been shown and the effectiveness of the control algorithm has been demonstrated through computer simulations

  7. Globally Coupled Chaotic Maps with Constant Force

    Li Jinghui

    2008-01-01

    We investigate the motion of the globally coupled maps (logistic map) with a constant force. It is shown that the constant force can cause multi-synchronization for the globally coupled chaotic maps studied by us.

  8. Multiswitching compound antisynchronization of four chaotic systems

    Ayub Khan

    2017-11-28

    Nov 28, 2017 ... systems, electrical engineering, information process- ... model. The synchronization problem among three or more chaotic ...... we perform numerical simulations in MATLAB using ... In the simulation process we assume α1 =.

  9. Synchronizing a class of uncertain chaotic systems

    Chen Maoyin; Zhou Donghua; Shang Yun

    2005-01-01

    This Letter deals with the synchronization of a class of uncertain chaotic systems in the drive-response framework. A robust adaptive observer based response system is designed to synchronize a given chaotic system with unknown parameters and external disturbances. Lyapunov stability ensures the global synchronization between the drive and response systems even if Lipschitz constants on function matrices and bounds on uncertainties are unknown. Numerical simulation of Genesio-Tesi system verifies the effectiveness of this scheme

  10. Anti-synchronization of chaotic oscillators

    Kim, Chil-Min; Rim, Sunghwan; Kye, Won-Ho; Ryu, Jung-Wan; Park, Young-Jai

    2003-01-01

    We have observed anti-synchronization phenomena in coupled identical chaotic oscillators. Anti-synchronization can be characterized by the vanishing of the sum of relevant variables. We have qualitatively analyzed its base mechanism by using the dynamics of the difference and the sum of the relevant variables in coupled chaotic oscillators. Near the threshold of the synchronization and anti-synchronization transition, we have obtained the novel characteristic relation

  11. Improvement on generalised synchronisation of chaotic systems

    Hui-Bin, Zhu; Fang, Qiu; Bao-Tong, Cui

    2010-01-01

    In this paper, the problem of generalised synchronisation of two different chaotic systems is investigated. Some less conservative conditions are derived using linear matrix inequality other than existing results. Furthermore, a simple adaptive control scheme is proposed to achieve the generalised synchronisation of chaotic systems. The proposed method is simple and easy to implement in practice and can be applied to secure communications. Numerical simulations are also given to demonstrate the effectiveness and feasibility of the theoretical analysis

  12. The Astrophysics of Merging Black Holes

    Schnittman, Jeremy D.

    2011-01-01

    When two supermassive black holes (SMBHs) approach within 1-10 mpc, gravitational wave (GW) losses begin to dominate the evolution of the binary, pushing the system to merge in a relatively small time. During this final inspiral regime, the system will emit copious energy in GWs, which should be directly detectable by pulsar timing arrays and space-based interferometers. At the same time, any gas or stars in the immediate vicinity of the merging 5MBHs can get heated and produce bright electromagnetic (EM) counterparts to the GW signals. We present here a number of possible mechanisms by which simultaneous EM and GW signals will yield valuable new information about galaxy evolution, accretion disk dynamics, and fundamental physics in the most extreme gravitational fields.

  13. Team Climate Inventory with a merged organization.

    Dackert, Ingrid; Brenner, Sten-Olof; Johansson, Curt R

    2002-10-01

    The present study examines the team climate for innovation in work teams within a newly merged organization. Four teams working at a regional head office of a Social Insurance organization answered the Team Climate Inventory. The results were compared to those of a study by Agrell and Gustafson of more stable teams. The comparison showed that participative safety and support for innovation were rated lower and that vision was rated higher in the newly merged teams. The 38-item original inventory was used and based on the results, a 1999 proposed shortened version of 14 items by Kivimäki and Elovainio was compared with the original one. Analysis indicated that the short version can be a valid alternative to the original version but that further testing of the short version is needed.

