Lyapunov exponents

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Barreira, Luís

2017-01-01

This book offers a self-contained introduction to the theory of Lyapunov exponents and its applications, mainly in connection with hyperbolicity, ergodic theory and multifractal analysis. It discusses the foundations and some of the main results and main techniques in the area, while also highlighting selected topics of current research interest. With the exception of a few basic results from ergodic theory and the thermodynamic formalism, all the results presented include detailed proofs. The book is intended for all researchers and graduate students specializing in dynamical systems who are looking for a comprehensive overview of the foundations of the theory and a sample of its applications.

Adaptive control of nonlinear in parameters chaotic system via Lyapunov exponents placement

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Ayati, Moosa [Department of Electrical Engineering, K.N. Toosi University of Technology, Sayyed Khandan Bridge, Shariati Street, Tehran (Iran, Islamic Republic of)], E-mail: Ayati@dena.kntu.ac.ir; Khaki-Sedigh, Ali [Department of Electrical Engineering, K.N. Toosi University of Technology, Sayyed Khandan Bridge, Shariati Street, Tehran (Iran, Islamic Republic of)], E-mail: sedigh@kntu.ac.ir

2009-08-30

This paper proposes a new method for the adaptive control of nonlinear in parameters (NLP) chaotic systems. A method based on Lagrangian of a cost function is used to identify the parameters of the system. Estimation results are used to calculate the Lyapunov exponents adaptively. Finally, the Lyapunov exponents placement method is used to assign the desired Lyapunov exponents of the closed loop system.

Adaptive control of nonlinear in parameters chaotic system via Lyapunov exponents placement

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Ayati, Moosa; Khaki-Sedigh, Ali

2009-01-01

This paper proposes a new method for the adaptive control of nonlinear in parameters (NLP) chaotic systems. A method based on Lagrangian of a cost function is used to identify the parameters of the system. Estimation results are used to calculate the Lyapunov exponents adaptively. Finally, the Lyapunov exponents placement method is used to assign the desired Lyapunov exponents of the closed loop system.

Multiscale Lyapunov exponent for 2-microlocal functions

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Dhifaoui, Zouhaier; Kortas, Hedi; Ammou, Samir Ben

2009-01-01

The Lyapunov exponent is an important indicator of chaotic dynamics. Using wavelet analysis, we define a multiscale representation of this exponent which we demonstrate the scale-wise dependence for functions belonging to C x 0 s,s ' spaces. An empirical study involving simulated processes and financial time series corroborates the theoretical findings.

Critical behavior of the Lyapunov exponent in type-III intermittency

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Alvarez-Llamoza, O. [Departamento de Fisica, FACYT, Universidad de Carabobo, Valencia (Venezuela); Centro de Fisica Fundamental, Grupo de Caos y Sistemas Complejos, Universidad de Los Andes, Merida 5251, Merida (Venezuela)], E-mail: llamoza@ula.ve; Cosenza, M.G. [Centro de Fisica Fundamental, Grupo de Caos y Sistemas Complejos, Universidad de Los Andes, Merida 5251, Merida (Venezuela); Ponce, G.A. [Departamento de Fisica, Universidad Nacional Autonoma de Honduras (Honduras); Departamento de Ciencias Naturales, Universidad Pedagogica Nacional Francisco Morazan, Tegucigalpa (Honduras)

2008-04-15

The critical behavior of the Lyapunov exponent near the transition to robust chaos via type-III intermittency is determined for a family of one-dimensional singular maps. Critical boundaries separating the region of robust chaos from the region where stable fixed points exist are calculated on the parameter space of the system. A critical exponent {beta} expressing the scaling of the Lyapunov exponent is calculated along the critical curve corresponding to the type-III intermittent transition to chaos. It is found that {beta} varies on the interval 0 {<=} {beta} < 1/2 as a function of the order of the singularity of the map. This contrasts with earlier predictions for the scaling behavior of the Lyapunov exponent in type-III intermittency. The variation of the critical exponent {beta} implies a continuous change in the nature of the transition to chaos via type-III intermittency, from a second-order, continuous transition to a first-order, discontinuous transition.

Critical behavior of the Lyapunov exponent in type-III intermittency

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Alvarez-Llamoza, O.; Cosenza, M.G.; Ponce, G.A.

2008-01-01

The critical behavior of the Lyapunov exponent near the transition to robust chaos via type-III intermittency is determined for a family of one-dimensional singular maps. Critical boundaries separating the region of robust chaos from the region where stable fixed points exist are calculated on the parameter space of the system. A critical exponent β expressing the scaling of the Lyapunov exponent is calculated along the critical curve corresponding to the type-III intermittent transition to chaos. It is found that β varies on the interval 0 ≤ β < 1/2 as a function of the order of the singularity of the map. This contrasts with earlier predictions for the scaling behavior of the Lyapunov exponent in type-III intermittency. The variation of the critical exponent β implies a continuous change in the nature of the transition to chaos via type-III intermittency, from a second-order, continuous transition to a first-order, discontinuous transition

Anisotropies in magnetic field evolution and local Lyapunov exponents

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Tang, X.Z.; Boozer, A.H.

2000-01-01

The natural occurrence of small scale structures and the extreme anisotropy in the evolution of a magnetic field embedded in a conducting flow is interpreted in terms of the properties of the local Lyapunov exponents along the various local characteristic (un)stable directions for the Lagrangian flow trajectories. The local Lyapunov exponents and the characteristic directions are functions of Lagrangian coordinates and time, which are completely determined once the flow field is specified. The characteristic directions that are associated with the spatial anisotropy of the problem, are prescribed in both Lagrangian and Eulerian frames. Coordinate transformation techniques are employed to relate the spatial distributions of the magnetic field, the induced current density, and the Lorentz force, which are usually followed in Eulerian frame, to those of the local Lyapunov exponents, which are naturally defined in Lagrangian coordinates

Local Lyapunov exponents for dissipative continuous systems

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Grond, Florian; Diebner, Hans H.

2005-01-01

We analyze a recently proposed algorithm for computing Lyapunov exponents focusing on its capability to calculate reliable local values for chaotic attractors. The averaging process of local contributions to the global measure becomes interpretable, i.e. they are related to the local topological structure in phase space. We compare the algorithm with the commonly used Wolf algorithm by means of analyzing correlations between coordinates of the chaotic attractor and local values of the Lyapunov exponents. The correlations for the new algorithm turn out to be significantly stronger than those for the Wolf algorithm. Since the usage of scalar measures to capture complex structures can be questioned we discuss these entities along with a more phenomenological description of scatter plots

Statistical-mechanical formulation of Lyapunov exponents

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Tanase-Nicola, Sorin; Kurchan, Jorge

2003-01-01

We show how the Lyapunov exponents of a dynamic system can, in general, be expressed in terms of the free energy of a (non-Hermitian) quantum many-body problem. This puts their study as a problem of statistical mechanics, whose intuitive concepts and techniques of approximation can hence be borrowed

Riemannian theory of Hamiltonian chaos and Lyapunov exponents

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Casetti, Lapo; Clementi, Cecilia; Pettini, Marco

1996-12-01

A nonvanishing Lyapunov exponent λ1 provides the very definition of deterministic chaos in the solutions of a dynamical system; however, no theoretical mean of predicting its value exists. This paper copes with the problem of analytically computing the largest Lyapunov exponent λ1 for many degrees of freedom Hamiltonian systems as a function of ɛ=E/N, the energy per degree of freedom. The functional dependence λ1(ɛ) is of great interest because, among other reasons, it detects the existence of weakly and strongly chaotic regimes. This aim, the analytic computation of λ1(ɛ), is successfully reached within a theoretical framework that makes use of a geometrization of Newtonian dynamics in the language of Riemannian differential geometry. An alternative point of view about the origin of chaos in these systems is obtained independently of the standard explanation based on homoclinic intersections. Dynamical instability (chaos) is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of the Jacobi-Levi-Civita equation (JLCE) for geodesic spread. In this paper it is shown how to derive from the JLCE an effective stability equation. Under general conditions, this effective equation formally describes a stochastic oscillator; an analytic formula for the instability growth rate of its solutions is worked out and applied to the Fermi-Pasta-Ulam β model and to a chain of coupled rotators. Excellent agreement is found between the theoretical prediction and numeric values of λ1(ɛ) for both models.

Controlled test for predictive power of Lyapunov exponents: their inability to predict epileptic seizures.

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Lai, Ying-Cheng; Harrison, Mary Ann F; Frei, Mark G; Osorio, Ivan

2004-09-01

Lyapunov exponents are a set of fundamental dynamical invariants characterizing a system's sensitive dependence on initial conditions. For more than a decade, it has been claimed that the exponents computed from electroencephalogram (EEG) or electrocorticogram (ECoG) signals can be used for prediction of epileptic seizures minutes or even tens of minutes in advance. The purpose of this paper is to examine the predictive power of Lyapunov exponents. Three approaches are employed. (1) We present qualitative arguments suggesting that the Lyapunov exponents generally are not useful for seizure prediction. (2) We construct a two-dimensional, nonstationary chaotic map with a parameter slowly varying in a range containing a crisis, and test whether this critical event can be predicted by monitoring the evolution of finite-time Lyapunov exponents. This can thus be regarded as a "control test" for the claimed predictive power of the exponents for seizure. We find that two major obstacles arise in this application: statistical fluctuations of the Lyapunov exponents due to finite time computation and noise from the time series. We show that increasing the amount of data in a moving window will not improve the exponents' detective power for characteristic system changes, and that the presence of small noise can ruin completely the predictive power of the exponents. (3) We report negative results obtained from ECoG signals recorded from patients with epilepsy. All these indicate firmly that, the use of Lyapunov exponents for seizure prediction is practically impossible as the brain dynamical system generating the ECoG signals is more complicated than low-dimensional chaotic systems, and is noisy. Copyright 2004 American Institute of Physics

On some properties of the discrete Lyapunov exponent

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Amigo, Jose M.; Kocarev, Ljupco; Szczepanski, Janusz

2008-01-01

One of the possible by-products of discrete chaos is the application of its tools, in particular of the discrete Lyapunov exponent, to cryptography. In this Letter we explore this question in a very general setting

Lyapunov exponents a tool to explore complex dynamics

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Pikovsky, Arkady

2016-01-01

Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept. Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances in applications to complex systems. Practical algorithms are thoroughly reviewed and their performance is discussed, while a broad set of examples illustrate the wide range of potential applications. The description of various numerical and analytical techniques for the computation of Lyapunov exponents offers an extensive array of tools for the characterization of phenomena such as synchronization, weak and global chaos in low and high-dimensional set-ups, and localization. This text equips readers with all the investigative expertise needed to fully explore the dynamical properties of complex systems, making it ideal for both graduate students and experienced researchers...

Lyapunov exponents for infinite dimensional dynamical systems

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Mhuiris, Nessan Mac Giolla

1987-01-01

Classically it was held that solutions to deterministic partial differential equations (i.e., ones with smooth coefficients and boundary data) could become random only through one mechanism, namely by the activation of more and more of the infinite number of degrees of freedom that are available to such a system. It is only recently that researchers have come to suspect that many infinite dimensional nonlinear systems may in fact possess finite dimensional chaotic attractors. Lyapunov exponents provide a tool for probing the nature of these attractors. This paper examines how these exponents might be measured for infinite dimensional systems.

On the relation between Lyapunov exponents and exponential decay of correlations

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Slipantschuk, Julia; Bandtlow, Oscar F; Just, Wolfram

2013-01-01

Chaotic dynamics with sensitive dependence on initial conditions may result in exponential decay of correlation functions. We show that for one-dimensional interval maps the corresponding quantities, that is, Lyapunov exponents and exponential decay rates, are related. More specifically, for piecewise linear expanding Markov maps observed via piecewise analytic functions, we show that the decay rate is bounded above by twice the Lyapunov exponent, that is, we establish lower bounds for the subleading eigenvalue of the corresponding Perron–Frobenius operator. In addition, we comment on similar relations for general piecewise smooth expanding maps. (paper)

Detection of the onset of numerical chaotic instabilities by lyapunov exponents

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Alicia Serfaty De Markus

2001-01-01

Full Text Available It is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided.

Conditional Lyapunov exponents and transfer entropy in coupled bursting neurons under excitation and coupling mismatch

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Soriano, Diogo C.; Santos, Odair V. dos; Suyama, Ricardo; Fazanaro, Filipe I.; Attux, Romis

2018-03-01

This work has a twofold aim: (a) to analyze an alternative approach for computing the conditional Lyapunov exponent (λcmax) aiming to evaluate the synchronization stability between nonlinear oscillators without solving the classical variational equations for the synchronization error dynamical system. In this first framework, an analytic reference value for λcmax is also provided in the context of Duffing master-slave scenario and precisely evaluated by the proposed numerical approach; (b) to apply this technique to the study of synchronization stability in chaotic Hindmarsh-Rose (HR) neuronal models under uni- and bi-directional resistive coupling and different excitation bias, which also considered the root mean square synchronization error, information theoretic measures and asymmetric transfer entropy in order to offer a better insight of the synchronization phenomenon. In particular, statistical and information theoretical measures were able to capture similarity increase between the neuronal oscillators just after a critical coupling value in accordance to the largest conditional Lyapunov exponent behavior. On the other hand, transfer entropy was able to detect neuronal emitter influence even in a weak coupling condition, i.e. under the increase of conditional Lyapunov exponent and apparently desynchronization tendency. In the performed set of numerical simulations, the synchronization measures were also evaluated for a two-dimensional parameter space defined by the neuronal coupling (emitter to a receiver neuron) and the (receiver) excitation current. Such analysis is repeated for different feedback couplings as well for different (emitter) excitation currents, revealing interesting characteristics of the attained synchronization region and conditions that facilitate the emergence of the synchronous behavior. These results provide a more detailed numerical insight of the underlying behavior of a HR in the excitation and coupling space, being in accordance

New prediction of chaotic time series based on local Lyapunov exponent

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Zhang Yong

2013-01-01

A new method of predicting chaotic time series is presented based on a local Lyapunov exponent, by quantitatively measuring the exponential rate of separation or attraction of two infinitely close trajectories in state space. After reconstructing state space from one-dimensional chaotic time series, neighboring multiple-state vectors of the predicting point are selected to deduce the prediction formula by using the definition of the local Lyapunov exponent. Numerical simulations are carried out to test its effectiveness and verify its higher precision over two older methods. The effects of the number of referential state vectors and added noise on forecasting accuracy are also studied numerically. (general)

An Isomorphism between Lyapunov Exponents and Shannon's Channel Capacity

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Friedland, Gerald [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Metere, Alfredo [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2017-06-07

We demonstrate that discrete Lyapunov exponents are isomorphic to numeric overflows of the capacity of an arbitrary noiseless and memoryless channel in a Shannon communication model with feedback. The isomorphism allows the understanding of Lyapunov exponents in terms of Information Theory, rather than the traditional definitions in chaos theory. The result also implies alternative approaches to the calculation of related quantities, such as the Kolmogorov Sinai entropy which has been linked to thermodynamic entropy. This work provides a bridge between fundamental physics and information theory. It suggests, among other things, that machine learning and other information theory methods can be employed at the core of physics simulations.

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

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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.

Application of the Lyapunov exponent to detect noise-induced chaos in oscillating microbial cultures

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Patnaik, P.R.

2005-01-01

Oscillating microbial processes can, under certain conditions, gravitate into chaotic behavior induced by external noise. Detection and control of chaos are important for the survival of the microorganisms and to operate a process usefully. In this study the largest Lyapunov exponent is recommended as a convenient and reliable index of chaos in continuous oscillating cultures. For the growth of Saccharomyces cerevisiae as a model system, the exponents increase with the oxygen mass transfer coefficient and decrease as the dilution rate increases. By comparing with the corresponding time-domain oscillations determined earlier, it is inferred that weakly oscillating cultures are less likely to be driven to chaotic behavior. The main carbon source, glucose, is quite robust to chaotic destabilization, thus enhancing its suitability as a manipulated variable for bioreactor control

Hyperchaos of four state autonomous system with three positive Lyapunov exponents

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Ge Zhengming; Yang, C-H.

2009-01-01

This Letter gives the results of numerical simulations of Quantum Cellular Neural Network (Quantum-CNN) autonomous system with four state variables. Three positive Lyapunov exponents confirm hyperchaotic nature of its dynamics

Effect of noise and filtering on largest Lyapunov exponent of time series associated with human walking.

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Mehdizadeh, Sina; Sanjari, Mohammad Ali

2017-11-07

This study aimed to determine the effect of added noise, filtering and time series length on the largest Lyapunov exponent (LyE) value calculated for time series obtained from a passive dynamic walker. The simplest passive dynamic walker model comprising of two massless legs connected by a frictionless hinge joint at the hip was adopted to generate walking time series. The generated time series was used to construct a state space with the embedding dimension of 3 and time delay of 100 samples. The LyE was calculated as the exponential rate of divergence of neighboring trajectories of the state space using Rosenstein's algorithm. To determine the effect of noise on LyE values, seven levels of Gaussian white noise (SNR=55-25dB with 5dB steps) were added to the time series. In addition, the filtering was performed using a range of cutoff frequencies from 3Hz to 19Hz with 2Hz steps. The LyE was calculated for both noise-free and noisy time series with different lengths of 6, 50, 100 and 150 strides. Results demonstrated a high percent error in the presence of noise for LyE. Therefore, these observations suggest that Rosenstein's algorithm might not perform well in the presence of added experimental noise. Furthermore, findings indicated that at least 50 walking strides are required to calculate LyE to account for the effect of noise. Finally, observations support that a conservative filtering of the time series with a high cutoff frequency might be more appropriate prior to calculating LyE. Copyright © 2017 Elsevier Ltd. All rights reserved.

Hyperbolicity and integral expression of the Lyapunov exponents for linear cocycles

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Dai, Xiongping

Consider in this paper a linear skew-product system (θ,Θ) :T×W×R→W×R; (t,w,x)↦(tw,Θ(t,w)ṡx) where T=R or Z, and θ :(t,w)↦tw is a topological dynamical system on a compact metrizable space W, and where Θ(t,w)∈GL(n,R) satisfies the cocycle condition based on θ and is continuously differentiable in t if T=R. We show that 'semi λ-exponential dichotomy' of (θ,Θ) implies ' λ-exponential dichotomy.' Precisely, if Θ has no Lyapunov exponent λ and is almost uniformly λ-contracting along the λ-stable direction E(w;λ) and if dimE(w;λ) is constant a.e., then Θ is almost λ-exponentially dichotomous. To prove this, we first use Liao's spectrum theorem, which gives integral expression of the Lyapunov exponents, and then use the semi-uniform ergodic theorem by Sturman and Stark, which allows one to derive uniform estimates from nonuniform ones. As a consequence, we obtain the open-and-dense hyperbolicity of eventual GL(2,R)-cocycles based on a uniquely ergodic endomorphism, and of GL(2,R)-cocycles based on a uniquely ergodic equi-continuous endomorphism, respectively. On the other hand, in the sense of C-topology we obtain the density of SL(2,R)-cocycles having positive Lyapunov exponent based on a minimal subshift satisfying the Boshernitzan condition.

Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents.

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Salceanu, Paul L

2011-07-01

This paper extends the work of Salceanu and Smith [12, 13] where Lyapunov exponents were used to obtain conditions for uniform persistence ina class of dissipative discrete-time dynamical systems on the positive orthant of R(m), generated by maps. Here a united approach is taken, for both discrete and continuous time, and the dissipativity assumption is relaxed. Sufficient conditions are given for compact subsets of an invariant part of the boundary of R(m+) to be robust uniform weak repellers. These conditions require Lyapunov exponents be positive on such sets. It is shown how this leads to robust uniform persistence. The results apply to the investigation of robust uniform persistence of the disease in host populations, as shown in an application.

Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map

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Gritli, Hassène; Belghith, Safya

2015-01-01

Highlights: • A numerical calculation method of the Lyapunov exponents in the compass-gait model under OGY control is proposed. • A new linearization method of the impulsive hybrid dynamics around a one-periodic hybrid limit cycle is achieved. • We develop a simple analytical expression of a controlled hybrid Poincaré map. • A dimension reduction of the hybrid Poincaré map is realized. • We describe the numerical computation procedure of the Lyapunov exponents via the designed hybrid Poincaré map. - Abstract: This paper aims at providing a numerical calculation method of the spectrum of Lyapunov exponents in a four-dimensional impulsive hybrid nonlinear dynamics of a passive compass-gait model under the OGY control approach by means of a controlled hybrid Poincaré map. We present a four-dimensional simplified analytical expression of such hybrid map obtained by linearizing the uncontrolled impulsive hybrid nonlinear dynamics around a desired one-periodic passive hybrid limit cycle. In order to compute the spectrum of Lyapunov exponents, a dimension reduction of the controlled hybrid Poincaré map is realized. The numerical calculation of the spectrum of Lyapunov exponents using the reduced-dimension controlled hybrid Poincaré map is given in detail. In order to show the effectiveness of the developed method, the spectrum of Lyapunov exponents is calculated as the slope (bifurcation) parameter varies and hence used to predict the walking dynamics behavior of the compass-gait model under the OGY control.

Lyapunov exponent as a metric for assessing the dynamic content and predictability of large-eddy simulations

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Nastac, Gabriel; Labahn, Jeffrey W.; Magri, Luca; Ihme, Matthias

2017-09-01

Metrics used to assess the quality of large-eddy simulations commonly rely on a statistical assessment of the solution. While these metrics are valuable, a dynamic measure is desirable to further characterize the ability of a numerical simulation for capturing dynamic processes inherent in turbulent flows. To address this issue, a dynamic metric based on the Lyapunov exponent is proposed which assesses the growth rate of the solution separation. This metric is applied to two turbulent flow configurations: forced homogeneous isotropic turbulence and a turbulent jet diffusion flame. First, it is shown that, despite the direct numerical simulation (DNS) and large-eddy simulation (LES) being high-dimensional dynamical systems with O (107) degrees of freedom, the separation growth rate qualitatively behaves like a lower-dimensional dynamical system, in which the dimension of the Lyapunov system is substantially smaller than the discretized dynamical system. Second, a grid refinement analysis of each configuration demonstrates that as the LES filter width approaches the smallest scales of the system the Lyapunov exponent asymptotically approaches a plateau. Third, a small perturbation is superimposed onto the initial conditions of each configuration, and the Lyapunov exponent is used to estimate the time required for divergence, thereby providing a direct assessment of the predictability time of simulations. By comparing inert and reacting flows, it is shown that combustion increases the predictability of the turbulent simulation as a result of the dilatation and increased viscosity by heat release. The predictability time is found to scale with the integral time scale in both the reacting and inert jet flows. Fourth, an analysis of the local Lyapunov exponent is performed to demonstrate that this metric can also determine flow-dependent properties, such as regions that are sensitive to small perturbations or conditions of large turbulence within the flow field. Finally

Modulational estimate for the maximal Lyapunov exponent in Fermi-Pasta-Ulam chains

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Dauxois, Thierry; Ruffo, Stefano; Torcini, Alessandro

1997-12-01

In the framework of the Fermi-Pasta-Ulam (FPU) model, we show a simple method to give an accurate analytical estimation of the maximal Lyapunov exponent at high energy density. The method is based on the computation of the mean value of the modulational instability growth rates associated to unstable modes. Moreover, we show that the strong stochasticity threshold found in the β-FPU system is closely related to a transition in tangent space, the Lyapunov eigenvector being more localized in space at high energy.

Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System.

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Rozenbaum, Efim B; Ganeshan, Sriram; Galitski, Victor

2017-02-24

It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because, in the semiclassical limit ℏ→0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator C(t) for the classical and quantum kicked rotor-a textbook driven chaotic system-and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are, in general, distinct quantities, corresponding to the logarithm of the phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength K, where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as K→0, while the OTOC's growth rate may decrease much slower, showing a higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time t_{E}: transitioning from a time-independent value of t^{-1}lnC(t) at ttime at t>t_{E}. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996)PRBMDO0163-182910.1103/PhysRevB.54.14423; Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.124101] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.

The brief time-reversibility of the local Lyapunov exponents for a small chaotic Hamiltonian system

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Waldner, Franz; Hoover, William G.; Hoover, Carol G.

2014-01-01

Highlights: •We consider the local Lyapunov spectrum for a four-dimensional Hamilton system. •Its stable periodic motion can be reversed for long times. •In the chaotic motion, time reversal occurs only for a short time. •Perturbations will change this short unstable case into a different stable case. •These observations might relate chaos to the Second Law of Thermodynamics. - Abstract: We consider the local (instantaneous) Lyapunov spectrum for a four-dimensional Hamiltonian system. Its stable periodic motion can be reversed for long times. Its unstable chaotic motion, with two symmetric pairs of exponents, cannot. In the latter case reversal occurs for more than a thousand fourth-order Runge–Kutta time steps, followed by a transition to a new set of paired Lyapunov exponents, unrelated to those seen in the forward time direction. The relation of the observed chaotic dynamics to the Second Law of Thermodynamics is discussed

Assessment of Effects of a Delay Block and a Nonlinear Block in Systems with Chaotic Behavior Using Lyapunov Exponents

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Pablo César Rodríguez Gómez

2017-05-01

Full Text Available Context: Because feedback systems are very common and widely used, studies of the structural characteristics under which chaotic behavior is generated have been developed. These can be separated into a nonlinear system and a linear system at least of the third order. Methods such as the descriptive function have been used for analysis. Method: A feedback system is proposed comprising a linear system, a nonlinear system and a delay block, in order to assess his behavior using Lyapunov exponents. It is evaluated with three different linear systems, different delay values and different values for parameters of nonlinear characteristic, aiming to reach chaotic behavior. Results: One hundred experiments were carried out for each of the three linear systems, by changing the value of some parameters, assessing their influence on the dynamics of the system. Contour plots that relate these parameters to the Largest Lyapunov exponent were obtained and analyzed. Conclusions: In spite non-linearity is a condition for the existence of chaos, this does not imply that any nonlinear characteristic generates a chaotic system, it is reflected by the contour plots showing the transitions between chaotic and no chaotic behavior of the feedback system. Language: English

Lyapunov exponent and topological entropy plateaus in piecewise linear maps

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Botella-Soler, V; Oteo, J A; Ros, J; Glendinning, P

2013-01-01

We consider a two-parameter family of piecewise linear maps in which the moduli of the two slopes take different values. We provide numerical evidence of the existence of some parameter regions in which the Lyapunov exponent and the topological entropy remain constant. Analytical proof of this phenomenon is also given for certain cases. Surprisingly however, the systems with that property are not conjugate as we prove by using kneading theory. (paper)

Comment on 'Exact analytical solution for the generalized Lyapunov exponent of the two-dimensional Anderson localization'

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Markos, P; Schweitzer, L; Weyrauch, M

2004-01-01

In a recent publication, Kuzovkov et al (2002 J. Phys.: Condens. Matter. 14 13777) announced an analytical solution of the two-dimensional Anderson localization problem via the calculation of a generalized Lyapunov exponent using signal theory. Surprisingly, for certain energies and small disorder strength they observed delocalized states. We study the transmission properties of the same model using well-known transfer matrix methods. Our results disagree with the findings obtained using signal theory. We point to the possible origin of this discrepancy and comment on the general strategy of using a generalized Lyapunov exponent for studying Anderson localization. (comment)

Adiabatic invariants and asymptotic behavior of Lyapunov exponents of the Schrodinger equation

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Delyon, F.; Foulon, P.

1986-01-01

We give an upper bound for the high-energy behavior of the Lyapunov exponent of the one-dimensional Schrodinger equation. We relate this behavior to the diffrentiability properties of the potential. As an application, this result provides an upper bound for the asymptotic length of the gaps of the Schrodinger equation

A Hyperchaotic Attractor with Multiple Positive Lyapunov Exponents

International Nuclear Information System (INIS)

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

Determination of the Lyapunov exponents and the information dimension in some dynamical systems

International Nuclear Information System (INIS)

Ziar, A.

1992-01-01

Classical phase space for some dynamical systems relevant in nuclear physics are studied. The nuclei is described by convex billiards or in the mean field theory. In both cases, besides the Poincare surface of sections which gives a qualitative description, each trajectory is characterized by its maximum Lyapunov exponent. The analytic monodromy matrix for a free particle in convex billiards rotating around an axis perpendicular to the plan of billiards, is determined, generalizing a previous result obtained for static billiards. In the frame of the mean field theory, it is shown an interesting alternative to the Lyapunov exponent, which is the dimension of the manifold in the phase space associated to the trajectory, leading to the evaluation of the relative chaotic volume in phase space as a function of the different parameters. The dimension appears as a character which could be determined easily for the rotating mean field, where the dimension of the manifold on which the trajectory is lying could be equal to 5 or 4 for chaotic trajectories, and less or equal to 3 for regular ones

Perturbation theory for Lyapunov exponents of an Anderson model on a strip

CERN Document Server

Schulz-Baldes, H

2003-01-01

It is proven that the localization length of an Anderson model on a strip of width $L$ is bounded above by $L/\\lambda^2$ for small values of the coupling constant $\\lambda$ of the disordered potential. For this purpose, a new formalism is developed in order to calculate the bottom Lyapunov exponent associated with random products of large symplectic matrices perturbatively in the coupling constant of the randomness.

Geodesic stability, Lyapunov exponents, and quasinormal modes

International Nuclear Information System (INIS)

Cardoso, Vitor; Miranda, Alex S.; Berti, Emanuele; Witek, Helvi; Zanchin, Vilson T.

2009-01-01

Geodesic motion determines important features of spacetimes. Null unstable geodesics are closely related to the appearance of compact objects to external observers and have been associated with the characteristic modes of black holes. By computing the Lyapunov exponent, which is the inverse of the instability time scale associated with this geodesic motion, we show that, in the eikonal limit, quasinormal modes of black holes in any dimensions are determined by the parameters of the circular null geodesics. This result is independent of the field equations and only assumes a stationary, spherically symmetric and asymptotically flat line element, but it does not seem to be easily extendable to anti-de Sitter spacetimes. We further show that (i) in spacetime dimensions greater than four, equatorial circular timelike geodesics in a Myers-Perry black-hole background are unstable, and (ii) the instability time scale of equatorial null geodesics in Myers-Perry spacetimes has a local minimum for spacetimes of dimension d≥6.

A new interpretation of zero Lyapunov exponents in BKL time for Mixmaster cosmology

International Nuclear Information System (INIS)

Wu Xin

2010-01-01

A global relationship between cosmological time and Belinskii-Khalatnikov-Lifshitz (BKL) time during the entire evolution of the Mixmaster Bianchi IX universe is used to explain why all the Lyapunov exponents are zero at the BKL time. The actual reason is that the domain of the cosmological time is finite as the BKL time runs from minus infinity to infinity.

Effective Power-Law Dependence of Lyapunov Exponents on the Central Mass in Galaxies

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Delis, N.; Efthymiopoulos, C.; Kalapotharakos, C.

2015-01-01

Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation L alpha m(sup p) between themean Lyapunov exponent L of stellar orbits chaotically scattered by a supermassive black hole (BH) in the centre of a galaxy and the mass parameter m, i.e. ratio of the mass of the BH over the mass of the galaxy. The exponent p is found numerically to obtain values in the range p approximately equals 0.3-0.5. We propose a theoretical interpretation of these exponents, based on estimates of local 'stretching numbers', i.e. local Lyapunov exponents at successive transits of the orbits through the BH's sphere of influence. We thus predict p = 2/3 - q with q approximately equaling 0.1-0.2. Our basic model refers to elliptical galaxy models with a central core. However, we find numerically that an effective power-law scaling of L with m holds also in models with central cusp, beyond a mass scale up to which chaos is dominated by the influence of the cusp itself. We finally show numerically that an analogous law exists also in disc galaxies with rotating bars. In the latter case, chaotic scattering by the BH affects mainly populations of thick tube-like orbits surrounding some low-order branches of the x(sub 1) family of periodic orbits, as well as its bifurcations at low-order resonances, mainly the inner Lindblad resonance and the 4/1 resonance. Implications of the correlations between L and m to determining the rate of secular evolution of galaxies are discussed.

Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows

International Nuclear Information System (INIS)

Novo, Sylvia; Obaya, Rafael; Sanz, Ana M

2013-01-01

Several results of uniform persistence above and below a minimal set of an abstract monotone skew-product semiflow are obtained. When the minimal set has a continuous separation the results are given in terms of the principal spectrum. In the case that the semiflow is generated by the solutions of a family of non-autonomous differential equations of ordinary, delay or parabolic type, the former results are strongly improved. A method of calculus of the upper Lyapunov exponent of the minimal set is also determined. (paper)

Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series

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A. M. López Jiménez

2002-01-01

Full Text Available The literature about non-linear dynamics offers a few recommendations, which sometimes are divergent, about the criteria to be used in order to select the optimal calculus parameters in the estimation of Lyapunov exponents by direct methods. These few recommendations are circumscribed to the analysis of chaotic systems. We have found no recommendation for the estimation of λ starting from the time series of classic systems. The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of non-linear terms from the analysis of time series. In this study we have centered in the dependence of the Lyapunov exponent, obtained by means of direct estimation, of the initial distance and the time evolution. We have used generated series of chaotic systems and generated series of classic systems with varying complexity. To generate the series we have used the logistic map.

Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system

International Nuclear Information System (INIS)

Zhang Jiangang; Li Xianfeng; Chu Yandong; Yu Jianning; Chang Yingxiang

2009-01-01

In this paper, complex dynamical behavior of a class of centrifugal flywheel governor system is studied. These systems have a rich variety of nonlinear behavior, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Bubbles of periodic orbits may also occur within the bifurcation sequence. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincare maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincare sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. This paper proposes a parametric open-plus-closed-loop approach to controlling chaos, which is capable of switching from chaotic motion to any desired periodic orbit. The theoretical work and numerical simulations of this paper can be extended to other systems. Finally, the results of this paper are of practical utility to designers of rotational machines.

Reliability of Lyapunov characteristic exponents computed by the two-particle method

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Mei, Lijie; Huang, Li

2018-03-01

For highly complex problems, such as the post-Newtonian formulation of compact binaries, the two-particle method may be a better, or even the only, choice to compute the Lyapunov characteristic exponent (LCE). This method avoids the complex calculations of variational equations compared with the variational method. However, the two-particle method sometimes provides spurious estimates to LCEs. In this paper, we first analyze the equivalence in the definition of LCE between the variational and two-particle methods for Hamiltonian systems. Then, we develop a criterion to determine the reliability of LCEs computed by the two-particle method by considering the magnitude of the initial tangent (or separation) vector ξ0 (or δ0), renormalization time interval τ, machine precision ε, and global truncation error ɛT. The reliable Lyapunov characteristic indicators estimated by the two-particle method form a V-shaped region, which is restricted by d0, ε, and ɛT. Finally, the numerical experiments with the Hénon-Heiles system, the spinning compact binaries, and the post-Newtonian circular restricted three-body problem strongly support the theoretical results.

Evaluating Noise Sensitivity on the Time Series Determination of Lyapunov Exponents Applied to the Nonlinear Pendulum

Directory of Open Access Journals (Sweden)

L.F.P. Franca

2003-01-01

Full Text Available This contribution presents an investigation on noise sensitivity of some of the most disseminated techniques employed to estimate Lyapunov exponents from time series. Since noise contamination is unavoidable in cases of data acquisition, it is important to recognize techniques that could be employed for a correct identification of chaos. State space reconstruction and the determination of Lyapunov exponents are carried out to investigate the response of a nonlinear pendulum. Signals are generated by numerical integration of the mathematical model, selecting a single variable of the system as a time series. In order to simulate experimental data sets, a random noise is introduced in the signal. Basically, the analyses of periodic and chaotic motions are carried out. Results obtained from mathematical model are compared with the one obtained from time series analysis, evaluating noise sensitivity. This procedure allows the identification of the best techniques to be employed in the analysis of experimental data.

Moment Lyapunov Exponent and Stochastic Stability of Binary Airfoil under Combined Harmonic and Non-Gaussian Colored Noise Excitations

Science.gov (United States)

Hu, D. L.; Liu, X. B.

Both periodic loading and random forces commonly co-exist in real engineering applications. However, the dynamic behavior, especially dynamic stability of systems under parametric periodic and random excitations has been reported little in the literature. In this study, the moment Lyapunov exponent and stochastic stability of binary airfoil under combined harmonic and non-Gaussian colored noise excitations are investigated. The noise is simplified to an Ornstein-Uhlenbeck process by applying the path-integral method. Via the singular perturbation method, the second-order expansions of the moment Lyapunov exponent are obtained, which agree well with the results obtained by the Monte Carlo simulation. Finally, the effects of the noise and parametric resonance (such as subharmonic resonance and combination additive resonance) on the stochastic stability of the binary airfoil system are discussed.

Analysis and control of chaotic behavior in boost converter by ramp compensation based on Lyapunov exponents assignment: theoretical and experimental investigation

International Nuclear Information System (INIS)

Zamani, Najmeh; Ataei, Mohammad; Niroomand, Mehdi

2015-01-01

Highlights: • Applying nonlinear analysis of complex dynamics displayed by current-mode controlled boost converter. • The ramp compensation method is used to control bifurcation and chaos in these converters based on bifurcation diagram and Lyapunov exponents assignment. • A discrete-time iterative nonlinear mapping model has been derived by inserting the ramp compensation parameter in the dynamical equations of the system. • A design methodology for chaos control is provided in this converter based on Lyapunov exponents assignment in desired values theoretically by proper selection of compensator slope. • Practical results are provided to confirm the theoretical analysis and simulations. - Abstract: Nonlinear analysis of complex dynamics displayed by current mode dc–dc converter and idea of Lyapunov exponents assignment by ramp compensator in order to control chaotic behavior is proposed in this article. A discrete-time iterative nonlinear mapping model is derived. The occurrence of the complex behaviors of bifurcation and chaos generated by varying the circuit parameters are investigated through numerical analysis and software implementation of the circuit. Next, in order to control bifurcation and chaos in these converters, the ramp compensation method is used. By inserting the ramp compensation parameter in the dynamical equations of the system, these complex behaviors are examined theoretically and numerically as well. It is proved that through this method, the stable period-one operation of the converter can be extended. By evaluating the Lyapunov exponents (LEs) of the system, the impact of the slope on the location of LEs are determined analytically. This leads to a design methodology for control of chaos in this converter based on LEs assignment in desired values by proper selection of compensator slope. By developing an experimental set up, practical results are obtained to confirm the theoretical analysis and simulations.