  14. Merging physical parameters and laboratory subjective ratings for the soundscape assessment of urban squares.

    Brambilla, Giovanni; Maffei, Luigi; Di Gabriele, Maria; Gallo, Veronica

    2013-07-01

    An experimental study was carried out in 20 squares in the center of Rome, covering a wide range of different uses, sonic environments, geometry, and architectural styles. Soundwalks along the perimeter of each square were performed during daylight and weekdays taking binaural and video recordings, as well as spot measurements of illuminance. The cluster analysis performed on the physical parameters, not only acoustic, provided two clusters that are in satisfactory agreement with the "a priori" classification. Applying the principal component analysis (PCA) to five physical parameters, two main components were obtained which might be associated to two environmental features, namely, "chaotic/calm" and "open/enclosed." On the basis of these two features, six squares were selected for the laboratory audio-video tests where 32 subjects took part filling in a questionnaire. The PCA performed on the subjective ratings on the sonic environment showed two main components which might be associated to two emotional meanings, namely, "calmness" and "vibrancy." The linear regression modeling between five objective parameters and the mean value of subjective ratings on chaotic/calm and enclosed/open attributes showed a good correlation. Notwithstanding these interesting results being limited to the specific data set, it is worth pointing out that the complexity of the soundscape quality assessment can be more comprehensively examined merging the field measurements of physical parameters with the subjective ratings provided by field and/or laboratory tests.

  15. Modeling merging behavior at lane drops : [tech transfer summary].

    2015-02-01

    A better understanding of the merging behavior of drivers will lead : to the development of better lane-drop traffic-control plans and : strategies, which will provide better guidance to drivers for safer : merging.

  16. Bump formation in a binary attractor neural network

    Koroutchev, Kostadin; Korutcheva, Elka

    2006-01-01

    The conditions for the formation of local bumps in the activity of binary attractor neural networks with spatially dependent connectivity are investigated. We show that these formations are observed when asymmetry between the activity during the retrieval and learning is imposed. An analytical approximation for the order parameters is derived. The corresponding phase diagram shows a relatively large and stable region where this effect is observed, although critical storage and information capacities drastically decrease inside that region. We demonstrate that the stability of the network, when starting from the bump formation, is larger than the stability when starting even from the whole pattern. Finally, we show a very good agreement between the analytical results and the simulations performed for different topologies of the network

  17. Topology and computational performance of attractor neural networks

    McGraw, Patrick N.; Menzinger, Michael

    2003-01-01

    To explore the relation between network structure and function, we studied the computational performance of Hopfield-type attractor neural nets with regular lattice, random, small-world, and scale-free topologies. The random configuration is the most efficient for storage and retrieval of patterns by the network as a whole. However, in the scale-free case retrieval errors are not distributed uniformly among the nodes. The portion of a pattern encoded by the subset of highly connected nodes is more robust and efficiently recognized than the rest of the pattern. The scale-free network thus achieves a very strong partial recognition. The implications of these findings for brain function and social dynamics are suggestive

  18. Explosive attractor solutions to a universal cubic delay equation

    Sanz-Orozco, D.; Berk, H. L.

    2017-05-01

    New explosive attractor solutions have been found in a universal cubic delay equation that has been studied in both the plasma and the fluid mechanics literature. Through computational simulations and analytic approximations, it is found that the oscillatory component of the explosive mode amplitude solutions are described through multi-frequency Fourier expansions with respect to a pseudo-time variable. The spectral dependence of these solutions as a function of a system parameter, ϕ , is studied. The mode amplitude that is described in the explosive regime has two main features: a well-known envelope ( t 0 - t ) - 5 / 2 , with t0 the blow-up time of the amplitude, and a spectrum of discrete oscillations with ever-increasing frequencies, which may give experimental information about the properties of a system's equilibrium.

  19. Detection of strong attractors in social media networks.

    Qasem, Ziyaad; Jansen, Marc; Hecking, Tobias; Hoppe, H Ulrich

    2016-01-01

    Detection of influential actors in social media such as Twitter or Facebook plays an important role for improving the quality and efficiency of work and services in many fields such as education and marketing. The work described here aims to introduce a new approach that characterizes the influence of actors by the strength of attracting new active members into a networked community. We present a model of influence of an actor that is based on the attractiveness of the actor in terms of the number of other new actors with which he or she has established relations over time. We have used this concept and measure of influence to determine optimal seeds in a simulation of influence maximization using two empirically collected social networks for the underlying graphs. Our empirical results on the datasets demonstrate that our measure stands out as a useful measure to define the attractors comparing to the other influence measures.