On the angle between the first and second Lyapunov vectors in spatio-temporal chaos

International Nuclear Information System (INIS)

Pazó, D; López, J M; Rodríguez, M A

2013-01-01

In a dynamical system the first Lyapunov vector (LV) is associated with the largest Lyapunov exponent and indicates—at any point on the attractor—the direction of maximal growth in tangent space. The LV corresponding to the second largest Lyapunov exponent generally points in a different direction, but tangencies between both vectors can in principle occur. Here we find that the probability density function (PDF) of the angle ψ spanned by the first and second LVs should be expected to be approximately symmetric around π/4 and to peak at 0 and π/2. Moreover, for small angles we uncover a scaling law for the PDF Q of ψ l = ln ψ with the system size L: Q(ψ l ) = L −1/2 f(ψ l L −1/2 ). We give a theoretical argument that justifies this scaling form and also explains why it should be universal (irrespective of the system details) for spatio-temporal chaos in one spatial dimension. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’. (paper)

Control of chaos in permanent magnet synchronous motor by using optimal Lyapunov exponents placement

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Ataei, Mohammad, E-mail: ataei@eng.ui.ac.i [Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Hezar-Jerib St., Postal Code 8174673441, Isfahan (Iran, Islamic Republic of); Kiyoumarsi, Arash, E-mail: kiyoumarsi@eng.ui.ac.i [Department of Electrical Engineering, Faculty of Engineering, University of Isfahan, Hezar-Jerib St., Postal Code 8174673441, Isfahan (Iran, Islamic Republic of); Ghorbani, Behzad, E-mail: behzad.ghorbani63@gmail.co [Department of Control Engineering, Najafabad Azad University, Najafabad, Isfahan (Iran, Islamic Republic of)

2010-09-13

Permanent Magnet Synchronous Motor (PMSM) experiences chaotic behavior for a certain range of its parameters. In this case, since the performance of the PMSM degrades, the chaos should be eliminated. In this Letter, the control of the undesirable chaos in PMSM using Lyapunov exponents (LEs) placement is proposed that is also improved by choosing optimal locations of the LEs in the sense of predefined cost function. Moreover, in order to provide the physical realization of the method, nonlinear parameter estimator for the system is suggested. Finally, to show the effectiveness of the proposed methodology, the simulation results for applying this control strategy are provided.

Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper's Model

Science.gov (United States)

Jitomirskaya, S.; Marx, C. A.

2012-11-01

We show how to extend (and with what limitations) Avila's global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for the extended Harper's model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila's global theory is also shown to imply continuous behavior of the LE on the space of analytic {M_2({C})}-cocycles. This includes rational approximation of the frequency, which so far has not been available.

Importance sampling with imperfect cloning for the computation of generalized Lyapunov exponents

Science.gov (United States)

Anteneodo, Celia; Camargo, Sabrina; Vallejos, Raúl O.

2017-12-01

We revisit the numerical calculation of generalized Lyapunov exponents, L (q ) , in deterministic dynamical systems. The standard method consists of adding noise to the dynamics in order to use importance sampling algorithms. Then L (q ) is obtained by taking the limit noise-amplitude → 0 after the calculation. We focus on a particular method that involves periodic cloning and pruning of a set of trajectories. However, instead of considering a noisy dynamics, we implement an imperfect (noisy) cloning. This alternative method is compared with the standard one and, when possible, with analytical results. As a workbench we use the asymmetric tent map, the standard map, and a system of coupled symplectic maps. The general conclusion of this study is that the imperfect-cloning method performs as well as the standard one, with the advantage of preserving the deterministic dynamics.

Heat conduction in one-dimensional chains and nonequilibrium Lyapunov spectrum

International Nuclear Information System (INIS)

Posch, H.A.; Hoover, W.G.

1998-01-01

We define and study the heat conductivity κ and the Lyapunov spectrum for a modified 'ding-a-ling' chain undergoing steady heat flow. Free and bound particles alternate along a chain. In the present work, we use a linear gravitational potential to bind all the even-numbered particles to their lattice sites. The chain is bounded by two stochastic heat reservoirs, one hot and one cold. The Fourier conductivity of the chain decreases smoothly to a finite large-system limit. Special treatment of satellite collisions with the stochastic boundaries is required to obtain Lyapunov spectra. The summed spectra are negative, and correspond to a relatively small contraction in phase space, with the formation of a multifractal strange attractor. The largest of the Lyapunov exponents for the ding-a-ling chain appears to converge to a limiting value with increasing chain length, so that the large-system Lyapunov spectrum has a finite limit. copyright 1998 The American Physical Society

Lyapunov spectra of density fluctuations in TBR-1

International Nuclear Information System (INIS)

Oiwa, N.N.; Fidler-Ferrara, N.

1993-01-01

The results for the Lyapunov exponents associated with density fluctuations measured by Langmuir probes placed in the scrape-off layer of the Tokamak TBR-1 are reported. By a judicious use of the Sano-Sawada and Eckmann-Ruelle algorithms conclusive values for the positive Lyapunov exponents for most of the analysed signals are used showing evidences of chaotic behavior. (author)

Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents

KAUST Repository

Athanassoulis, Agissilaos; Katsaounis, Theodoros; Kyza, Irene

2016-01-01

Semiclassical asymptotics for Schrödinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P.L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as -|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posteriori error control. Thus rigorous upper bounds for the asymptotic error in concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for -|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM. © 2016 International Press.

Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents

KAUST Repository

Athanassoulis, Agissilaos

2016-08-30

Semiclassical asymptotics for Schrödinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P.L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as -|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posteriori error control. Thus rigorous upper bounds for the asymptotic error in concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for -|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM. © 2016 International Press.

Designing Hyperchaotic Cat Maps With Any Desired Number of Positive Lyapunov Exponents.

Science.gov (United States)

Hua, Zhongyun; Yi, Shuang; Zhou, Yicong; Li, Chengqing; Wu, Yue

2018-02-01

Generating chaotic maps with expected dynamics of users is a challenging topic. Utilizing the inherent relation between the Lyapunov exponents (LEs) of the Cat map and its associated Cat matrix, this paper proposes a simple but efficient method to construct an -dimensional ( -D) hyperchaotic Cat map (HCM) with any desired number of positive LEs. The method first generates two basic -D Cat matrices iteratively and then constructs the final -D Cat matrix by performing similarity transformation on one basic -D Cat matrix by the other. Given any number of positive LEs, it can generate an -D HCM with desired hyperchaotic complexity. Two illustrative examples of -D HCMs were constructed to show the effectiveness of the proposed method, and to verify the inherent relation between the LEs and Cat matrix. Theoretical analysis proves that the parameter space of the generated HCM is very large. Performance evaluations show that, compared with existing methods, the proposed method can construct -D HCMs with lower computation complexity and their outputs demonstrate strong randomness and complex ergodicity.

A statistical approach to estimate the LYAPUNOV spectrum in disc brake squeal

Science.gov (United States)

Oberst, S.; Lai, J. C. S.

2015-01-01

The estimation of squeal propensity of a brake system from the prediction of unstable vibration modes using the linear complex eigenvalue analysis (CEA) in the frequency domain has its fair share of successes and failures. While the CEA is almost standard practice for the automotive industry, time domain methods and the estimation of LYAPUNOV spectra have not received much attention in brake squeal analyses. One reason is the challenge in estimating the true LYAPUNOV exponents and their discrimination against spurious ones in experimental data. A novel method based on the application of the ECKMANN-RUELLE matrices is proposed here to estimate LYAPUNOV exponents by using noise in a statistical procedure. It is validated with respect to parameter variations and dimension estimates. By counting the number of non-overlapping confidence intervals for LYAPUNOV exponent distributions obtained by moving a window of increasing size over bootstrapped same-length estimates of an observation function, a dispersion measure's width is calculated and fed into a BAYESIAN beta-binomial model. Results obtained using this method for benchmark models of white and pink noise as well as the classical HENON map indicate that true LYAPUNOV exponents can be isolated from spurious ones with high confidence. The method is then applied to accelerometer and microphone data obtained from brake squeal tests. Estimated LYAPUNOV exponents indicate that the pad's out-of-plane vibration behaves quasi-periodically on the brink to chaos while the microphone's squeal signal remains periodic.

Lyapunov analysis: from dynamical systems theory to applications

Science.gov (United States)

Cencini, Massimo; Ginelli, Francesco

2013-06-01

[17], von Neumann [18], Krylov [19]3 and Asonov and Sinai [20] on ergodic theory. Lyapunov exponents quantify exponential sensitivity to initial conditions and provide direct access to the entropy production in ergodic systems via the Pesin theory [21]. Further advances have been made possible by the introduction of proper physical invariant measures for certain dissipative systems due to Sinai [22], Ruelle [23] and Bowen [24, 25]. However, it was necessary to wait until the end of the 1970s before the independent works of Shimada and Nagashima [26] and Benettin et al [27] introduced the numerical algorithms required to compute Lyapunov exponents beyond the largest one. The availability of such algorithms and also, at about the same time, of those necessary for the computation of fractal dimensions and entropies by Grassberger and Procaccia [28], made possible the study of chaotic behavior in physically relevant models. Lyapunov analysis, applied to experimental systems [29], was also made possible by a combination of these numerical methods with ideas from nonlinear time series analysis [30]. As a result, it is nowadays widely recognized that Lyapunov exponents are a central tool of chaos theory, crucial for characterizing a number of interesting physical properties including dynamical entropies and fractal dimensions [31]. Their pivotal role in modern dynamical systems theory has been established by a fruitful exchange between a rigorous (and beautiful) mathematical theory and the algorithmic approaches essential for understanding many physical phenomena. From the 1990s to the present, with the concomitant progress in both theoretical understanding and computer capabilities, there has been a progressive shift of interest from low dimensional towards high dimensional systems. This shift towards dynamics characterized by many degrees of freedom, possibly spatially organized and/or with several characteristic temporal scales, has been accompanied by the need for

Influence of finite-time Lyapunov exponents on winter precipitation over the Iberian Peninsula

Science.gov (United States)

Garaboa-Paz, Daniel; Lorenzo, Nieves; Pérez-Muñuzuri, Vicente

2017-05-01

Seasonal forecasts have improved during the last decades, mostly due to an increase in understanding of the coupled ocean-atmosphere dynamics, and the development of models able to predict the atmosphere variability. Correlations between different teleconnection patterns and severe weather in different parts of the world are constantly evolving and changing. This paper evaluates the connection between winter precipitation over the Iberian Peninsula and the large-scale tropospheric mixing over the eastern Atlantic Ocean. Finite-time Lyapunov exponents (FTLEs) have been calculated from 1979 to 2008 to evaluate this mixing. Our study suggests that significant negative correlations exist between summer FTLE anomalies and winter precipitation over Portugal and Spain. To understand the mechanisms behind this correlation, summer anomalies of the FTLE have also been correlated with other climatic variables such as the sea surface temperature (SST), the sea level pressure (SLP) or the geopotential. The East Atlantic (EA) teleconnection index correlates with the summer FTLE anomalies, confirming their role as a seasonal predictor for winter precipitation over the Iberian Peninsula.

Dynamic stability of running: The effects of speed and leg amputations on the maximal Lyapunov exponent

International Nuclear Information System (INIS)

Look, Nicole; Arellano, Christopher J.; Grabowski, Alena M.; Kram, Rodger; McDermott, William J.; Bradley, Elizabeth

2013-01-01

In this paper, we study dynamic stability during running, focusing on the effects of speed, and the use of a leg prosthesis. We compute and compare the maximal Lyapunov exponents of kinematic time-series data from subjects with and without unilateral transtibial amputations running at a wide range of speeds. We find that the dynamics of the affected leg with the running-specific prosthesis are less stable than the dynamics of the unaffected leg and also less stable than the biological legs of the non-amputee runners. Surprisingly, we find that the center-of-mass dynamics of runners with two intact biological legs are slightly less stable than those of runners with amputations. Our results suggest that while leg asymmetries may be associated with instability, runners may compensate for this effect by increased control of their center-of-mass dynamics

The Fermi-Pasta-Ulam Problem and Its Underlying Integrable Dynamics: An Approach Through Lyapunov Exponents

Science.gov (United States)

Benettin, G.; Pasquali, S.; Ponno, A.

2018-05-01

FPU models, in dimension one, are perturbations either of the linear model or of the Toda model; perturbations of the linear model include the usual β -model, perturbations of Toda include the usual α +β model. In this paper we explore and compare two families, or hierarchies, of FPU models, closer and closer to either the linear or the Toda model, by computing numerically, for each model, the maximal Lyapunov exponent χ . More precisely, we consider statistically typical trajectories and study the asymptotics of χ for large N (the number of particles) and small ɛ (the specific energy E / N), and find, for all models, asymptotic power laws χ ˜eq Cɛ ^a, C and a depending on the model. The asymptotics turns out to be, in general, rather slow, and producing accurate results requires a great computational effort. We also revisit and extend the analytic computation of χ introduced by Casetti, Livi and Pettini, originally formulated for the β -model. With great evidence the theory extends successfully to all models of the linear hierarchy, but not to models close to Toda.

Characterization of coherent structures in three-dimensional turbulent flows using the finite-size Lyapunov exponent

International Nuclear Information System (INIS)

Bettencourt, João H; López, Cristóbal; Hernández-García, Emilio

2013-01-01

In this paper, we use the finite-size Lyapunov exponent (FSLE) to characterize Lagrangian coherent structures in three-dimensional (3D) turbulent flows. Lagrangian coherent structures act as the organizers of transport in fluid flows and are crucial to understand their stirring and mixing properties. Generalized maxima (ridges) of the FSLE fields are used to locate these coherent structures. 3D FSLE fields are calculated in two phenomenologically distinct turbulent flows: a wall-bounded flow (channel flow) and a regional oceanic flow obtained by the numerical solution of the primitive equations where two-dimensional (2D) turbulence dominates. In the channel flow, autocorrelations of the FSLE field show that the structure is substantially different from the near wall to the mid-channel region and relates well to the more widely studied Eulerian coherent structure of the turbulent channel flow. The ridges of the FSLE field have complex shapes due to the 3D character of the turbulent fluctuations. In the oceanic flow, strong horizontal stirring is present and the flow regime is similar to that of 2D turbulence where the domain is populated by coherent eddies that interact strongly. This in turn results in the presence of high FSLE lines throughout the domain leading to strong non-local mixing. The ridges of the FSLE field are quasi-vertical surfaces, indicating that the horizontal dynamics dominates the flow. Indeed, due to rotation and stratification, vertical motions in the ocean are much less intense than horizontal ones. This suppression is absent in the channel flow, as the 3D character of the FSLE ridges shows. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’. (paper)

Competitive autocatalytic reactions in chaotic flows with diffusion: Prediction using finite-time Lyapunov exponents

International Nuclear Information System (INIS)

Schlick, Conor P.; Umbanhowar, Paul B.; Ottino, Julio M.; Lueptow, Richard M.

2014-01-01

We investigate chaotic advection and diffusion in autocatalytic reactions for time-periodic sine flow computationally using a mapping method with operator splitting. We specifically consider three different autocatalytic reaction schemes: a single autocatalytic reaction, competitive autocatalytic reactions, which can provide insight into problems of chiral symmetry breaking and homochirality, and competitive autocatalytic reactions with recycling. In competitive autocatalytic reactions, species B and C both undergo an autocatalytic reaction with species A such that A+B→2B and A+C→2C. Small amounts of initially spatially localized B and C and a large amount of spatially homogeneous A are advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that local finite-time Lyapunov exponents (FTLEs) can accurately predict the final average concentrations of B and C after the reaction completes. The species that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If B and C start in regions with similar FTLEs, their average concentrations at the end of the reaction will also be similar. When a recycling reaction is added, the system evolves towards a single species state, with the FTLE often being useful in predicting which species fills the entire domain and which is depleted. The FTLE approach is also demonstrated for competitive autocatalytic reactions in journal bearing flow, an experimentally realizable flow that generates chaotic dynamics

Resonant forcing of multidimensional chaotic map dynamics.

Science.gov (United States)

Foster, Glenn; Hübler, Alfred W; Dahmen, Karin

2007-03-01

We study resonances of chaotic map dynamics. We use the calculus of variations to determine the additive forcing function that induces the largest response. We find that resonant forcing functions complement the separation of nearby trajectories, in that the product of the displacement of nearby trajectories and the resonant forcing is a conserved quantity. As a consequence, the resonant function will have the same periodicity as the displacement dynamics, and if the displacement dynamics is irregular, then the resonant forcing function will be irregular as well. Furthermore, we show that resonant forcing functions of chaotic systems decrease exponentially, where the rate equals the negative of the largest Lyapunov exponent of the unperturbed system. We compare the response to optimal forcing with random forcing and find that the optimal forcing is particularly effective if the largest Lyapunov exponent is significantly larger than the other Lyapunov exponents. However, if the largest Lyapunov exponent is much larger than unity, then the optimal forcing decreases rapidly and is only as effective as a single-push forcing.

A perturbation method to the tent map based on Lyapunov exponent and its application

Science.gov (United States)

Cao, Lv-Chen; Luo, Yu-Ling; Qiu, Sen-Hui; Liu, Jun-Xiu

2015-10-01

Perturbation imposed on a chaos system is an effective way to maintain its chaotic features. A novel parameter perturbation method for the tent map based on the Lyapunov exponent is proposed in this paper. The pseudo-random sequence generated by the tent map is sent to another chaos function — the Chebyshev map for the post processing. If the output value of the Chebyshev map falls into a certain range, it will be sent back to replace the parameter of the tent map. As a result, the parameter of the tent map keeps changing dynamically. The statistical analysis and experimental results prove that the disturbed tent map has a highly random distribution and achieves good cryptographic properties of a pseudo-random sequence. As a result, it weakens the phenomenon of strong correlation caused by the finite precision and effectively compensates for the digital chaos system dynamics degradation. Project supported by the Guangxi Provincial Natural Science Foundation, China (Grant No. 2014GXNSFBA118271), the Research Project of Guangxi University, China (Grant No. ZD2014022), the Fund from Guangxi Provincial Key Laboratory of Multi-source Information Mining & Security, China (Grant No. MIMS14-04), the Fund from the Guangxi Provincial Key Laboratory of Wireless Wideband Communication & Signal Processing, China (Grant No. GXKL0614205), the Education Development Foundation and the Doctoral Research Foundation of Guangxi Normal University, the State Scholarship Fund of China Scholarship Council (Grant No. [2014]3012), and the Innovation Project of Guangxi Graduate Education, China (Grant No. YCSZ2015102).

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

Directory of Open Access Journals (Sweden)

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.

Universality in chaos: Lyapunov spectrum and random matrix theory.

Science.gov (United States)

Hanada, Masanori; Shimada, Hidehiko; Tezuka, Masaki

2018-02-01

We propose the existence of a new universality in classical chaotic systems when the number of degrees of freedom is large: the statistical property of the Lyapunov spectrum is described by random matrix theory. We demonstrate it by studying the finite-time Lyapunov exponents of the matrix model of a stringy black hole and the mass-deformed models. The massless limit, which has a dual string theory interpretation, is special in that the universal behavior can be seen already at t=0, while in other cases it sets in at late time. The same pattern is demonstrated also in the product of random matrices.

Universality in chaos: Lyapunov spectrum and random matrix theory

Science.gov (United States)

Hanada, Masanori; Shimada, Hidehiko; Tezuka, Masaki

2018-02-01

We propose the existence of a new universality in classical chaotic systems when the number of degrees of freedom is large: the statistical property of the Lyapunov spectrum is described by random matrix theory. We demonstrate it by studying the finite-time Lyapunov exponents of the matrix model of a stringy black hole and the mass-deformed models. The massless limit, which has a dual string theory interpretation, is special in that the universal behavior can be seen already at t =0 , while in other cases it sets in at late time. The same pattern is demonstrated also in the product of random matrices.