  20. Strange attractor of Henon map and its basin

    曹永罗

    1995-01-01

    In this paper, Henon map is considered. For a positive measure set of parameters (a, b), we construct a trapping region G of topologically transitive strange attractor Aa,b for Ta,b, and prove that Aa,b= ∩n≥0Ta,bnG, and the basin B(Aa,b) of Aa,b is exactly the union of domain whose boundary is contained in w5(p) ∪wu(p) and ws(p). Therefore, that the conjecture posed by Benedicks and Carleson about the basin of strange attactor is true is proved. Furthermore, B(Aa,b) is simply connected and path-connected, w4(p2) is contained in the attainable boundary set of B(Aa,b) (where p2 is another hyperbolic fixed point of Ta,b).

  1. Noether symmetry approach in the cosmological alpha-attractors

    Kaewkhao, Narakorn; Kanesom, Thanyagamon; Channuie, Phongpichit

    2018-06-01

    In cosmological framework, Noether symmetry technique has revealed a useful tool in order to examine exact solutions. In this work, we first introduce the Jordan-frame Lagrangian and apply the conformal transformation in order to obtain the Lagrangian equivalent to Einstein-frame form. We then analyze the dynamics of the field in the cosmological alpha-attractors using the Noether symmetry approach by focusing on the single field scenario in the Einstein-frame form. We show that with a Noether symmetry the corresponding dynamical system can be completely integrated and the potential exhibited by the symmetry can be exactly obtained. With the proper choice of parameters, the behavior of the scale factor displays an exponential (de Sitter) behavior at the present epoch. Moreover, we discover that the Hubble parameters strongly depends on the initial values of parameters exhibited by the Noether symmetry. Interestingly, it can retardedly evolve and becomes a constant in the present epoch in all cases.

  2. Merging Digital Surface Models Implementing Bayesian Approaches

    Sadeq, H.; Drummond, J.; Li, Z.

    2016-06-01

    In this research different DSMs from different sources have been merged. The merging is based on a probabilistic model using a Bayesian Approach. The implemented data have been sourced from very high resolution satellite imagery sensors (e.g. WorldView-1 and Pleiades). It is deemed preferable to use a Bayesian Approach when the data obtained from the sensors are limited and it is difficult to obtain many measurements or it would be very costly, thus the problem of the lack of data can be solved by introducing a priori estimations of data. To infer the prior data, it is assumed that the roofs of the buildings are specified as smooth, and for that purpose local entropy has been implemented. In addition to the a priori estimations, GNSS RTK measurements have been collected in the field which are used as check points to assess the quality of the DSMs and to validate the merging result. The model has been applied in the West-End of Glasgow containing different kinds of buildings, such as flat roofed and hipped roofed buildings. Both quantitative and qualitative methods have been employed to validate the merged DSM. The validation results have shown that the model was successfully able to improve the quality of the DSMs and improving some characteristics such as the roof surfaces, which consequently led to better representations. In addition to that, the developed model has been compared with the well established Maximum Likelihood model and showed similar quantitative statistical results and better qualitative results. Although the proposed model has been applied on DSMs that were derived from satellite imagery, it can be applied to any other sourced DSMs.

  3. MERGING DIGITAL SURFACE MODELS IMPLEMENTING BAYESIAN APPROACHES

    H. Sadeq

    2016-06-01

    Full Text Available In this research different DSMs from different sources have been merged. The merging is based on a probabilistic model using a Bayesian Approach. The implemented data have been sourced from very high resolution satellite imagery sensors (e.g. WorldView-1 and Pleiades. It is deemed preferable to use a Bayesian Approach when the data obtained from the sensors are limited and it is difficult to obtain many measurements or it would be very costly, thus the problem of the lack of data can be solved by introducing a priori estimations of data. To infer the prior data, it is assumed that the roofs of the buildings are specified as smooth, and for that purpose local entropy has been implemented. In addition to the a priori estimations, GNSS RTK measurements have been collected in the field which are used as check points to assess the quality of the DSMs and to validate the merging result. The model has been applied in the West-End of Glasgow containing different kinds of buildings, such as flat roofed and hipped roofed buildings. Both quantitative and qualitative methods have been employed to validate the merged DSM. The validation results have shown that the model was successfully able to improve the quality of the DSMs and improving some characteristics such as the roof surfaces, which consequently led to better representations. In addition to that, the developed model has been compared with the well established Maximum Likelihood model and showed similar quantitative statistical results and better qualitative results. Although the proposed model has been applied on DSMs that were derived from satellite imagery, it can be applied to any other sourced DSMs.

  4. Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks.

    Vlachas, Pantelis R; Byeon, Wonmin; Wan, Zhong Y; Sapsis, Themistoklis P; Koumoutsakos, Petros

    2018-05-01

    We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.