Regeneration cycle and the covariant Lyapunov vectors in a minimal wall turbulence.

Science.gov (United States)

Inubushi, Masanobu; Takehiro, Shin-ichi; Yamada, Michio

2015-08-01

Considering a wall turbulence as a chaotic dynamical system, we study regeneration cycles in a minimal wall turbulence from the viewpoint of orbital instability by employing the covariant Lyapunov analysis developed by [F. Ginelli et al. Phys. Rev. Lett. 99, 130601 (2007)]. We divide the regeneration cycle into two phases and characterize them with the local Lyapunov exponents and the covariant Lyapunov vectors of the Navier-Stokes turbulence. In particular, we show numerically that phase (i) is dominated by instabilities related to the sinuous mode and the streamwise vorticity, and there is no instability in phase (ii). Furthermore, we discuss a mechanism of the regeneration cycle, making use of an energy budget analysis.

Appropriateness of dynamical systems for the comparison of different embedding methods via calculation of the maximum Lyapunov exponent

International Nuclear Information System (INIS)

Franchi, M; Ricci, L

2014-01-01

The embedding of time series provides a valuable, and sometimes indispensable, tool in order to analyze the dynamical properties of a chaotic system. To this purpose, the choice of the embedding dimension and lag is decisive. The scientific literature describes several methods for selecting the most appropriate parameter pairs. Unfortunately, no conclusive criterion to decide which method – and thus which embedding pair – is the best has been so far devised. A widely employed quantity to compare different methods is the maximum Lyapunov exponent (MLE) because, for chaotic systems that have explicit analytic representations, MLE can be numerically evaluated independently of the embedding dimension and lag. Within this framework, we investigated the dependence on the calculated MLE on the embedding dimension and lag in the case of three dynamical systems that are also widespreadly used as reference systems, namely the Lorenz, Rössler and Mackey-Glass attractors. By also taking into account the statistical fluctuations of the calculated MLE, we propose a new method to assess which systems provide suitable test benches for the comparison of different embedding methods via MLE calculation. For example we found that, despite of its popularity in this scientific context, the Rössler attractor is not a reliable workbench to test the validity of an embedding method

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

Energy Technology Data Exchange (ETDEWEB)

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.

Linear versus non-linear measures of temporal variability in finger tapping and their relation to performance on open- versus closed-loop motor tasks: comparing standard deviations to Lyapunov exponents.

Science.gov (United States)

Christman, Stephen D; Weaver, Ryan

2008-05-01

The nature of temporal variability during speeded finger tapping was examined using linear (standard deviation) and non-linear (Lyapunov exponent) measures. Experiment 1 found that right hand tapping was characterised by lower amounts of both linear and non-linear measures of variability than left hand tapping, and that linear and non-linear measures of variability were often negatively correlated with one another. Experiment 2 found that increased non-linear variability was associated with relatively enhanced performance on a closed-loop motor task (mirror tracing) and relatively impaired performance on an open-loop motor task (pointing in a dark room), especially for left hand performance. The potential uses and significance of measures of non-linear variability are discussed.

Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate models

Science.gov (United States)

De Cruz, Lesley; Schubert, Sebastian; Demaeyer, Jonathan; Lucarini, Valerio; Vannitsem, Stéphane

2018-05-01

The stability properties of intermediate-order climate models are investigated by computing their Lyapunov exponents (LEs). The two models considered are PUMA (Portable University Model of the Atmosphere), a primitive-equation simple general circulation model, and MAOOAM (Modular class="text">Arbitrary-Order Ocean-Atmosphere Model), a quasi-geostrophic coupled ocean-class="text">atmosphere model on a β-plane. We wish to investigate the effect of the different levels of filtering on the instabilities and dynamics of the atmospheric flows. Moreover, we assess the impact of the oceanic coupling, the dissipation scheme, and the resolution on the spectra of LEs. The PUMA Lyapunov spectrum is computed for two different values of the meridional temperature gradient defining the Newtonian forcing to the temperature field. The increase in the gradient gives rise to a higher baroclinicity and stronger instabilities, corresponding to a larger dimension of the unstable manifold and a larger first LE. The Kaplan-Yorke dimension of the attractor increases as well. The convergence rate of the rate function for the large deviation law of the finite-time Lyapunov exponents (FTLEs) is fast for all exponents, which can be interpreted as resulting from the absence of a clear-cut atmospheric timescale separation in such a model. The MAOOAM spectra show that the dominant atmospheric instability is correctly represented even at low resolutions. However, the dynamics of the central manifold, which is mostly associated with the ocean dynamics, is not fully resolved because of its associated long timescales, even at intermediate orders. As expected, increasing the mechanical atmosphere-ocean coupling coefficient or introducing a turbulent diffusion parametrisation reduces the Kaplan-Yorke dimension and Kolmogorov-Sinai entropy. In all considered configurations, we are not yet in the regime in which one can robustly define large deviation laws describing the statistics of the FTLEs. This

Análisis de estabilidad de controladores borrosos tipo Mamdani mediante el cálculo del exponente de Lyapunov

Directory of Open Access Journals (Sweden)

Leonardo-Alonso Martínez Rivera

2015-10-01

overcome this problem, we propose to employ the Lyapunov Exponent in order to determine whether fuzzy controllers are stable. The Lyapunov exponent is calculated through a numerical method on the time series obtained experimentally by having the fuzzy controller in closed loop with the plant dynamics. In this paper, the plant is the inverted pendulum, which is a benchmark plant to test complex control laws. Sixteen experiments were carried by modifying the rule base structure of Mamdani fuzzy controllers, which were also tested under normal and disturbed conditions. In all the cases, the Lyapunov Exponent is negative, indicating that the analyzed Mamdani controllers are indeed dissipative systems. Future applications on adaptive control are presented because the Lyapunov serves as a quantitative metric to determine controllers’ performance. Palabras clave: Estabilidad, Exponente de Lyapunov, Control Borroso, Bases de Reglas Mamdani, Series de Tiempo, Keywords: Stability Analysis, Lyapunov Exponent, Fuzzy Controllers, Mamdani Rule Base.

Lyapunov vectors and assimilation in the unstable subspace: theory and applications

International Nuclear Information System (INIS)

Palatella, Luigi; Carrassi, Alberto; Trevisan, Anna

2013-01-01

Based on a limited number of noisy observations, estimation algorithms provide a complete description of the state of a system at current time. Estimation algorithms that go under the name of assimilation in the unstable subspace (AUS) exploit the nonlinear stability properties of the forecasting model in their formulation. Errors that grow due to sensitivity to initial conditions are efficiently removed by confining the analysis solution in the unstable and neutral subspace of the system, the subspace spanned by Lyapunov vectors with positive and zero exponents, while the observational noise does not disturb the system along the stable directions. The formulation of the AUS approach in the context of four-dimensional variational assimilation (4DVar-AUS) and the extended Kalman filter (EKF-AUS) and its application to chaotic models is reviewed. In both instances, the AUS algorithms are at least as efficient but simpler to implement and computationally less demanding than their original counterparts. As predicted by the theory when error dynamics is linear, the optimal subspace dimension for 4DVar-AUS is given by the number of positive and null Lyapunov exponents, while the EKF-AUS algorithm, using the same unstable and neutral subspace, recovers the solution of the full EKF algorithm, but dealing with error covariance matrices of a much smaller dimension and significantly reducing the computational burden. Examples of the application to a simplified model of the atmospheric circulation and to the optimal velocity model for traffic dynamics are given. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’. (paper)

Collective Lyapunov modes

International Nuclear Information System (INIS)

Takeuchi, Kazumasa A; Chaté, Hugues

2013-01-01

We show, using covariant Lyapunov vectors in addition to standard Lyapunov analysis, that there exists a set of collective Lyapunov modes in large chaotic systems exhibiting collective dynamics. Associated with delocalized Lyapunov vectors, they act collectively on the trajectory and hence characterize the instability of its collective dynamics. We further develop, for globally coupled systems, a connection between these collective modes and the Lyapunov modes in the corresponding Perron–Frobenius equation. We thereby address the fundamental question of the effective dimension of collective dynamics and discuss the extensivity of chaos in the presence of collective dynamics. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’. (paper)

Low-dimensional chaos in a hydrodynamic system

International Nuclear Information System (INIS)

Brandstater, A.; Swift, J.; Swinney, H.L.; Wolf, A.; Farmer, J.D.; Jen, E.; Crutchfield, J.P.

1983-01-01

Evidence is presented for low-dimensional strange attractors in Couette-Taylor flow data. Computations of the largest Lyapunov exponent and metric entropy show that the system displays sensitive dependence on initial conditions. Although the phase space is very high dimensional, analysis of experimental data shows that motion is restricted to an attractor of dimension less than 5 for Reynolds numbers up to 30% above the onset of chaos. The Lyapunov exponent, entropy, and dimension all generally increase with Reynolds number

A New Feigenbaum-Like Chaotic 3D System

Directory of Open Access Journals (Sweden)

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.

Relative Lyapunov Center Bifurcations

DEFF Research Database (Denmark)

Wulff, Claudia; Schilder, Frank

2014-01-01

Relative equilibria (REs) and relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur, for example, in celestial mechanics, molecular dynamics, and rigid body motion. REs are equilibria, and RPOs are periodic orbits of the symmetry reduced system. Relative Lyapunov...... center bifurcations are bifurcations of RPOs from REs corresponding to Lyapunov center bifurcations of the symmetry reduced dynamics. In this paper we first prove a relative Lyapunov center theorem by combining recent results on the persistence of RPOs in Hamiltonian systems with a symmetric Lyapunov...... center theorem of Montaldi, Roberts, and Stewart. We then develop numerical methods for the detection of relative Lyapunov center bifurcations along branches of RPOs and for their computation. We apply our methods to Lagrangian REs of the N-body problem....

[Analysis of the heart sound with arrhythmia based on nonlinear chaos theory].

Science.gov (United States)

Ding, Xiaorong; Guo, Xingming; Zhong, Lisha; Xiao, Shouzhong

2012-10-01

In this paper, a new method based on the nonlinear chaos theory was proposed to study the arrhythmia with the combination of the correlation dimension and largest Lyapunov exponent, through computing and analyzing these two parameters of 30 cases normal heart sound and 30 cases with arrhythmia. The results showed that the two parameters of the heart sounds with arrhythmia were higher than those with the normal, and there was significant difference between these two kinds of heart sounds. That is probably due to the irregularity of the arrhythmia which causes the decrease of predictability, and it's more complex than the normal heart sound. Therefore, the correlation dimension and the largest Lyapunov exponent can be used to analyze the arrhythmia and for its feature extraction.

Lyapunov spectrum of the separated flow around the NACA 0012 airfoil and its dependence on numerical discretization

International Nuclear Information System (INIS)

Fernandez, P.; Wang, Q.

2017-01-01

We investigate the impact of numerical discretization on the Lyapunov spectrum of separated flow simulations. The two-dimensional chaotic flow around the NACA 0012 airfoil at a low Reynolds number and large angle of attack is considered to that end. Time, space and accuracy-order refinement studies are performed to examine each of these effects separately. Numerical results show that the time discretization has a small impact on the dynamics of the system, whereas the spatial discretization can dramatically change them. Also, the finite-time Lyapunov exponents associated to unstable modes are shown to be positively skewed, and quasi-homoclinic tangencies are observed in the attractor of the system. The implications of these results on flow physics and sensitivity analysis of chaotic flows are discussed.

Lyapunov spectrum of the separated flow around the NACA 0012 airfoil and its dependence on numerical discretization

Science.gov (United States)

Fernandez, P.; Wang, Q.

2017-12-01

We investigate the impact of numerical discretization on the Lyapunov spectrum of separated flow simulations. The two-dimensional chaotic flow around the NACA 0012 airfoil at a low Reynolds number and large angle of attack is considered to that end. Time, space and accuracy-order refinement studies are performed to examine each of these effects separately. Numerical results show that the time discretization has a small impact on the dynamics of the system, whereas the spatial discretization can dramatically change them. Also, the finite-time Lyapunov exponents associated to unstable modes are shown to be positively skewed, and quasi-homoclinic tangencies are observed in the attractor of the system. The implications of these results on flow physics and sensitivity analysis of chaotic flows are discussed.

Recognizing brain motor imagery activities by identifying chaos properties of oxy-hemoglobin dynamics time series

International Nuclear Information System (INIS)

Khoa, Truong Quang Dang; Yuichi, Nakamura; Masahiro, Nakagawa

2009-01-01

In recent years, functional near-infrared spectroscopy (NIRS) has been introduced as a new neuroimaging modality with which to conduct functional brain-imaging studies. With its advanced features, NIRS signal processing has become a very attractive field in computational science. This work explores nonlinear physical aspects of cerebral hemodynamic changes over the time series of NIRS. Detecting the presence of chaos in a dynamical system is an important problem in studying the irregular or chaotic motion that is generated by nonlinear systems whose dynamical laws uniquely determine the time of evolution of a state of the system. The strategy results directly from the definition of the largest Lyapunov exponent. The Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. The method is an application of the Rosenstein algorithm, an efficient method for calculating the largest Lyapunov exponent from an experimental time series. In the present paper, the authors focus mainly on the detection of chaos characteristics of the time series associated to hemoglobin dynamics. Furthermore, the chaos parameters obtained can be used to identify the active state period of the human brain.

Long Range Dependence Prognostics for Bearing Vibration Intensity Chaotic Time Series

Directory of Open Access Journals (Sweden)

Qing Li

2016-01-01

Full Text Available According to the chaotic features and typical fractional order characteristics of the bearing vibration intensity time series, a forecasting approach based on long range dependence (LRD is proposed. In order to reveal the internal chaotic properties, vibration intensity time series are reconstructed based on chaos theory in phase-space, the delay time is computed with C-C method and the optimal embedding dimension and saturated correlation dimension are calculated via the Grassberger–Procaccia (G-P method, respectively, so that the chaotic characteristics of vibration intensity time series can be jointly determined by the largest Lyapunov exponent and phase plane trajectory of vibration intensity time series, meanwhile, the largest Lyapunov exponent is calculated by the Wolf method and phase plane trajectory is illustrated using Duffing-Holmes Oscillator (DHO. The Hurst exponent and long range dependence prediction method are proposed to verify the typical fractional order features and improve the prediction accuracy of bearing vibration intensity time series, respectively. Experience shows that the vibration intensity time series have chaotic properties and the LRD prediction method is better than the other prediction methods (largest Lyapunov, auto regressive moving average (ARMA and BP neural network (BPNN model in prediction accuracy and prediction performance, which provides a new approach for running tendency predictions for rotating machinery and provide some guidance value to the engineering practice.

Jamming and chaotic dynamics in different granular systems

Science.gov (United States)

Maghsoodi, Homayoon; Luijten, Erik

Although common in nature and industry, the jamming transition has long eluded a concrete, mechanistic explanation. Recently, Banigan et al. (Nat. Phys. 9, 288-292, 2013) proposed a method for characterizing this transition in a granular system in terms of the system's chaotic properties, as quantified by the largest Lyapunov exponent. They demonstrated that in a two-dimensional shear cell the jamming transition coincides with the bulk density at which the system's largest Lyapunov exponent changes sign, indicating a transition between chaotic and non-chaotic regimes. To examine the applicability of this observation to realistic granular systems, we study a model that includes frictional forces within an expanded phase space. Furthermore, we test the generality of the relation between chaos and jamming by investigating the relationship between jamming and the chaotic properties of several other granular systems, notably sheared systems (Howell, D., Behringer R. P., Veje C., Phys. Rev. Lett. 82, 5241-5244, 1999) and systems with a free boundary. Finally, we quantify correlations between the largest Lyapunov vector and collective rearrangements of the system to demonstrate the predictive capabilities enabled by adopting this perspective of jamming.

Extending the length and time scales of Gram–Schmidt Lyapunov vector computations

Energy Technology Data Exchange (ETDEWEB)

Costa, Anthony B., E-mail: acosta@northwestern.edu [Department of Chemistry, Northwestern University, Evanston, IL 60208 (United States); Green, Jason R., E-mail: jason.green@umb.edu [Department of Chemistry, Northwestern University, Evanston, IL 60208 (United States); Department of Chemistry, University of Massachusetts Boston, Boston, MA 02125 (United States)

2013-08-01

Lyapunov vectors have found growing interest recently due to their ability to characterize systems out of thermodynamic equilibrium. The computation of orthogonal Gram–Schmidt vectors requires multiplication and QR decomposition of large matrices, which grow as N{sup 2} (with the particle count). This expense has limited such calculations to relatively small systems and short time scales. Here, we detail two implementations of an algorithm for computing Gram–Schmidt vectors. The first is a distributed-memory message-passing method using Scalapack. The second uses the newly-released MAGMA library for GPUs. We compare the performance of both codes for Lennard–Jones fluids from N=100 to 1300 between Intel Nahalem/Infiniband DDR and NVIDIA C2050 architectures. To our best knowledge, these are the largest systems for which the Gram–Schmidt Lyapunov vectors have been computed, and the first time their calculation has been GPU-accelerated. We conclude that Lyapunov vector calculations can be significantly extended in length and time by leveraging the power of GPU-accelerated linear algebra.

Extending the length and time scales of Gram–Schmidt Lyapunov vector computations

International Nuclear Information System (INIS)

Costa, Anthony B.; Green, Jason R.

2013-01-01

Lyapunov vectors have found growing interest recently due to their ability to characterize systems out of thermodynamic equilibrium. The computation of orthogonal Gram–Schmidt vectors requires multiplication and QR decomposition of large matrices, which grow as N 2 (with the particle count). This expense has limited such calculations to relatively small systems and short time scales. Here, we detail two implementations of an algorithm for computing Gram–Schmidt vectors. The first is a distributed-memory message-passing method using Scalapack. The second uses the newly-released MAGMA library for GPUs. We compare the performance of both codes for Lennard–Jones fluids from N=100 to 1300 between Intel Nahalem/Infiniband DDR and NVIDIA C2050 architectures. To our best knowledge, these are the largest systems for which the Gram–Schmidt Lyapunov vectors have been computed, and the first time their calculation has been GPU-accelerated. We conclude that Lyapunov vector calculations can be significantly extended in length and time by leveraging the power of GPU-accelerated linear algebra

Dynamic Analysis and Adaptive Sliding Mode Controller for a Chaotic Fractional Incommensurate Order Financial System

Science.gov (United States)

Hajipour, Ahmad; Tavakoli, Hamidreza

2017-12-01

In this study, the dynamic behavior and chaos control of a chaotic fractional incommensurate-order financial system are investigated. Using well-known tools of nonlinear theory, i.e. Lyapunov exponents, phase diagrams and bifurcation diagrams, we observe some interesting phenomena, e.g. antimonotonicity, crisis phenomena and route to chaos through a period doubling sequence. Adopting largest Lyapunov exponent criteria, we find that the system yields chaos at the lowest order of 2.15. Next, in order to globally stabilize the chaotic fractional incommensurate order financial system with uncertain dynamics, an adaptive fractional sliding mode controller is designed. Numerical simulations are used to demonstrate the effectiveness of the proposed control method.

Chaotic dynamics from interspike intervals

DEFF Research Database (Denmark)

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...

Finite-time braiding exponents

Science.gov (United States)

Budišić, Marko; Thiffeault, Jean-Luc

2015-08-01

Topological entropy of a dynamical system is an upper bound for the sum of positive Lyapunov exponents; in practice, it is strongly indicative of the presence of mixing in a subset of the domain. Topological entropy can be computed by partition methods, by estimating the maximal growth rate of material lines or other material elements, or by counting the unstable periodic orbits of the flow. All these methods require detailed knowledge of the velocity field that is not always available, for example, when ocean flows are measured using a small number of floating sensors. We propose an alternative calculation, applicable to two-dimensional flows, that uses only a sparse set of flow trajectories as its input. To represent the sparse set of trajectories, we use braids, algebraic objects that record how trajectories exchange positions with respect to a projection axis. Material curves advected by the flow are represented as simplified loop coordinates. The exponential rate at which a braid stretches loops over a finite time interval is the Finite-Time Braiding Exponent (FTBE). We study FTBEs through numerical simulations of the Aref Blinking Vortex flow, as a representative of a general class of flows having a single invariant component with positive topological entropy. The FTBEs approach the value of the topological entropy from below as the length and number of trajectories is increased; we conjecture that this result holds for a general class of ergodic, mixing systems. Furthermore, FTBEs are computed robustly with respect to the numerical time step, details of braid representation, and choice of initial conditions. We find that, in the class of systems we describe, trajectories can be re-used to form different braids, which greatly reduces the amount of data needed to assess the complexity of the flow.

Riemannian geometry of Hamiltonian chaos: hints for a general theory.

Science.gov (United States)

Cerruti-Sola, Monica; Ciraolo, Guido; Franzosi, Roberto; Pettini, Marco

2008-10-01

We aim at assessing the validity limits of some simplifying hypotheses that, within a Riemmannian geometric framework, have provided an explanation of the origin of Hamiltonian chaos and have made it possible to develop a method of analytically computing the largest Lyapunov exponent of Hamiltonian systems with many degrees of freedom. Therefore, a numerical hypotheses testing has been performed for the Fermi-Pasta-Ulam beta model and for a chain of coupled rotators. These models, for which analytic computations of the largest Lyapunov exponents have been carried out in the mentioned Riemannian geometric framework, appear as paradigmatic examples to unveil the reason why the main hypothesis of quasi-isotropy of the mechanical manifolds sometimes breaks down. The breakdown is expected whenever the topology of the mechanical manifolds is nontrivial. This is an important step forward in view of developing a geometric theory of Hamiltonian chaos of general validity.

Robust lyapunov controller for uncertain systems

KAUST Repository

Laleg-Kirati, Taous-Meriem

2017-02-23

Various examples of systems and methods are provided for Lyapunov control for uncertain systems. In one example, a system includes a process plant and a robust Lyapunov controller configured to control an input of the process plant. The robust Lyapunov controller includes an inner closed loop Lyapunov controller and an outer closed loop error stabilizer. In another example, a method includes monitoring a system output of a process plant; generating an estimated system control input based upon a defined output reference; generating a system control input using the estimated system control input and a compensation term; and adjusting the process plant based upon the system control input to force the system output to track the defined output reference. An inner closed loop Lyapunov controller can generate the estimated system control input and an outer closed loop error stabilizer can generate the system control input.

Using genetic programming to find Lyapunov functions

NARCIS (Netherlands)

Soute, I.A.C.; Molengraft, van de M.J.G.; Angelis, G.Z.; Ryan, C; Spector, L.

2001-01-01

In this paper Genetic Programming is used to find Lyapunov functions for (non)linear dif ferential equations of autonomous systems. As Lyapunov functions can be difficult to find, we use OP to make the decisions concerning the form of the Lyapunov function. As an e5cample two systems are taken to

The deterministic chaos and random noise in turbulent jet

International Nuclear Information System (INIS)

Yao, Tian-Liang; Liu, Hai-Feng; Xu, Jian-Liang; Li, Wei-Feng

2014-01-01

A turbulent flow is usually treated as a superposition of coherent structure and incoherent turbulence. In this paper, the largest Lyapunov exponent and the random noise in the near field of round jet and plane jet are estimated with our previously proposed method of chaotic time series analysis [T. L. Yao, et al., Chaos 22, 033102 (2012)]. The results show that the largest Lyapunov exponents of the round jet and plane jet are in direct proportion to the reciprocal of the integral time scale of turbulence, which is in accordance with the results of the dimensional analysis, and the proportionality coefficients are equal. In addition, the random noise of the round jet and plane jet has the same linear relation with the Kolmogorov velocity scale of turbulence. As a result, the random noise may well be from the incoherent disturbance in turbulence, and the coherent structure in turbulence may well follow the rule of chaotic motion

Robust lyapunov controller for uncertain systems

KAUST Repository

Laleg-Kirati, Taous-Meriem; Elmetennani, Shahrazed

2017-01-01

Various examples of systems and methods are provided for Lyapunov control for uncertain systems. In one example, a system includes a process plant and a robust Lyapunov controller configured to control an input of the process plant. The robust

Interaction of Lyapunov vectors in the formulation of the nonlinear extension of the Kalman filter.

Science.gov (United States)

Palatella, Luigi; Trevisan, Anna

2015-04-01

When applied to strongly nonlinear chaotic dynamics the extended Kalman filter (EKF) is prone to divergence due to the difficulty of correctly forecasting the forecast error probability density function. In operational forecasting applications ensemble Kalman filters circumvent this problem with empirical procedures such as covariance inflation. This paper presents an extension of the EKF that includes nonlinear terms in the evolution of the forecast error estimate. This is achieved starting from a particular square-root implementation of the EKF with assimilation confined in the unstable subspace (EKF-AUS), that is, the span of the Lyapunov vectors with non-negative exponents. When the error evolution is nonlinear, the space where it is confined is no more restricted to the unstable and neutral subspace causing filter divergence. The algorithm presented here, denominated EKF-AUS-NL, includes the nonlinear terms in the error dynamics: These result from the nonlinear interaction among the leading Lyapunov vectors and account for all directions where the error growth may take place. Numerical results show that with the nonlinear terms included, filter divergence can be avoided. We test the algorithm on the Lorenz96 model, showing very promising results.

Lyapunov Function Synthesis - Algorithm and Software

DEFF Research Database (Denmark)

Leth, Tobias; Sloth, Christoffer; Wisniewski, Rafal

2016-01-01

In this paper we introduce an algorithm for the synthesis of polynomial Lyapunov functions for polynomial vector fields. The Lyapunov function is a continuous piecewisepolynomial defined on simplices, which compose a collection of simplices. The algorithm is elaborated and crucial features are ex...

An algorithm for constructing Lyapunov functions

Directory of Open Access Journals (Sweden)

Sigurdur Freyr Hafstein

2007-08-01

Full Text Available In this monograph we develop an algorithm for constructing Lyapunov functions for arbitrary switched dynamical systems $dot{mathbf{x}} = mathbf{f}_sigma(t,mathbf{x}$, possessing a uniformly asymptotically stable equilibrium. Let $dot{mathbf{x}}=mathbf{f}_p(t,mathbf{x}$, $pinmathcal{P}$, be the collection of the ODEs, to which the switched system corresponds. The number of the vector fields $mathbf{f}_p$ on the right-hand side of the differential equation is assumed to be finite and we assume that their components $f_{p,i}$ are $mathcal{C}^2$ functions and that we can give some bounds, not necessarily close, on their second-order partial derivatives. The inputs of the algorithm are solely a finite number of the function values of the vector fields $mathbf{f}_p$ and these bounds. The domain of the Lyapunov function constructed by the algorithm is only limited by the size of the equilibrium's region of attraction. Note, that the concept of a Lyapunov function for the arbitrary switched system $dot{mathbf{x}} = mathbf{f}_sigma(t,mathbf{x}$ is equivalent to the concept of a common Lyapunov function for the systems $dot{mathbf{x}}=mathbf{f}_p(t,mathbf{x}$, $pinmathcal{P}$, and that if $mathcal{P}$ contains exactly one element, then the switched system is just a usual ODE $dot{mathbf{x}}=mathbf{f}(t,mathbf{x}$. We give numerous examples of Lyapunov functions constructed by our method at the end of this monograph.

Lyapunov functionals and stability of stochastic functional differential equations

CERN Document Server

Shaikhet, Leonid

2013-01-01

Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for discrete- and continuous-time difference equations. The text begins with a description of the peculiarities of deterministic and stochastic functional differential equations. There follow basic definitions for stability theory of stochastic hereditary systems, and a formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of di...

Covariant Lyapunov vectors

International Nuclear Information System (INIS)

Ginelli, Francesco; Politi, Antonio; Chaté, Hugues; Livi, Roberto

2013-01-01

Recent years have witnessed a growing interest in covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here, we review the basic results of ergodic theory, with a specific reference to the implications of Oseledets’ theorem for the properties of the CLVs. We then present a detailed description of a ‘dynamical’ algorithm to compute the CLVs and show that it generically converges exponentially in time. We also discuss its numerical performance and compare it with other algorithms presented in the literature. We finally illustrate how CLVs can be used to quantify deviations from hyperbolicity with reference to a dissipative system (a chain of Hénon maps) and a Hamiltonian model (a Fermi–Pasta–Ulam chain). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’. (paper)

Stability Analysis of Interconnected Fuzzy Systems Using the Fuzzy Lyapunov Method

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Ken Yeh

2010-01-01

Full Text Available The fuzzy Lyapunov method is investigated for use with a class of interconnected fuzzy systems. The interconnected fuzzy systems consist of J interconnected fuzzy subsystems, and the stability analysis is based on Lyapunov functions. Based on traditional Lyapunov stability theory, we further propose a fuzzy Lyapunov method for the stability analysis of interconnected fuzzy systems. The fuzzy Lyapunov function is defined in fuzzy blending quadratic Lyapunov functions. Some stability conditions are derived through the use of fuzzy Lyapunov functions to ensure that the interconnected fuzzy systems are asymptotically stable. Common solutions can be obtained by solving a set of linear matrix inequalities (LMIs that are numerically feasible. Finally, simulations are performed in order to verify the effectiveness of the proposed stability conditions in this paper.

On the pth moment stability of the binary airfoil induced by bounded noise

International Nuclear Information System (INIS)

Wu, Jiancheng; Li, Xuan; Liu, Xianbin

2017-01-01

Highlights: • We obtain finite pth moment Lyapunov exponent for binary airfoil subject to a bounded noise. • Based on perturbation approach and Green's functions method, second differential eigenvalue equation governing moment Lyapunov exponent is established. • The types of singular points are investigated. • The eigenvalue problem is solved analytically and numerically. • The effects of noise and system parameters on the moment Lyapunov exponent and the stochastic stability of the system are discussed. - Abstract: In the paper, the stochastic stability of the binary airfoil subject to the effect of a bounded noise is studied through the determination of moment Lyapunov exponents. The noise excitation here is often used to model a realistic model of noise in many engineering application. The partial differential eigenvalue problem governing the moment Lyapunov exponent is established. Via the Feller boundary classification, the types of singular points are discussed here, and for the system discussed, the singular points only exist in end points. The fundamental methods used are the perturbation approach and the Green's functions method. With these methods, the second-order expansions of the moment Lyapunov exponents are obtained, which are shown to be in good agreement with those obtained using Monte Carlo simulation. The effects of noise and system parameters on the moment Lyapunov exponent and the stochastic stability of the binary airfoil system are discussed.

Neglected chaos in international stock markets: Bayesian analysis of the joint return-volatility dynamical system

Science.gov (United States)

Tsionas, Mike G.; Michaelides, Panayotis G.

2017-09-01

We use a novel Bayesian inference procedure for the Lyapunov exponent in the dynamical system of returns and their unobserved volatility. In the dynamical system, computation of largest Lyapunov exponent by traditional methods is impossible as the stochastic nature has to be taken explicitly into account due to unobserved volatility. We apply the new techniques to daily stock return data for a group of six countries, namely USA, UK, Switzerland, Netherlands, Germany and France, from 2003 to 2014, by means of Sequential Monte Carlo for Bayesian inference. The evidence points to the direction that there is indeed noisy chaos both before and after the recent financial crisis. However, when a much simpler model is examined where the interaction between returns and volatility is not taken into consideration jointly, the hypothesis of chaotic dynamics does not receive much support by the data ("neglected chaos").

Synthetic jet flow control of two-dimensional NACA 65(1)-412 airfoil flow with finite-time lyapunov exponent analysis of Lagrangian coherent structures

Science.gov (United States)

Jeong, Peter Inuk

Synthetic jet (SJ) control of a low-Reynolds number, unsteady, compressible, viscous flow over a NACA 65-(1)412 airfoil, typical for unmanned air vehicles and gas turbines, has been investigated computationally. A particular focus was placed in the development and control of Lagrangian Coherent Structures (LCS) and the associated Finite-Time Lyapunov Exponent (FTLE) fields. The FTLE fields quantitatively measure of the repulsion rate in forward-time and the attraction rate in backward-time, and provide a unique perspective on effective flow control. A Discontinuous-Galerkin (DG) methods, high-fidelity Navier-Stokes solver performs direct numerical simulation (DNS) of the airfoil flow. Three SJ control strategies have been investigated: immediately downstream of flow separation, normal to the separated shear layer; near the leading edge, normal to the airfoil suction side; near the trailing edge, normal to the airfoil pressure side. A finite difference algorithm computes the FTLE from DNS velocity data. A baseline flow without SJ control is compared to SJ actuated flows. The baseline flow forms a regular, time-periodic, asymmetric von Karman vortex street in the wake. The SJ downstream of flow separation increases recirculation region vorticity and reduces the effective angle of attack. This decreases the time-averaged lift by 2:98% and increases the time-averaged drag by 5:21%. The leading edge SJ produces small vortices that deflect the shear layer downwards, and decreases the effective angle of attack. This reduces the time-averaged lift by 1:80%, and the time-averaged drag by 1:84%. The trailing edge SJ produces perturbations that add to pressure side vortices without affecting global flow characteristics. The time-averaged lift decreases by 0:47%, and the time-averaged drag increases by 0:20%. For all SJ cases, the aerodynamic performance is much more dependent on changes to the pressure distribution than changes to the skin friction distribution. No proposed

Period doubling phenomenon in a class of time delay equations

International Nuclear Information System (INIS)

Oliveira, C.R. de; Malta, C.P.

1985-01-01

The properties of the solution of a nonlinear time delayed differential equation (infinite dimension) as function of two parameters: the time delay tau and another parameter A (nonlinearity) are investigated. After a Hopf bifurcation period doubling may occur and is characterized by Feigenbaum's delta. A strange atractor is obtained after the period doubling cascade and the largest Lyapunov exponent is calculated indicating that the attractor has low dimension. The behaviour of this Liapunov exponent as function of tau is different from its behaviour as function of A. (Author) [pt

Hall magnetohydrodynamics: Conservation laws and Lyapunov stability

International Nuclear Information System (INIS)

Holm, D.D.

1987-01-01

Hall electric fields produce circulating mass flow in confined ideal-fluid plasmas. The conservation laws, Hamiltonian structure, equilibrium state relations, and Lyapunov stability conditions are presented here for ideal Hall magnetohydrodynamics (HMHD) in two and three dimensions. The approach here is to use the remarkable array of nonlinear conservation laws for HMHD that follow from its Hamiltonian structure in order to construct explicit Lyapunov functionals for the HMHD equilibrium states. In this way, the Lyapunov stability analysis provides classes of HMHD equilibria that are stable and whose linearized initial-value problems are well posed (in the sense of possessing continuous dependence on initial conditions). Several examples are discussed in both two and three dimensions

Application of autoregressive methods and Lyapunov coefficients for instability studies of nuclear reactors

International Nuclear Information System (INIS)

Aruquipa Coloma, Wilmer

2017-01-01

Nuclear reactors are susceptible to instability, causing oscillations in reactor power in specific working regions characterized by determined values of power and coolant mass flow. During reactor startup, there is a greater probability that these regions of instability will be present; another reason may be due to transient processes in some reactor parameters. The analysis of the temporal evolution of the power reveals a stable or unstable process after the disturbance in a light water reactor of type BWR (Boiling Water Reactor). In this work, the instability problem was approached in two ways. The first form is based on the ARMA (Autoregressive Moving Average models) model. This model was used to calculate the Decay Ratio (DR) and natural frequency (NF) of the oscillations, parameters that indicate if the one power signal is stable or not. In this sense, the DRARMA code was developed. In the second form, the problems of instability were analyzed using the classical concepts of non-linear systems, such as Lyapunov exponents, phase space and attractors. The Lyapunov exponents quantify the exponential divergence of the trajectories initially close to the phase space and estimate the amount of chaos in a system; the phase space and the attractors describe the dynamic behavior of the system. The main aim of the instability phenomena studies in nuclear reactors is to try to identify points or regions of operation that can lead to power oscillations conditions. The two approaches were applied to two sets of signals. The first set comes from signals of instability events of the commercial Forsmark reactors 1 and 2 and were used to validate the DRARMA code. The second set was obtained from the simulation of transient events of the Peach Bottom reactor; for the simulation, the PARCS and RELAP5 codes were used for the neutronic/thermal hydraulic coupling calculation. For all analyzes made in this work, the Matlab software was used due to its ease of programming and

Lyapunov matrices approach to the parametric optimization of time-delay systems

Directory of Open Access Journals (Sweden)

Duda Józef

2015-09-01

Full Text Available In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix

Lyapunov functions for the fixed points of the Lorenz model

International Nuclear Information System (INIS)

Bakasov, A.A.; Govorkov, B.B. Jr.

1992-11-01

We have shown how the explicit Lyapunov functions can be constructed in the framework of a regular procedure suggested and completed by Lyapunov a century ago (''method of critical cases''). The method completely covers all practically encountering subtle cases of stability study for ordinary differential equations when the linear stability analysis fails. These subtle cases, ''the critical cases'', according to Lyapunov, include both bifurcations of solutions and solutions of systems with symmetry. Being properly specialized and actually powerful in case of ODE's, this Lyapunov's method is formulated in simple language and should attract a wide interest of the physical audience. The method leads to inevitable construction of the explicit Lyapunov function, takes automatically into account the Fredholm alternative and avoids infinite step calculations. Easy and apparent physical interpretation of the Lyapunov function as a potential or as a time-dependent entropy provides one with more details about the local dynamics of the system at non-equilibrium phase transition points. Another advantage is that this Lyapunov's method consists of a set of very detailed explicit prescriptions which allow one to easy programmize the method for a symbolic processor. The application of the Lyapunov theory for critical cases has been done in this work to the real Lorenz equations and it is shown, in particular, that increasing σ at the Hopf bifurcation point suppresses the contribution of one of the variables to the destabilization of the system. The relation of the method to contemporary methods and its place among them have been clearly and extensively discussed. Due to Appendices, the paper is self-contained and does not require from a reader to approach results published only in Russian. (author). 38 refs

Application of autoregressive methods and Lyapunov coefficients for instability studies of nuclear reactors; Aplicação de métodos autorregressivos e coeficientes de Lyapunov para estudos de instabilidades em reatores nucleares

Energy Technology Data Exchange (ETDEWEB)

Aruquipa Coloma, Wilmer

2017-07-01

Nuclear reactors are susceptible to instability, causing oscillations in reactor power in specific working regions characterized by determined values of power and coolant mass flow. During reactor startup, there is a greater probability that these regions of instability will be present; another reason may be due to transient processes in some reactor parameters. The analysis of the temporal evolution of the power reveals a stable or unstable process after the disturbance in a light water reactor of type BWR (Boiling Water Reactor). In this work, the instability problem was approached in two ways. The first form is based on the ARMA (Autoregressive Moving Average models) model. This model was used to calculate the Decay Ratio (DR) and natural frequency (NF) of the oscillations, parameters that indicate if the one power signal is stable or not. In this sense, the DRARMA code was developed. In the second form, the problems of instability were analyzed using the classical concepts of non-linear systems, such as Lyapunov exponents, phase space and attractors. The Lyapunov exponents quantify the exponential divergence of the trajectories initially close to the phase space and estimate the amount of chaos in a system; the phase space and the attractors describe the dynamic behavior of the system. The main aim of the instability phenomena studies in nuclear reactors is to try to identify points or regions of operation that can lead to power oscillations conditions. The two approaches were applied to two sets of signals. The first set comes from signals of instability events of the commercial Forsmark reactors 1 and 2 and were used to validate the DRARMA code. The second set was obtained from the simulation of transient events of the Peach Bottom reactor; for the simulation, the PARCS and RELAP5 codes were used for the neutronic/thermal hydraulic coupling calculation. For all analyzes made in this work, the Matlab software was used due to its ease of programming and

Period-doubling bifurcation and chaos control in a discrete-time mosquito model

Directory of Open Access Journals (Sweden)

Qamar Din

2017-12-01

Full Text Available This article deals with the study of some qualitative properties of a discrete-time mosquito Model. It is shown that there exists period-doubling bifurcation for wide range of bifurcation parameter for the unique positive steady-state of given system. In order to control the bifurcation we introduced a feedback strategy. For further confirmation of complexity and chaotic behavior largest Lyapunov exponents are plotted.