  5. In-depth analysis of drivers' merging behavior and rear-end crash risks in work zone merging areas.

    Weng, Jinxian; Xue, Shan; Yang, Ying; Yan, Xuedong; Qu, Xiaobo

    2015-04-01

    This study investigates the drivers' merging behavior and the rear-end crash risk in work zone merging areas during the entire merging implementation period from the time of starting a merging maneuver to that of completing the maneuver. With the merging traffic data from a work zone site in Singapore, a mixed probit model is developed to describe the merging behavior, and two surrogate safety measures including the time to collision (TTC) and deceleration rate to avoid the crash (DRAC) are adopted to compute the rear-end crash risk between the merging vehicle and its neighboring vehicles. Results show that the merging vehicle has a bigger probability of completing a merging maneuver quickly under one of the following situations: (i) the merging vehicle moves relatively fast; (ii) the merging lead vehicle is a heavy vehicle; and (iii) there is a sizable gap in the adjacent through lane. Results indicate that the rear-end crash risk does not monotonically increase as the merging vehicle speed increases. The merging vehicle's rear-end crash risk is also affected by the vehicle type. There is a biggest increment of rear-end crash risk if the merging lead vehicle belongs to a heavy vehicle. Although the reduced remaining distance to work zone could urge the merging vehicle to complete a merging maneuver quickly, it might lead to an increased rear-end crash risk. Interestingly, it is found that the rear-end crash risk could be generally increased over the elapsed time after the merging maneuver being triggered. Copyright © 2015 Elsevier Ltd. All rights reserved.

  6. Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains

    Wang, Xiaohu; Lu, Kening; Wang, Bixiang

    2018-01-01

    In this paper, we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of the stochastic reaction-diffusion equation driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of stochastic reaction-diffusion equation. Then, we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic reaction-diffusion equation for both additive and multiplicative noise.

  7. How organisms do the right thing: The attractor hypothesis

    Emlen, J.M.; Freeman, D.C.; Mills, A.; Graham, J.H.

    1998-01-01

    Neo-Darwinian theory is highly successful at explaining the emergence of adaptive traits over successive generations. However, there are reasons to doubt its efficacy in explaining the observed, impressively detailed adaptive responses of organisms to day-to-day changes in their surroundings. Also, the theory lacks a clear mechanism to account for both plasticity and canalization. In effect, there is a growing sentiment that the neo-Darwinian paradigm is incomplete, that something more than genetic structure, mutation, genetic drift, and the action of natural selection is required to explain organismal behavior. In this paper we extend the view of organisms as complex self-organizing entities by arguing that basic physical laws, coupled with the acquisitive nature of organisms, makes adaptation all but tautological. That is, much adaptation is an unavoidable emergent property of organisms' complexity and, to some a significant degree, occurs quite independently of genomic changes wrought by natural selection. For reasons that will become obvious, we refer to this assertion as the attractor hypothesis. The arguments also clarify the concept of "adaptation." Adaptation across generations, by natural selection, equates to the (game theoretic) maximization of fitness (the success with which one individual produces more individuals), while self-organizing based adaptation, within generations, equates to energetic efficiency and the matching of intake and biosynthesis to need. Finally, we discuss implications of the attractor hypothesis for a wide variety of genetical and physiological phenomena, including genetic architecture, directed mutation, genetic imprinting, paramutation, hormesis, plasticity, optimality theory, genotype-phenotype linkage and puncuated equilibrium, and present suggestions for tests of the hypothesis. ?? 1998 American Institute of Physics.

  8. Changes in thunderstorm characteristics due to feeder cloud merging

    Sinkevich, Andrei A.; Krauss, Terrence W.

    2014-06-01

    Cumulus cloud merging is a complex dynamical and microphysical process in which two convective cells merge into a single cell. Previous radar observations and numerical simulations have shown a substantial increase in the maximum area, maximum echo top and maximum reflectivity as a result of the merging process. Although the qualitative aspects of merging have been well documented, the quantitative effects on storm properties remain less defined. Therefore, a statistical assessment of changes in storm characteristics due to merging is of importance. Further investigation into the effects of cloud merging on precipitation flux (Pflux) in a statistical manner provided the motivation for this study in the Asir region of Saudi Arabia. It was confirmed that merging has a strong effect on storm development in this region. The data analysis shows that an increase in the median of the distribution of maximum reflectivity was observed just after merging and was equal to 3.9 dBZ. A detailed analysis of the individual merge cases compared the merged storm Pflux and mass to the sum of the individual Feeder and Storm portions just before merging for each case. The merged storm Pflux increased an average of 106% over the 20-min period after merging, and the mass increased on average 143%. The merged storm clearly became larger and more severe than the sum of the two parts prior to merging. One consequence of this study is that any attempts to evaluate the precipitation enhancement effects of cloud seeding must also include the issue of cloud mergers because merging can have a significant effect on the results.