Chaos in a complex plasma

International Nuclear Information System (INIS)

Sheridan, T.E.

2005-01-01

Chaotic dynamics is observed experimentally in a complex (dusty) plasma of three particles. A low-frequency sinusoidal modulation of the plasma density excites both the center-of-mass and breathing modes. Low-dimensional chaos is seen for a 1:2 resonance between these modes. A strange attractor with a dimension of 2.48±0.05 is observed. The largest Lyapunov exponent is positive

A survey of quantum Lyapunov control methods.

Science.gov (United States)

Cong, Shuang; Meng, Fangfang

2013-01-01

The condition of a quantum Lyapunov-based control which can be well used in a closed quantum system is that the method can make the system convergent but not just stable. In the convergence study of the quantum Lyapunov control, two situations are classified: nondegenerate cases and degenerate cases. For these two situations, respectively, in this paper the target state is divided into four categories: the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian. For these four categories, the quantum Lyapunov control methods for the closed quantum systems are summarized and analyzed. Particularly, the convergence of the control system to the different target states is reviewed, and how to make the convergence conditions be satisfied is summarized and analyzed.

Time-delay effects and simplified control fields in quantum Lyapunov control

International Nuclear Information System (INIS)

Yi, X X; Wu, S L; Wu, Chunfeng; Feng, X L; Oh, C H

2011-01-01

Lyapunov-based quantum control has the advantage that it is free from the measurement-induced decoherence and it includes the instantaneous information of the system in the control. The Lyapunov control is often confronted with time delay in the control fields and difficulty in practical implementations of the control. In this paper, we study the effect of time delay on the Lyapunov control and explore the possibility of replacing the control field with a pulse train or a bang-bang signal. The efficiency of the Lyapunov control is also presented through examining the convergence time of the system. These results suggest that the Lyapunov control is robust against time delay, easy to realize and effective for high-dimensional quantum systems.

Small-world networks exhibit pronounced intermittent synchronization

Science.gov (United States)

Choudhary, Anshul; Mitra, Chiranjit; Kohar, Vivek; Sinha, Sudeshna; Kurths, Jürgen

2017-11-01

We report the phenomenon of temporally intermittently synchronized and desynchronized dynamics in Watts-Strogatz networks of chaotic Rössler oscillators. We consider topologies for which the master stability function (MSF) predicts stable synchronized behaviour, as the rewiring probability (p) is tuned from 0 to 1. MSF essentially utilizes the largest non-zero Lyapunov exponent transversal to the synchronization manifold in making stability considerations, thereby ignoring the other Lyapunov exponents. However, for an N-node networked dynamical system, we observe that the difference in its Lyapunov spectra (corresponding to the N - 1 directions transversal to the synchronization manifold) is crucial and serves as an indicator of the presence of intermittently synchronized behaviour. In addition to the linear stability-based (MSF) analysis, we further provide global stability estimate in terms of the fraction of state-space volume shared by the intermittently synchronized state, as p is varied from 0 to 1. This fraction becomes appreciably large in the small-world regime, which is surprising, since this limit has been otherwise considered optimal for synchronized dynamics. Finally, we characterize the nature of the observed intermittency and its dominance in state-space as network rewiring probability (p) is varied.

Stability of time-delay systems via Lyapunov functions

Directory of Open Access Journals (Sweden)

Carlos F. Alastruey

2002-01-01

Full Text Available In this paper, a Lyapunov function candidate is introduced for multivariable systems with inner delays, without assuming a priori stability for the nondelayed subsystem. By using this Lyapunov function, a controller is deduced. Such a controller utilizes an input–output description of the original system, a circumstance that facilitates practical applications of the proposed approach.

Can a polynomial interpolation improve on the Kaplan-Yorke dimension?

International Nuclear Information System (INIS)

Richter, Hendrik

2008-01-01

The Kaplan-Yorke dimension can be derived using a linear interpolation between an h-dimensional Lyapunov exponent λ (h) >0 and an h+1-dimensional Lyapunov exponent λ (h+1) <0. In this Letter, we use a polynomial interpolation to obtain generalized Lyapunov dimensions and study the relationships among them for higher-dimensional systems

A New 3-D Piecewise-Linear System for Chaos Generation

Directory of Open Access Journals (Sweden)

Z. Elhadj

2007-06-01

Full Text Available We propose in this paper a new simple continuous-time piecewise-linear three dimensional system for chaos generation. Nonlinearity in this model is introduced by the characteristic function of the Chua's circuit given in [1]. Simulated results of some chaotic attractors are shown and justified numerically via computing the largest Lyapunov exponent. The possibility and the robustness of the circuitry realization is also given and discussed.

Chaos analysis of the electrical signal time series evoked by acupuncture

International Nuclear Information System (INIS)

Wang Jiang; Sun Li; Fei Xiangyang; Zhu Bing

2007-01-01

This paper employs chaos theory to analyze the time series of electrical signal which are evoked by different acupuncture methods applied to the Zusanli point. The phase space is reconstructed and the embedding parameters are obtained by the mutual information and Cao's methods. Subsequently, the largest Lyapunov exponent is calculated. From the analyses we can conclude that the time series are chaotic. In addition, differences between various acupuncture methods are discussed

Lyapunov spectra and conjugate-pairing rule for confined atomic fluids

DEFF Research Database (Denmark)

Bernadi, Stefano; Todd, B.D.; Hansen, Jesper Schmidt

2010-01-01

In this work we present nonequilibrium molecular dynamics simulation results for the Lyapunov spectra of atomic fluids confined in narrow channels of the order of a few atomic diameters. We show the effect that realistic walls have on the Lyapunov spectra. All the degrees of freedom of the confin...... evolved Lyapunov vectors projected into a reduced dimensional phase space. We finally observe that the phase-space compression due to the thermostat remains confined into the wall region and does not significantly affect the purely Newtonian fluid region....

Observer-based design of set-point tracking adaptive controllers for nonlinear chaotic systems

International Nuclear Information System (INIS)

Khaki-Sedigh, A.; Yazdanpanah-Goharrizi, A.

2006-01-01

A gradient based approach for the design of set-point tracking adaptive controllers for nonlinear chaotic systems is presented. In this approach, Lyapunov exponents are used to select the controller gain. In the case of unknown or time varying chaotic plants, the Lyapunov exponents may vary during the plant operation. In this paper, an effective adaptive strategy is used for online identification of Lyapunov exponents and adaptive control of nonlinear chaotic plants. Also, a nonlinear observer for estimation of the states is proposed. Simulation results are provided to show the effectiveness of the proposed methodology

Observer-based design of set-point tracking adaptive controllers for nonlinear chaotic systems

Energy Technology Data Exchange (ETDEWEB)

Khaki-Sedigh, A. [Department of Electrical Engineering, K.N. Toosi University of Technology, Sayyed Khandan Bridge, Shariati Street, Tehran 16314 (Iran, Islamic Republic of)]. E-mail: sedigh@kntu.ac.ir; Yazdanpanah-Goharrizi, A. [Department of Electrical Engineering, K.N. Toosi University of Technology, Sayyed Khandan Bridge, Shariati Street, Tehran 16314 (Iran, Islamic Republic of)]. E-mail: yazdanpanah@ee.kntu.ac.ir

2006-09-15

A gradient based approach for the design of set-point tracking adaptive controllers for nonlinear chaotic systems is presented. In this approach, Lyapunov exponents are used to select the controller gain. In the case of unknown or time varying chaotic plants, the Lyapunov exponents may vary during the plant operation. In this paper, an effective adaptive strategy is used for online identification of Lyapunov exponents and adaptive control of nonlinear chaotic plants. Also, a nonlinear observer for estimation of the states is proposed. Simulation results are provided to show the effectiveness of the proposed methodology.

Using Covariant Lyapunov Vectors to Understand Spatiotemporal Chaos in Fluids

Science.gov (United States)

Paul, Mark; Xu, Mu; Barbish, Johnathon; Mukherjee, Saikat

2017-11-01

The spatiotemporal chaos of fluids present many difficult and fascinating challenges. Recent progress in computing covariant Lyapunov vectors for a variety of model systems has made it possible to probe fundamental ideas from dynamical systems theory including the degree of hyperbolicity, the fractal dimension, the dimension of the inertial manifold, and the decomposition of the dynamics into a finite number of physical modes and spurious modes. We are interested in building upon insights such as these for fluid systems. We first demonstrate the power of covariant Lyapunov vectors using a system of maps on a lattice with a nonlinear coupling. We then compute the covariant Lyapunov vectors for chaotic Rayleigh-Bénard convection for experimentally accessible conditions. We show that chaotic convection is non-hyperbolic and we quantify the spatiotemporal features of the spectrum of covariant Lyapunov vectors. NSF DMS-1622299 and DARPA/DSO Models, Dynamics, and Learning (MoDyL).

Lyapunov Functions to Caputo Fractional Neural Networks with Time-Varying Delays

Directory of Open Access Journals (Sweden)

Ravi Agarwal

2018-05-01

Full Text Available One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable. In connection with the Lyapunov fractional method we present a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of neural networks with variable coefficients and time-varying delays. We show that quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability of equilibrium of nonlinear Caputo fractional neural networks with time dependent transmission delays, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as nonLipschitz activation functions are studied. We illustrate our theory on particular nonlinear Caputo fractional neural networks.

Chaos analysis of the electrical signal time series evoked by acupuncture

Energy Technology Data Exchange (ETDEWEB)

Wang Jiang [School of Electrical Engineering, Tianjin University, Tianjin 300072 (China)]. E-mail: jiangwang@tju.edu.cn; Sun Li [School of Electrical Engineering, Tianjin University, Tianjin 300072 (China); Fei Xiangyang [School of Electrical Engineering, Tianjin University, Tianjin 300072 (China); Zhu Bing [Institute of Acupuncture and Moxibustion, China Academy of Traditional Chinese Medicine, Beijing 100700 (China)

2007-08-15

This paper employs chaos theory to analyze the time series of electrical signal which are evoked by different acupuncture methods applied to the Zusanli point. The phase space is reconstructed and the embedding parameters are obtained by the mutual information and Cao's methods. Subsequently, the largest Lyapunov exponent is calculated. From the analyses we can conclude that the time series are chaotic. In addition, differences between various acupuncture methods are discussed.

A New Fractional-Order Chaotic Complex System and Its Antisynchronization

Directory of Open Access Journals (Sweden)

Cuimei Jiang

2014-01-01

with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.

Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks

DEFF Research Database (Denmark)

Anderson, David F; Craciun, Gheorghe; Gopalkrishnan, Manoj

2015-01-01

We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potent...

Construction of Lyapunov Function for Dissipative Gyroscopic System

International Nuclear Information System (INIS)

Xu Wei; Ao Ping; Yuan Bo

2011-01-01

We introduce a force decomposition to construct a potential function in deterministic dynamics described by ordinary differential equations in the context of dissipative gyroscopic systems. Such a potential function serves as the corresponding Lyapunov function for the dynamics, hence it gives both quantitative and qualitative descriptions for stability of motion. As an example we apply our force decomposition to a four-dimensional dissipative gyroscopic system. We explicitly obtain the potential function for all parameter regimes in the linear limit, including those regimes where the Lyapunov function was previously believed not to exist. (general)

On the existence of polynomial Lyapunov functions for rationally stable vector fields

DEFF Research Database (Denmark)

Leth, Tobias; Wisniewski, Rafal; Sloth, Christoffer

2018-01-01

This paper proves the existence of polynomial Lyapunov functions for rationally stable vector fields. For practical purposes the existence of polynomial Lyapunov functions plays a significant role since polynomial Lyapunov functions can be found algorithmically. The paper extents an existing result...... on exponentially stable vector fields to the case of rational stability. For asymptotically stable vector fields a known counter example is investigated to exhibit the mechanisms responsible for the inability to extend the result further....

Exploring conservative islands using correlated and uncorrelated noise

Science.gov (United States)

da Silva, Rafael M.; Manchein, Cesar; Beims, Marcus W.

2018-02-01

In this work, noise is used to analyze the penetration of regular islands in conservative dynamical systems. For this purpose we use the standard map choosing nonlinearity parameters for which a mixed phase space is present. The random variable which simulates noise assumes three distributions, namely equally distributed, normal or Gaussian, and power law (obtained from the same standard map but for other parameters). To investigate the penetration process and explore distinct dynamical behaviors which may occur, we use recurrence time statistics (RTS), Lyapunov exponents and the occupation rate of the phase space. Our main findings are as follows: (i) the standard deviations of the distributions are the most relevant quantity to induce the penetration; (ii) the penetration of islands induce power-law decays in the RTS as a consequence of enhanced trapping; (iii) for the power-law correlated noise an algebraic decay of the RTS is observed, even though sticky motion is absent; and (iv) although strong noise intensities induce an ergodic-like behavior with exponential decays of RTS, the largest Lyapunov exponent is reminiscent of the regular islands.

Regularity and chaos in cavity QED

International Nuclear Information System (INIS)

Bastarrachea-Magnani, Miguel Angel; López-del-Carpio, Baldemar; Chávez-Carlos, Jorge; Lerma-Hernández, Sergio; Hirsch, Jorge G

2017-01-01

The interaction of a quantized electromagnetic field in a cavity with a set of two-level atoms inside it can be described with algebraic Hamiltonians of increasing complexity, from the Rabi to the Dicke models. Their algebraic character allows, through the use of coherent states, a semiclassical description in phase space, where the non-integrable Dicke model has regions associated with regular and chaotic motion. The appearance of classical chaos can be quantified calculating the largest Lyapunov exponent over the whole available phase space for a given energy. In the quantum regime, employing efficient diagonalization techniques, we are able to perform a detailed quantitative study of the regular and chaotic regions, where the quantum participation ratio (P R ) of coherent states on the eigenenergy basis plays a role equivalent to the Lyapunov exponent. It is noted that, in the thermodynamic limit, dividing the participation ratio by the number of atoms leads to a positive value in chaotic regions, while it tends to zero in the regular ones. (paper)

Lyapunov stability analysis of magnetohydrodynamic plasma equilibria with axisymmetric toroidal flow

International Nuclear Information System (INIS)

Almaguer, J.A.; Hameiri, E.; Herrera, J.; Holm, D.D.

1988-01-01

Lyapunov stability conditions for ideal magnetohydrodynamic (MHD) plasmas with mass flow in axisymmetric toroidal geometry are determined in the Eulerian representation. Axisymmetric equilibrium solutions of ideal MHD are associated to critical points of a nonlinearly conserved Lyapunov functional consisting of the sum of the total energy and the following flux-weighted quantities: the circulation along field lines, the angular momentum, the toroidal flux, and the mass content within each flux tube. Conditions sufficient for Lyapunov stability of these equilibria against axisymmetric perturbations are found by taking advantage of the Hamiltonian formalism for ideal MHD. In particular [see Eq. (60)], it is sufficient for Lyapunov stability under linearized dynamics that an axisymmetric equilibrium be subsonic in the appropriate rotating frame, lie in the first elliptic regime of the Bernoulli--Grad--Shafranov (BGS) system of equations, and satisfy one additional, more complicated, condition. Effects of boundary conditions, nonlinearity, and three-dimensionality on MHD stability are also discussed

Dynamical behaviors determined by the Lyapunov function in competitive Lotka-Volterra systems

Science.gov (United States)

Tang, Ying; Yuan, Ruoshi; Ma, Yian

2013-01-01

Dynamical behaviors of the competitive Lotka-Volterra system even for 3 species are not fully understood. In this paper, we study this problem from the perspective of the Lyapunov function. We construct explicitly the Lyapunov function using three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-species case, (2) a 3-species model, and (3) the model of May-Leonard. The basins of attraction for these examples are demonstrated, including cases with bistability and cyclical behavior. The first two examples are the generalized gradient system, where the energy dissipation may not follow the gradient of the Lyapunov function. In addition, under a new type of stochastic interpretation, the Lyapunov function also leads to the Boltzmann-Gibbs distribution on the final steady state when multiplicative noise is added.

Relating Lagrangian passive scalar scaling exponents to Eulerian scaling exponents in turbulence

OpenAIRE

Schmitt , François G

2005-01-01

Intermittency is a basic feature of fully developed turbulence, for both velocity and passive scalars. Intermittency is classically characterized by Eulerian scaling exponent of structure functions. The same approach can be used in a Lagrangian framework to characterize the temporal intermittency of the velocity and passive scalar concentration of a an element of fluid advected by a turbulent intermittent field. Here we focus on Lagrangian passive scalar scaling exponents, and discuss their p...

Inertia theorems for operator Lyapunov inequalities

NARCIS (Netherlands)

Sasane, AJ; Curtain, RF

2001-01-01

We study operator Lyapunov inequalities and equations for which the infinitesimal generator is not necessarily stable, but it satisfies the spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Moreover, the input or output operators are not necessarily bounded,

Lyapunov stability of ideal compressible and incompressible fluid equilibria in three dimensions

International Nuclear Information System (INIS)

Holm, D.D.

1985-08-01

Linearized stability of ideal compressible and incompressible fluid equilibria in three dimensions is analyzed using Lyapunov's direct method. An action principle is given for the Eulerian and Lagrangian fluid descriptions and the family of constants of motion due to symmetry under fluid-particle relabelling is derived in the form of Ertel's theorem for each description. In an augmented Euleriah description, the steady equilibrium flows of these two fluids theories are identified as critical points of the conserved Lyapunov functionals defined by the sum, H + C, of the energy H, and the Ertel constants of motion, C. It turns out that unconditional linear Lyapunov stability of these flows in the norm provided by the second variation of H + C is precluded by vortex-particle stretching, even for otherwise shear-stable flows. Conditional Lyapunov stability of these flows is discussed. 24 refs

Construction of a Smooth Lyapunov Function for the Robust and Exact Second-Order Differentiator

Directory of Open Access Journals (Sweden)

Tonametl Sanchez

2016-01-01

Full Text Available Differentiators play an important role in (continuous feedback control systems. In particular, the robust and exact second-order differentiator has shown some very interesting properties and it has been used successfully in sliding mode control, in spite of the lack of a Lyapunov based procedure to design its gains. As contribution of this paper, we provide a constructive method to determine a differentiable Lyapunov function for such a differentiator. Moreover, the Lyapunov function is used to provide a procedure to design the differentiator’s parameters. Also, some sets of such parameters are provided. The determination of the positive definiteness of the Lyapunov function and negative definiteness of its derivative is converted to the problem of solving a system of inequalities linear in the parameters of the Lyapunov function candidate and also linear in the gains of the differentiator, but bilinear in both.

Nonlinear generalized synchronization of chaotic systems by pure error dynamics and elaborate nondiagonal Lyapunov function

International Nuclear Information System (INIS)

Ge Zhengming; Chang Chingming

2009-01-01

By applying pure error dynamics and elaborate nondiagonal Lyapunov function, the nonlinear generalized synchronization is studied in this paper. Instead of current mixed error dynamics in which master state variables and slave state variables are presented, the nonlinear generalized synchronization can be obtained by pure error dynamics without auxiliary numerical simulation. The elaborate nondiagonal Lyapunov function is applied rather than current monotonous square sum Lyapunov function deeply weakening the powerfulness of Lyapunov direct method. Both autonomous and nonautonomous double Mathieu systems are used as examples with numerical simulations.

A sampling approach to constructing Lyapunov functions for nonlinear continuous–time systems

NARCIS (Netherlands)

Bobiti, R.V.; Lazar, M.

2016-01-01

The problem of constructing a Lyapunov function for continuous-time nonlinear dynamical systems is tackled in this paper via a sampling-based approach. The main idea of the sampling-based method is to verify a Lyapunov-type inequality for a finite number of points (known state vectors) in the

Thermodynamic stability of elementary chemical reactions proceeding at finite rates revisited using Lyapunov function analysis

International Nuclear Information System (INIS)

Burande, Chandrakant S.; Bhalekar, Anil A.

2005-01-01

The thermodynamic stability of a few representative elementary chemical reactions proceeding at finite rates has been investigated using the recently proposed thermodynamic Lyapunov function and following the steps of Lyapunov's second method (also termed as the direct method) of stability of motion. The thermodynamic Lyapunov function; L s , used herein is the excess rate of entropy production in the thermodynamic perturbation space, which thereby inherits the dictates of the second law of thermodynamics. This Lyapunov function is not the same as the excess entropy rate that one encounters in thermodynamic (irreversible) literature. The model chemical conversions studied in this presentation are A+B→v x X and A+B↔ν x X. For the sake of simplicity, the thermal effects of chemical reactions have been considered as not adding to the perturbation as our main aim was to demonstrate how one should use systematically the proposed thermodynamic Lyapunov function following the steps of Lyapunov's second method of stability of motion. The domains of thermodynamic stability under the constantly acting small disturbances, thermodynamic asymptotic stability and thermodynamic instability in these model systems get established

Piecewise quadratic Lyapunov functions for stability verification of approximate explicit MPC

Directory of Open Access Journals (Sweden)

Morten Hovd

2010-04-01

Full Text Available Explicit MPC of constrained linear systems is known to result in a piecewise affine controller and therefore also piecewise affine closed loop dynamics. The complexity of such analytic formulations of the control law can grow exponentially with the prediction horizon. The suboptimal solutions offer a trade-off in terms of complexity and several approaches can be found in the literature for the construction of approximate MPC laws. In the present paper a piecewise quadratic (PWQ Lyapunov function is used for the stability verification of an of approximate explicit Model Predictive Control (MPC. A novel relaxation method is proposed for the LMI criteria on the Lyapunov function design. This relaxation is applicable to the design of PWQ Lyapunov functions for discrete-time piecewise affine systems in general.

Chaos in a fractional-order Roessler system

International Nuclear Information System (INIS)

Zhang Weiwei; Zhou Shangbo; Li Hua; Zhu Hao

2009-01-01

The dynamic behaviors in the fractional-order Roessler equations were numerically studied. Basic properties of the system have been analyzed by means of Lyapunov exponents and bifurcation diagrams. The parameter and the derivative order ranges used were relatively broad. Regular motions (including period-3 motion) and chaotic motions were examined. The chaotic motion identified was validated by the positive Lyapunov exponent.

Complex dynamics of a stochastic discrete modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting

Directory of Open Access Journals (Sweden)

A. Elhassanein

2014-06-01

Full Text Available This paper introduced a stochastic discretized version of the modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting. The dynamical behavior of the proposed model was investigated. The existence and stability of the equilibria of the skeleton were studied. Numerical simulations were employed to show the model's complex dynamics by means of the largest Lyapunov exponents, bifurcations, time series diagrams and phase portraits. The effects of noise intensity on its dynamics and the intermittency phenomenon were also discussed via simulation.

Monte Carlo-based tail exponent estimator

Science.gov (United States)

Barunik, Jozef; Vacha, Lukas

2010-11-01

In this paper we propose a new approach to estimation of the tail exponent in financial stock markets. We begin the study with the finite sample behavior of the Hill estimator under α-stable distributions. Using large Monte Carlo simulations, we show that the Hill estimator overestimates the true tail exponent and can hardly be used on samples with small length. Utilizing our results, we introduce a Monte Carlo-based method of estimation for the tail exponent. Our proposed method is not sensitive to the choice of tail size and works well also on small data samples. The new estimator also gives unbiased results with symmetrical confidence intervals. Finally, we demonstrate the power of our estimator on the international world stock market indices. On the two separate periods of 2002-2005 and 2006-2009, we estimate the tail exponent.

Fault Diagnosis of Broken Rotor Bars in Squirrel-Cage Induction Motor of Hoister Based on Duffing Oscillator and Multifractal Dimension

Directory of Open Access Journals (Sweden)

Zhike Zhao

2014-07-01

Full Text Available This paper is to propose a novel fault diagnosis method for broken rotor bars in squirrel-cage induction motor of hoister, which is based on duffing oscillator and multifractal dimension. Firstly, based on the analysis of the structure and performance of modified duffing oscillator, the end of transitional slope from chaotic area to large-scale cycle area is selected as the optimal critical threshold of duffing oscillator by bifurcation diagrams and Lyapunov exponent. Secondly, the phase transformation duffing oscillator from chaos to intermittent chaos is sensitive to the signals, whose frequency difference is quite weak from the reference signal. The spectrums of the largest Lyapunov exponents and bifurcation diagrams of the duffing oscillator are utilized to analyze the variance in different parameters of frequency. Finally, this paper is to analyze the characteristics of both single fractal (box-counting dimension and multifractal and make a comparison between them. Multifractal detrended fluctuation analysis is applied to detect extra frequency component of current signal. Experimental results reveal that the method is effective for early detection of broken rotor bars in squirrel-cage induction motor of hoister.

Quantum critical Hall exponents

CERN Document Server

Lütken, C A

2014-01-01

We investigate a finite size "double scaling" hypothesis using data from an experiment on a quantum Hall system with short range disorder [1-3]. For Hall bars of width w at temperature T the scaling form is w(-mu)T(-kappa), where the critical exponent mu approximate to 0.23 we extract from the data is comparable to the multi-fractal exponent alpha(0) - 2 obtained from the Chalker-Coddington (CC) model [4]. We also use the data to find the approximate location (in the resistivity plane) of seven quantum critical points, all of which closely agree with the predictions derived long ago from the modular symmetry of a toroidal sigma-model with m matter fields [5]. The value nu(8) = 2.60513 ... of the localisation exponent obtained from the m = 8 model is in excellent agreement with the best available numerical value nu(num) = 2.607 +/- 0.004 derived from the CC-model [6]. Existing experimental data appear to favour the m = 9 model, suggesting that the quantum Hall system is not in the same universality class as th...

Generalized decompositions of dynamic systems and vector Lyapunov functions

Science.gov (United States)

Ikeda, M.; Siljak, D. D.

1981-10-01

The notion of decomposition is generalized to provide more freedom in constructing vector Lyapunov functions for stability analysis of nonlinear dynamic systems. A generalized decomposition is defined as a disjoint decomposition of a system which is obtained by expanding the state-space of a given system. An inclusion principle is formulated for the solutions of the expansion to include the solutions of the original system, so that stability of the expansion implies stability of the original system. Stability of the expansion can then be established by standard disjoint decompositions and vector Lyapunov functions. The applicability of the new approach is demonstrated using the Lotka-Volterra equations.

Amputation effects on the underlying complexity within transtibial amputee ankle motion

Energy Technology Data Exchange (ETDEWEB)

Wurdeman, Shane R., E-mail: shanewurdeman@gmail.com [Nebraska Biomechanics Core Facility, University of Nebraska at Omaha, Omaha, Nebraska 68182 (United States); Advanced Prosthetics Center, Omaha, Nebraska 68134 (United States); Myers, Sara A. [Nebraska Biomechanics Core Facility, University of Nebraska at Omaha, Omaha, Nebraska 68182 (United States); Stergiou, Nicholas [Nebraska Biomechanics Core Facility, University of Nebraska at Omaha, Omaha, Nebraska 68182 (United States); College of Public Health, University of Nebraska Medical Center, Omaha, Nebraska 68198 (United States)

2014-03-15

The presence of chaos in walking is considered to provide a stable, yet adaptable means for locomotion. This study examined whether lower limb amputation and subsequent prosthetic rehabilitation resulted in a loss of complexity in amputee gait. Twenty-eight individuals with transtibial amputation participated in a 6 week, randomized cross-over design study in which they underwent a 3 week adaptation period to two separate prostheses. One prosthesis was deemed “more appropriate” and the other “less appropriate” based on matching/mismatching activity levels of the person and the prosthesis. Subjects performed a treadmill walking trial at self-selected walking speed at multiple points of the adaptation period, while kinematics of the ankle were recorded. Bilateral sagittal plane ankle motion was analyzed for underlying complexity through the pseudoperiodic surrogation analysis technique. Results revealed the presence of underlying deterministic structure in both prostheses and both the prosthetic and sound leg ankle (discriminant measure largest Lyapunov exponent). Results also revealed that the prosthetic ankle may be more likely to suffer loss of complexity than the sound ankle, and a “more appropriate” prosthesis may be better suited to help restore a healthy complexity of movement within the prosthetic ankle motion compared to a “less appropriate” prosthesis (discriminant measure sample entropy). Results from sample entropy results are less likely to be affected by the intracycle periodic dynamics as compared to the largest Lyapunov exponent. Adaptation does not seem to influence complexity in the system for experienced prosthesis users.

Evaluation of mental stress by physiological indices derived from finger plethysmography.

Science.gov (United States)

Minakuchi, Emiko; Ohnishi, Eriko; Ohnishi, Junji; Sakamoto, Shigeko; Hori, Miyo; Motomura, Miwa; Hoshino, Junichi; Murakami, Kazuo; Kawaguchi, Takayasu

2013-10-12

Quantitative evaluation of mental stress is important to prevent stress-related disorders. Finger plethysmography (FPG) is a simple noninvasive method to monitor peripheral circulation, and provides many physiological indices. Our purpose is to investigate how FPG-derived indices reflect on mental stress, and to clarify any association between these physiological indices and subjective indices of mental stress. Thirty-one healthy women (mean age, 22 years ± 2) participated. The participants rested by sitting on a chair for 10 min. They then performed a computerized version of the Stroop color-word conflict test (CWT) for 10 min. Finally, they rested for 10 min. FPG was recorded throughout the experiment. The participants completed a brief form of the Profile of Mood States (POMS) questionnaire before and after the test. Using the FPG data, we conducted chaos analysis and fast Fourier transform analysis, and calculated chaotic attractors, the largest Lyapunov exponent, a high-frequency (HF) component, a low-to-high-frequency (LF/HF) ratio, finger pulse rate and finger pulse wave amplitude. The HF component decreased and the LF/HF ratio increased significantly during the test (P stress. Our findings indicate that FPG is one of the easiest methods to evaluate mental stress quantitatively. In particular, the largest Lyapunov exponent and the LF/HF ratio might be associated with acute mental stress. Farther examination is needed to find any association between the physiological indices and various types of mental stress.

Amputation effects on the underlying complexity within transtibial amputee ankle motion

International Nuclear Information System (INIS)

Wurdeman, Shane R.; Myers, Sara A.; Stergiou, Nicholas

2014-01-01

The presence of chaos in walking is considered to provide a stable, yet adaptable means for locomotion. This study examined whether lower limb amputation and subsequent prosthetic rehabilitation resulted in a loss of complexity in amputee gait. Twenty-eight individuals with transtibial amputation participated in a 6 week, randomized cross-over design study in which they underwent a 3 week adaptation period to two separate prostheses. One prosthesis was deemed “more appropriate” and the other “less appropriate” based on matching/mismatching activity levels of the person and the prosthesis. Subjects performed a treadmill walking trial at self-selected walking speed at multiple points of the adaptation period, while kinematics of the ankle were recorded. Bilateral sagittal plane ankle motion was analyzed for underlying complexity through the pseudoperiodic surrogation analysis technique. Results revealed the presence of underlying deterministic structure in both prostheses and both the prosthetic and sound leg ankle (discriminant measure largest Lyapunov exponent). Results also revealed that the prosthetic ankle may be more likely to suffer loss of complexity than the sound ankle, and a “more appropriate” prosthesis may be better suited to help restore a healthy complexity of movement within the prosthetic ankle motion compared to a “less appropriate” prosthesis (discriminant measure sample entropy). Results from sample entropy results are less likely to be affected by the intracycle periodic dynamics as compared to the largest Lyapunov exponent. Adaptation does not seem to influence complexity in the system for experienced prosthesis users

Nuclear multifragmentation critical exponents

International Nuclear Information System (INIS)

Bauer, W.; Friedman, W.A.; Univ. of Wisconsin, Madison, WI

1995-01-01

In a recent Letter, cited in a reference, the EoS collaboration presented data of fragmentation of 1 A GeV gold nuclei incident on carbon. By analyzing moments of the fragment charge distribution, the authors claim to determine the values of the critical exponents γ, β, and τ for finite nuclei. These data represent a crucial step forward in the understanding of the physics of nuclear fragmentation. However, as shown in this paper, the analysis presented in the cited reference is not sufficient to support the claim that the critical exponents for nuclear fragmentation have been unambiguously determined

Study of multiparameter respiratory pattern complexity in surgical critically ill patients during weaning trials

Directory of Open Access Journals (Sweden)

Maglaveras Nikos K

2011-01-01

Full Text Available Abstract Background Separation from mechanical ventilation is a difficult task, whereas conventional predictive indices have not been proven accurate enough, so far. A few studies have explored changes of breathing pattern variability for weaning outcome prediction, with conflicting results. In this study, we tried to assess respiratory complexity during weaning trials, using different non-linear methods derived from theory of complex systems, in a cohort of surgical critically ill patients. Results Thirty two patients were enrolled in the study. There were 22 who passed and 10 who failed a weaning trial. Tidal volume and mean inspiratory flow were analyzed for 10 minutes during two phases: 1. pressure support (PS ventilation (15-20 cm H2O and 2. weaning trials with PS: 5 cm H2O. Sample entropy (SampEn, detrended fluctuation analysis (DFA exponent, fractal dimension (FD and largest lyapunov exponents (LLE of the two respiratory parameters were computed in all patients and during the two phases of PS. Weaning failure patients exhibited significantly decreased respiratory pattern complexity, reflected in reduced sample entropy and lyapunov exponents and increased DFA exponents of respiratory flow time series, compared to weaning success subjects (p 0.1, SampEn and LLE predicted better weaning outcome compared with RSBI, P0.1 and RSBI* P0.1 (conventional model, R2 = 0.874 vs 0.643, p Conclusions We suggest that complexity analysis of respiratory signals can assess inherent breathing pattern dynamics and has increased prognostic impact upon weaning outcome in surgical patients.

International Congress NONLINEAR DYNAMICAL ANALYSIS 2007 dedicated to the 150th Anniversary of Academician A. M. Lyapunov

Science.gov (United States)

2010-05-14

Mikhailovich Lyapunov is discussed. Main attention is focused on the first Lyapunov method. LYAPUNOV BUNDLES IN CYCLIC FEEDBACK SYSTEMS WITH DELAYS George ...Lyapunov frequently discussed this problem with Henry Poincare (1854-1912) and George Darwin (1845 - 1912). They both considered the "pear-form" figure as... Cantor -type set. Neither can the existence of such systems be excluded. The results we present are discussed in a joint paper with K. Bjerkloev. МЕТОДЫ А.М

Lyapunov vs. geometrical stability analysis of the Kepler and the restricted three body problems

International Nuclear Information System (INIS)

Yahalom, A.; Levitan, J.; Lewkowicz, M.; Horwitz, L.

2011-01-01

In this Letter we show that although the application of standard Lyapunov analysis predicts that completely integrable Kepler motion is unstable, the geometrical analysis of Horwitz et al. predicts the observed stability. This seems to us to provide evidence for both the incompleteness of the standard Lyapunov analysis and the strength of the geometrical analysis. Moreover, we apply this approach to the three body problem in which the third body is restricted to move on a circle of large radius which induces an adiabatic time dependent potential on the second body. This causes the second body to move in a very interesting and intricate but periodic trajectory; however, the standard Lyapunov analysis, as well as methods based on the parametric variation of curvature associated with the Jacobi metric, incorrectly predict chaotic behavior. The geometric approach predicts the correct stable motion in this case as well. - Highlights: → Lyapunov analysis predicts Kepler motion to be unstable. → Geometrical analysis predicts the observed stability. → Lyapunov analysis predicts chaotic behavior in restricted three body problem. → The geometric approach predicts the correct stable motion in restricted three body problem.

Lyapunov functionals and stability of stochastic difference equations

CERN Document Server

Shaikhet, Leonid

2011-01-01

This book offers a general method of Lyapunov functional construction which lets researchers analyze the degree to which the stability properties of differential equations are preserved in their difference analogues. Includes examples from physical systems.

Diophantine exponents for mildly restricted approximation

DEFF Research Database (Denmark)

Bugeaud, Yann; Kristensen, Simon

We are studying the Diophantine exponent defined for integers and a vector by letting , where is the scalar product and denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first coordinates. We show that the exponent...

Comment on the presence of chaos in mean field dynamics inside the spinodal region

International Nuclear Information System (INIS)

Jacquot, B.; Colonna, M.; Chomaz, Ph.; Guarnera, A.

1995-01-01

The role of chaos in the mean field simulations of spinodal decomposition is analysed. It is demonstrated that, conversely to recent publications, the RPA unstable modes are only weakly non-linear and weakly coupled. The Lyapunov exponents are shown to be nothing by the largest imaginary RPA frequencies and the early mean field evolution, even of a randomly initialized trajectory, is shown to be mostly understandable within the framework of simple linear instabilities. Finally, it is recalled how the time scales are related to the use of a reasonable range for the nuclear force. (author)

Dynamic complexities in a parasitoid-host-parasitoid ecological model

International Nuclear Information System (INIS)

Yu Hengguo; Zhao Min; Lv Songjuan; Zhu Lili

2009-01-01

Chaotic dynamics have been observed in a wide range of population models. In this study, the complex dynamics in a discrete-time ecological model of parasitoid-host-parasitoid are presented. The model shows that the superiority coefficient not only stabilizes the dynamics, but may strongly destabilize them as well. Many forms of complex dynamics were observed, including pitchfork bifurcation with quasi-periodicity, period-doubling cascade, chaotic crisis, chaotic bands with narrow or wide periodic window, intermittent chaos, and supertransient behavior. Furthermore, computation of the largest Lyapunov exponent demonstrated the chaotic dynamic behavior of the model

Spatio-temporal intermittency on the sandpile

International Nuclear Information System (INIS)

Erzan, A.; Sinha, S.