  9. A novel image block cryptosystem based on a spatiotemporal chaotic system and a chaotic neural network

    Wang Xing-Yuan; Bao Xue-Mei

    2013-01-01

    In this paper, we propose a novel block cryptographic scheme based on a spatiotemporal chaotic system and a chaotic neural network (CNN). The employed CNN comprises a 4-neuron layer called a chaotic neuron layer (CNL), where the spatiotemporal chaotic system participates in generating its weight matrix and other parameters. The spatiotemporal chaotic system used in our scheme is the typical coupled map lattice (CML), which can be easily implemented in parallel by hardware. A 160-bit-long binary sequence is used to generate the initial conditions of the CML. The decryption process is symmetric relative to the encryption process. Theoretical analysis and experimental results prove that the block cryptosystem is secure and practical, and suitable for image encryption. (general)

  10. Composing chaotic music from the letter m

    Sotiropoulos, Anastasios D.

    Chaotic music is composed from a proposed iterative map depicting the letter m, relating the pitch, duration and loudness of successive steps. Each of the two curves of the letter m is based on the classical logistic map. Thus, the generating map is xn+1 = r xn(1/2 - xn) for xn between 0 and 1/2 defining the first curve, and xn+1 = r (xn - 1/2)(1 - xn) for xn between 1/2 and 1 representing the second curve. The parameter r which determines the height(s) of the letter m varies from 2 to 16, the latter value ensuring fully developed chaotic solutions for the whole letter m; r = 8 yielding full chaotic solutions only for its first curve. The m-model yields fixed points, bifurcation points and chaotic regions for each separate curve, as well as values of the parameter r greater than 8 which produce inter-fixed points, inter-bifurcation points and inter-chaotic regions from the interplay of the two curves. Based on this, music is composed from mapping the m- recurrence model solutions onto actual notes. The resulting musical score strongly depends on the sequence of notes chosen by the composer to define the musical range corresponding to the range of the chaotic mathematical solutions x from 0 to 1. Here, two musical ranges are used; one is the middle chromatic scale and the other is the seven- octaves range. At the composer's will and, for aesthetics, within the same composition, notes can be the outcome of different values of r and/or shifted in any octave. Compositions with endings of non-repeating note patterns result from values of r in the m-model that do not produce bifurcations. Scores of chaotic music composed from the m-model and the classical logistic model are presented.

  11. Noise activated bistable sensor based on chaotic system with output defined by temporal coding and firing rate.

    Korneta, Wojciech; Gomes, Iacyel

    2017-11-01

    Traditional bistable sensors use external bias signal to drive its response between states and their detection strategy is based on the output power spectral density or the residence time difference (RTD) in two sensor states. Recently, the noise activated nonlinear dynamic sensors driven only by noise based on RTD technique have been proposed. Here, we present experimental results of dc voltage measurements by noise-driven bistable sensor based on electronic Chua's circuit operating in a chaotic regime where two single scroll attractors coexist. The output of the sensor is quantified by the proportion of the time the sensor stays in one state to the total observation time and by the spike-count rate with spikes defined by crossings between attractors. The relationship between the stimuli and particular observable for different noise intensities is obtained, the usefulness of each coding scheme is discussed, and the optimal noise intensity for detection is indicated. It is shown that the obtained relationship is the same for any observation time when population coding is used. The optimal time window for both detection and the number of units in population coding is found. Our results may be useful for analyses and understanding of the neural activity and in designing bistable storage elements at length scales where thermal fluctuations drastically increase and the effect of noise must be taken into consideration.

  12. Image Encryption and Chaotic Cellular Neural Network

    Peng, Jun; Zhang, Du

    Machine learning has been playing an increasingly important role in information security and assurance. One of the areas of new applications is to design cryptographic systems by using chaotic neural network due to the fact that chaotic systems have several appealing features for information security applications. In this chapter, we describe a novel image encryption algorithm that is based on a chaotic cellular neural network. We start by giving an introduction to the concept of image encryption and its main technologies, and an overview of the chaotic cellular neural network. We then discuss the proposed image encryption algorithm in details, which is followed by a number of security analyses (key space analysis, sensitivity analysis, information entropy analysis and statistical analysis). The comparison with the most recently reported chaos-based image encryption algorithms indicates that the algorithm proposed in this chapter has a better security performance. Finally, we conclude the chapter with possible future work and application prospects of the chaotic cellular neural network in other information assurance and security areas.

  13. Qualitative feature extractions of chaotic systems

    Vicha, T.; Dohnal, M.

    2008-01-01

    The theory of chaos offers useful tools for systems analysis. However, models of complex systems are based on a network of inconsistent, space and uncertain knowledge items. Traditional quantitative methods of chaos analysis are therefore not applicable. The paper by the same authors [Vicha T, Dohnal M. Qualitative identification of chaotic systems behaviours. Chaos, Solitons and Fractals, in press, [Log. No. 601019] ] presents qualitative interpretation of some chaos concepts. There are only three qualitative values positive/increasing, negative/decreasing and zero/constant. It means that any set of qualitative multidimensional descriptions of unsteady state behaviours is discrete and finite. A finite upper limit exists for the total number of qualitatively distinguishable scenarios. A set of 21 published chaotic models is solved qualitatively and 21 sets of all existing qualitative scenarios are presented. The intersection of all 21 scenario sets is empty. There is no such a behaviour which is common for all 21 models. The set of 21 qualitative models (e.g. Lorenz, Roessler) can be used to compare chaotic behaviours of an unknown qualitative model with them to evaluate if its chaotic behaviours is close to e.g. Lorenz chaotic model and how much

  14. The dynamical and statistical properties of cognitive strategies: relations between strategies, attractors, and latent classes

    van der Maas, H.L.J.; Newell, K.; Molenaar, P.C.M.

    1998-01-01

    Cognitive developmental psychology is faced with new developments in the mathematical theory of nonlinear dynamic systems and in psychometrics. This chapter addresses: the relation between the strategy concept in cognitive developmental psychology and the concept of attractor in nonlinear dynamic

  15. Random Attractors for the Stochastic Navier-Stokes Equations on the 2D Unit Sphere

    Brzeźniak, Z.; Goldys, B.; Le Gia, Q. T.

    2018-03-01

    In this paper we prove the existence of random attractors for the Navier-Stokes equations on 2 dimensional sphere under random forcing irregular in space and time. We also deduce the existence of an invariant measure.

  16. Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay

    Chueshov, I.; Rezunenko, Oleksandr

    2015-01-01

    Roč. 14, č. 5 (2015), s. 1685-1704 ISSN 1534-0392 R&D Projects: GA ČR GAP103/12/2431 Institutional support: RVO:67985556 Keywords : Parabolic evolution equations * state-dependent delay * global attractor * finite-dimension * exponential attractor Subject RIV: BC - Control Systems Theory Impact factor: 0.926, year: 2015 http://library.utia.cas.cz/separaty/2015/AS/rezunenko-0444705.pdf

  17. DETECTION OF FLUX EMERGENCE, SPLITTING, MERGING, AND CANCELLATION OF NETWORK FIELD. I. SPLITTING AND MERGING

    Iida, Y.; Yokoyama, T. [Department of Earth and Planetary Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033 (Japan); Hagenaar, H. J. [Lockheed Martin Advanced Technology Center, Org. ADBS, Building 252, 3251 Hanover Street, Palo Alto, CA 94304 (United States)

    2012-06-20

    Frequencies of magnetic patch processes on the supergranule boundary, namely, flux emergence, splitting, merging, and cancellation, are investigated through automatic detection. We use a set of line-of-sight magnetograms taken by the Solar Optical Telescope (SOT) on board the Hinode satellite. We found 1636 positive patches and 1637 negative patches in the data set, whose time duration is 3.5 hr and field of view is 112'' Multiplication-Sign 112''. The total numbers of magnetic processes are as follows: 493 positive and 482 negative splittings, 536 positive and 535 negative mergings, 86 cancellations, and 3 emergences. The total numbers of emergence and cancellation are significantly smaller than those of splitting and merging. Further, the frequency dependence of the merging and splitting processes on the flux content are investigated. Merging has a weak dependence on the flux content with a power-law index of only 0.28. The timescale for splitting is found to be independent of the parent flux content before splitting, which corresponds to {approx}33 minutes. It is also found that patches split into any flux contents with the same probability. This splitting has a power-law distribution of the flux content with an index of -2 as a time-independent solution. These results support that the frequency distribution of the flux content in the analyzed flux range is rapidly maintained by merging and splitting, namely, surface processes. We suggest a model for frequency distributions of cancellation and emergence based on this idea.

  18. From determinism and probability to chaos: chaotic evolution towards philosophy and methodology of chaotic optimization.

    Pei, Yan

    2015-01-01

    We present and discuss philosophy and methodology of chaotic evolution that is theoretically supported by chaos theory. We introduce four chaotic systems, that is, logistic map, tent map, Gaussian map, and Hénon map, in a well-designed chaotic evolution algorithm framework to implement several chaotic evolution (CE) algorithms. By comparing our previous proposed CE algorithm with logistic map and two canonical differential evolution (DE) algorithms, we analyse and discuss optimization performance of CE algorithm. An investigation on the relationship between optimization capability of CE algorithm and distribution characteristic of chaotic system is conducted and analysed. From evaluation result, we find that distribution of chaotic system is an essential factor to influence optimization performance of CE algorithm. We propose a new interactive EC (IEC) algorithm, interactive chaotic evolution (ICE) that replaces fitness function with a real human in CE algorithm framework. There is a paired comparison-based mechanism behind CE search scheme in nature. A simulation experimental evaluation is conducted with a pseudo-IEC user to evaluate our proposed ICE algorithm. The evaluation result indicates that ICE algorithm can obtain a significant better performance than or the same performance as interactive DE. Some open topics on CE, ICE, fusion of these optimization techniques, algorithmic notation, and others are presented and discussed.

  19. From Determinism and Probability to Chaos: Chaotic Evolution towards Philosophy and Methodology of Chaotic Optimization

    2015-01-01

    We present and discuss philosophy and methodology of chaotic evolution that is theoretically supported by chaos theory. We introduce four chaotic systems, that is, logistic map, tent map, Gaussian map, and Hénon map, in a well-designed chaotic evolution algorithm framework to implement several chaotic evolution (CE) algorithms. By comparing our previous proposed CE algorithm with logistic map and two canonical differential evolution (DE) algorithms, we analyse and discuss optimization performance of CE algorithm. An investigation on the relationship between optimization capability of CE algorithm and distribution characteristic of chaotic system is conducted and analysed. From evaluation result, we find that distribution of chaotic system is an essential factor to influence optimization performance of CE algorithm. We propose a new interactive EC (IEC) algorithm, interactive chaotic evolution (ICE) that replaces fitness function with a real human in CE algorithm framework. There is a paired comparison-based mechanism behind CE search scheme in nature. A simulation experimental evaluation is conducted with a pseudo-IEC user to evaluate our proposed ICE algorithm. The evaluation result indicates that ICE algorithm can obtain a significant better performance than or the same performance as interactive DE. Some open topics on CE, ICE, fusion of these optimization techniques, algorithmic notation, and others are presented and discussed. PMID:25879067

  20. From Determinism and Probability to Chaos: Chaotic Evolution towards Philosophy and Methodology of Chaotic Optimization

    Yan Pei

    2015-01-01

    Full Text Available We present and discuss philosophy and methodology of chaotic evolution that is theoretically supported by chaos theory. We introduce four chaotic systems, that is, logistic map, tent map, Gaussian map, and Hénon map, in a well-designed chaotic evolution algorithm framework to implement several chaotic evolution (CE algorithms. By comparing our previous proposed CE algorithm with logistic map and two canonical differential evolution (DE algorithms, we analyse and discuss optimization performance of CE algorithm. An investigation on the relationship between optimization capability of CE algorithm and distribution characteristic of chaotic system is conducted and analysed. From evaluation result, we find that distribution of chaotic system is an essential factor to influence optimization performance of CE algorithm. We propose a new interactive EC (IEC algorithm, interactive chaotic evolution (ICE that replaces fitness function with a real human in CE algorithm framework. There is a paired comparison-based mechanism behind CE search scheme in nature. A simulation experimental evaluation is conducted with a pseudo-IEC user to evaluate our proposed ICE algorithm. The evaluation result indicates that ICE algorithm can obtain a significant better performance than or the same performance as interactive DE. Some open topics on CE, ICE, fusion of these optimization techniques, algorithmic notation, and others are presented and discussed.