1990-08-01

The self-organized critical state exhibited by a sandpile model is shown to correspond to motion on an attractor characterized by an invariant distribution of the height variable. The largest Lyapunov exponent is equal to zero. The model nonetheless displays intermittent chaos, with a multifractal distribution of local expansion coefficients in history space. Laminar spatio-temporal regions are interrupted by chaotic bursts caused by avalanches. We introduce the concept of local histories in configuration space and show that their expansion parameters also exhibit a multifractal distribution in time and space. (author). 22 refs, 5 figs

STRANGE ATTRACTORS ON PSEUDOSPECTRAL SOLUTIONS FOR DISSIPATIVE ZAKHAROV EQUATIONS

Institute of Scientific and Technical Information of China (English)

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.

Chaotic Boltzmann machines

Science.gov (United States)

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

Dynamic complexities in a parasitoid-host-parasitoid ecological model

Energy Technology Data Exchange (ETDEWEB)

Yu Hengguo [School of Mathematic and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035 (China); Zhao Min [School of Life and Environmental Science, Wenzhou University, Wenzhou, Zhejiang 325027 (China)], E-mail: zmcn@tom.com; Lv Songjuan; Zhu Lili [School of Mathematic and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035 (China)

2009-01-15

Chaotic dynamics have been observed in a wide range of population models. In this study, the complex dynamics in a discrete-time ecological model of parasitoid-host-parasitoid are presented. The model shows that the superiority coefficient not only stabilizes the dynamics, but may strongly destabilize them as well. Many forms of complex dynamics were observed, including pitchfork bifurcation with quasi-periodicity, period-doubling cascade, chaotic crisis, chaotic bands with narrow or wide periodic window, intermittent chaos, and supertransient behavior. Furthermore, computation of the largest Lyapunov exponent demonstrated the chaotic dynamic behavior of the model.

Image Encryption Technology Based on Fractional Two-Dimensional Triangle Function Combination Discrete Chaotic Map Coupled with Menezes-Vanstone Elliptic Curve Cryptosystem

Directory of Open Access Journals (Sweden)

Zeyu Liu

2018-01-01

Full Text Available A new fractional two-dimensional triangle function combination discrete chaotic map (2D-TFCDM with the discrete fractional difference is proposed. We observe the bifurcation behaviors and draw the bifurcation diagrams, the largest Lyapunov exponent plot, and the phase portraits of the proposed map, respectively. On the application side, we apply the proposed discrete fractional map into image encryption with the secret keys ciphered by Menezes-Vanstone Elliptic Curve Cryptosystem (MVECC. Finally, the image encryption algorithm is analysed in four main aspects that indicate the proposed algorithm is better than others.

A feedback control strategy for the airfoil system under non-Gaussian colored noise excitation

Energy Technology Data Exchange (ETDEWEB)

Huang, Yong, E-mail: hy@njust.edu.cn, E-mail: taogang@njust.edu.cn; Tao, Gang, E-mail: hy@njust.edu.cn, E-mail: taogang@njust.edu.cn [School of Energy and Power Engineering, Nanjing University of Science and Technology, 200 XiaoLingwei Street, Nanjing 210094 (China)

2014-09-01

The stability of a binary airfoil with feedback control under stochastic disturbances, a non-Gaussian colored noise, is studied in this paper. First, based on some approximated theories and methods the non-Gaussian colored noise is simplified to an Ornstein-Uhlenbeck process. Furthermore, via the stochastic averaging method and the logarithmic polar transformation, one dimensional diffusion process can be obtained. At last by applying the boundary conditions, the largest Lyapunov exponent which can determine the almost-sure stability of the system and the effective region of control parameters is calculated.

A feedback control strategy for the airfoil system under non-Gaussian colored noise excitation.

Science.gov (United States)

Huang, Yong; Tao, Gang

2014-09-01

The stability of a binary airfoil with feedback control under stochastic disturbances, a non-Gaussian colored noise, is studied in this paper. First, based on some approximated theories and methods the non-Gaussian colored noise is simplified to an Ornstein-Uhlenbeck process. Furthermore, via the stochastic averaging method and the logarithmic polar transformation, one dimensional diffusion process can be obtained. At last by applying the boundary conditions, the largest Lyapunov exponent which can determine the almost-sure stability of the system and the effective region of control parameters is calculated.

Merit exponents and control area diagrams in materials selection

International Nuclear Information System (INIS)

Zander, Johan; Sandstroem, Rolf

2011-01-01

Highlights: → Merit exponents are introduced to generalise the merit indices commonly used in materials selection. → The merit exponents can rank materials in general design situations. → To allow identification of the active merit exponent(s), control area diagrams are used. → Principles for generating the control area diagrams are presented. -- Abstract: Merit indices play a fundamental role in materials selection, since they enable ranking of materials. However, the conventional formulation of merit indices is associated with severe limitations. They are dependent on the explicit solution of the variables in the equations for the constraints from the design criteria. Furthermore, it is not always easy to determine which the controlling merit index is. To enable the ranking of materials in more general design cases, merit exponents are introduced as generalisations of the merit indices. Procedures are presented for how to compute the merit exponents numerically without having to solve equations algebraically. Merit exponents (and indices) are only valid in a certain range of property values. To simplify the identification of the controlling merit exponent, it is suggested that so called control area diagrams are used. These diagrams consist of a number of domains, each showing the active constraints and the controlling merit exponent. It is shown that the merit exponents play a crucial role when the control area diagram (CAD) is set up. The principles in the paper are developed for mechanically loaded components and are illustrated for engineering beams with two or three geometric variables.

On chaos in quantum mechanics: The two meanings of sensitive dependence

International Nuclear Information System (INIS)

Ingraham, R.L.; Luna Acosta, G.A.

1993-08-01

Sensitive dependence on initial conditions, the most important signature of chaos, can mean failure of Lyapunov stability, the primary meaning adopted in dynamical systems theory, or the presence of positive Lyapunov exponents, the meaning favored in physics. These are not equivalent in general. We show that there is sensitive dependence in quantum mechanics in the sense of violation of Lyapunov stability for maps of the state vector like involving unbounded operators A. This is true even for bounded quantum systems, where the corresponding Lyapunov exponents are all zero. Experiments to reveal this sensitive dependence, a definite though unfamiliar prediction of quantum mechanics, should be devised. It may also invalidate the usual assumption of linear response theory in quantum statistical mechanics in some cases. (author) 13 refs

Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

OpenAIRE

Bernal Reza, Miguel Ángel; Sala, Antonio; JAADARI, ABDELHAFIDH; Guerra, Thierry-Marie

2011-01-01

In this paper, the stability of continuous-time polynomial fuzzy models by means of a polynomial generalization of fuzzy Lyapunov functions is studied. Fuzzy Lyapunov functions have been fruitfully used in the literature for local analysis of Takagi-Sugeno models, a particular class of the polynomial fuzzy ones. Based on a recent Taylor-series approach which allows a polynomial fuzzy model to exactly represent a nonlinear model in a compact set of the state space, it is shown that a refinemen...

Uniting Control Lyapunov and Control Barrier Functions

NARCIS (Netherlands)

Romdlony, Zakiyullah; Jayawardhana, Bayu

2014-01-01

In this paper, we propose a nonlinear control design for solving the problem of stabilization with guaranteed safety. The design is based on the merging of a Control Lyapunov Function and a Control Barrier Function. The proposed control method allows us to combine the design of a stabilizer based on

Stability of dynamical systems on the role of monotonic and non-monotonic Lyapunov functions

CERN Document Server

Michel, Anthony N; Liu, Derong

2015-01-01

The second edition of this textbook provides a single source for the analysis of system models represented by continuous-time and discrete-time, finite-dimensional and infinite-dimensional, and continuous and discontinuous dynamical systems.  For these system models, it presents results which comprise the classical Lyapunov stability theory involving monotonic Lyapunov functions, as well as corresponding contemporary stability results involving non-monotonicLyapunov functions.Specific examples from several diverse areas are given to demonstrate the applicability of the developed theory to many important classes of systems, including digital control systems, nonlinear regulator systems, pulse-width-modulated feedback control systems, and artificial neural networks.   The authors cover the following four general topics:   -          Representation and modeling of dynamical systems of the types described above -          Presentation of Lyapunov and Lagrange stability theory for dynamical sy...

Fractal analysis and nonlinear forecasting of indoor 222Rn time series

International Nuclear Information System (INIS)

Pausch, G.; Bossew, P.; Hofmann, W.; Steger, F.

1998-01-01

Fractal analyses of indoor 222 Rn time series were performed using different chaos theory based measurements such as time delay method, Hurst's rescaled range analysis, capacity (fractal) dimension, and Lyapunov exponent. For all time series we calculated only positive Lyapunov exponents which is a hint to chaos, while the Hurst exponents were well below 0.5, indicating antipersistent behaviour (past trends tend to reverse in the future). These time series were also analyzed with a nonlinear prediction method which allowed an estimation of the embedding dimensions with some restrictions, limiting the prediction to about three relative time steps. (orig.)

Lyapunov equation for infinite-dimensional discrete bilinear systems

International Nuclear Information System (INIS)

Costa, O.L.V.; Kubrusly, C.S.

1991-03-01

Mean-square stability for discrete systems requires that uniform convergence is preserved between input and state correlation sequences. Such a convergence preserving property holds for an infinite-dimensional bilinear system if and only if the associate Lyapunov equation has a unique strictly positive solution. (author)

Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility

KAUST Repository

Korobeinikov, Andrei; Melnik, Andrey V.

2013-01-01

We consider global asymptotic properties for the SIR and SEIR age structured models for infectious diseases where the susceptibility depends on the age. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of a unique endemic steady state and the infection-free steady state.

Sampled-Data Control of Spacecraft Rendezvous with Discontinuous Lyapunov Approach

Directory of Open Access Journals (Sweden)

Zhuoshi Li

2013-01-01

Full Text Available This paper investigates the sampled-data stabilization problem of spacecraft relative positional holding with improved Lyapunov function approach. The classical Clohessy-Wiltshire equation is adopted to describe the relative dynamic model. The relative position holding problem is converted into an output tracking control problem using sampling signals. A time-dependent discontinuous Lyapunov functionals approach is developed, which will lead to essentially less conservative results for the stability analysis and controller design of the corresponding closed-loop system. Sufficient conditions for the exponential stability analysis and the existence of the proposed controller are provided, respectively. Finally, a simulation result is established to illustrate the effectiveness of the proposed control scheme.

Nonlinear analysis of dynamic signature

Science.gov (United States)

Rashidi, S.; Fallah, A.; Towhidkhah, F.

2013-12-01

Signature is a long trained motor skill resulting in well combination of segments like strokes and loops. It is a physical manifestation of complex motor processes. The problem, generally stated, is that how relative simplicity in behavior emerges from considerable complexity of perception-action system that produces behavior within an infinitely variable biomechanical and environmental context. To solve this problem, we present evidences which indicate that motor control dynamic in signing process is a chaotic process. This chaotic dynamic may explain a richer array of time series behavior in motor skill of signature. Nonlinear analysis is a powerful approach and suitable tool which seeks for characterizing dynamical systems through concepts such as fractal dimension and Lyapunov exponent. As a result, they can be analyzed in both horizontal and vertical for time series of position and velocity. We observed from the results that noninteger values for the correlation dimension indicates low dimensional deterministic dynamics. This result could be confirmed by using surrogate data tests. We have also used time series to calculate the largest Lyapunov exponent and obtain a positive value. These results constitute significant evidence that signature data are outcome of chaos in a nonlinear dynamical system of motor control.

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

International Nuclear Information System (INIS)

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

Non-universal spreading exponents in a catalytic reaction model

International Nuclear Information System (INIS)

De Andrade, Marcelo F; Figueiredo, W

2011-01-01

We investigated the dependence of the spreading critical exponents and the ultimate survival probability exponent on the initial configuration of a nonequilibrium catalytic reaction model. The model considers the competitive reactions between two different monomers, A and B, where we take into account the energy couplings between nearest neighbor monomers, and the adsorption energies, as well as the temperature T of the catalyst. For each value of T the model shows distinct absorbing states, with different concentrations of the two monomers. Employing an epidemic analysis, we established the behavior of the spreading exponents as we started the Monte Carlo simulations with different concentrations of the monomers. The exponents were determined as a function of the initial concentration ρ A, ini of A monomers. We have also considered initial configurations with correlations for a fixed concentration of A monomers. From the determination of three spreading exponents, and the ultimate survival probability exponent, we checked the validity of the generalized hyperscaling relation for a continuous set of initial states, random and correlated, which are dependent on the temperature of the catalyst

The Critical Exponent is Computable for Automatic Sequences

Directory of Open Access Journals (Sweden)

Jeffrey Shallit

2011-08-01

Full Text Available The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. This generalizes or recovers previous results of Krieger and others. Our technique is applicable to other situations; e.g., the computation of the optimal recurrence constant for a linearly recurrent k-automatic sequence.

Intermittent Chaos in the Bray-Liebhafsky Oscillator. Dependence of Dynamic States on the Iodate Concentration

Science.gov (United States)

Bubanja, I. N.; Ivanović-Šašić, A.; Čupić, Ž.; Anić, S.; Kolar-Anić, Lj.

2017-12-01

Chaotic dynamic states with intermittent oscillations were generated in a Bray-Liebhafsky (BL) oscillatory reaction in an isothermal open reactor i.e., in the continuously-fed well-stirred tank reactor (CSTR) when the inflow concentration of potassium iodate was the control parameter. They are found between periodic oscillations obtained when [KIO3]0 4.10 × 10-2 M. It was shown that the most chaotic states obtained experimentally somewhere in the middle of this region are in high correlation with results obtained by means of largest Lyapunov exponents and phenomenological analysis based on the quantitative characteristics of intermittent oscillations.

Dynamical behaviour of the coupled diffusion map lattice

International Nuclear Information System (INIS)

Wei Wang; Cerdeira, H.A.

1993-10-01

In this paper we report the dynamical study of a coupled diffusive map lattice with the coupling between the elements only through the bifurcation parameter of the mapping function. The diffusive process of the lattice from an initially random distribution state to a homogeneous one and the stable range of the diffusive homogeneous attractor are discussed. For various coupling strengths we find that there are several types of spatio-temporal structures. In addition, the evolution of the lattice into chaos is studied and a largest Lyapunov exponent is used to characterize the dynamical behaviour. (author). 22 refs, 9 figs

Dynamics in the Parameter Space of a Neuron Model

Science.gov (United States)

Paulo, C. Rech

2012-06-01

Some two-dimensional parameter-space diagrams are numerically obtained by considering the largest Lyapunov exponent for a four-dimensional thirteen-parameter Hindmarsh—Rose neuron model. Several different parameter planes are considered, and it is shown that depending on the combination of parameters, a typical scenario can be preserved: for some choice of two parameters, the parameter plane presents a comb-shaped chaotic region embedded in a large periodic region. It is also shown that there exist regions close to these comb-shaped chaotic regions, separated by the comb teeth, organizing themselves in period-adding bifurcation cascades.

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

International Nuclear Information System (INIS)

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

Experimental dynamical characterization of five autonomous chaotic oscillators with tunable series resistance

Energy Technology Data Exchange (ETDEWEB)

Minati, Ludovico, E-mail: lminati@ieee.org, E-mail: ludovico.minati@unitn.it [MR-Lab, Center for Mind/Brain Science, University of Trento, Trento, Italy and Scientific Department, Fondazione IRCCS Istituto Neurologico Carlo Besta, Milan (Italy)

2014-09-01

In this paper, an experimental characterization of the dynamical properties of five autonomous chaotic oscillators, based on bipolar-junction transistors and obtained de-novo through a genetic algorithm in a previous study, is presented. In these circuits, a variable resistor connected in series to the DC voltage source acts as control parameter, for a range of which the largest Lyapunov exponent, correlation dimension, approximate entropy, and amplitude variance asymmetry are calculated, alongside bifurcation diagrams and spectrograms. Numerical simulations are compared to experimental measurements. The oscillators can generate a considerable variety of regular and chaotic sine-like and spike-like signals.

Dynamic analysis of a new chaotic system with fractional order and its generalized projective synchronization

International Nuclear Information System (INIS)

Niu Yu-Jun; Wang Xing-Yuan; Nian Fu-Zhong; Wang Ming-Jun

2010-01-01

Based on the stability theory of the fractional order system, the dynamic behaviours of a new fractional order system are investigated theoretically. The lowest order we found to have chaos in the new three-dimensional system is 2.46, and the period routes to chaos in the new fractional order system are also found. The effectiveness of our analysis results is further verified by numerical simulations and positive largest Lyapunov exponent. Furthermore, a nonlinear feedback controller is designed to achieve the generalized projective synchronization of the fractional order chaotic system, and its validity is proved by Laplace transformation theory. (general)

Posture as a chaotic system and an application to the Parkinson's disease

International Nuclear Information System (INIS)

Pascolo, Paolo B.; Marini, Alfio; Carniel, Roberto; Barazza, Fausto

2005-01-01

In this work we investigate and compare a number of time series of stabilograms of healthy subjects and Parkinsonians. This is carried out by means of the chaos paradigm through the preliminary computation of the first minimum of the mutual information function and the embedding dimension (using false nearest neighbours) in order to obtain the correlation dimension as well as the largest Lyapunov exponent. We show that the postural act is indeed chaotic and especially that the latter two parameters do not allow to discriminate healthy subjects from parkinsonians. Moreover we report a discrepancy of our values with those found in previous works

Velocity width of the resonant domain in wave-particle interaction

International Nuclear Information System (INIS)

Firpo, Marie-Christine; Doveil, Fabrice

2002-01-01

Wave-particle interaction is a ubiquitous physical mechanism exhibiting locality in velocity space. A single-wave Hamiltonian provides a rich model by which to study the self-consistent interaction between one electrostatic wave and N quasiresonant particles. For the simplest nonintegrable Hamiltonian coupling two particles to one wave, we analytically derive the particle velocity borders separating quasi-integrable motions from chaotic ones. These estimates are fully retrieved through computation of the largest Lyapunov exponent. For the large-N particle self-consistent case, we numerically investigate the localization of stochasticity in velocity space and test a qualitative estimate of the borders of chaos

Gottwald Melborune (0–1 test for chaos in a plasma

Directory of Open Access Journals (Sweden)

D. R. Chowdhury

2012-01-01

Full Text Available Plasma is a highly complex system exhibiting a rich variety of nonlinear dynamical phenomena. In the last two decades or so there has been a spurt of growth in exploring unconventional nonlinear dynamical methods of analysis, like chaos theory, multi fractal analysis, self organized criticality etc. of experimental data from different plasma systems. Investigation of fluctuating plasma parameters is very important since they are correlated with transport of particles, and energy. In time series analysis, it is considered of key importance to determine whether the data measured from the system is regular, deterministically chaotic, or random. The two important parameters that are in general estimated are the correlation dimension and the Lyapunov exponent. Though correlation dimension helps in determining the complexity of a system, Lyapunov exponent reveals if the system is chaotic or not and also helps in prediction to some extent. In spite of its extensive usage, estimation of Lyapunov exponent can be quite tedious and sometimes suffers from some disadvantages like reliability in the presence of noise, requirement of phase space reconstruction etc., and hence it is necessary to explore other possibilities of estimating the chaoticity of a data. In this paper we have analysed for chaoticity, the nonlinear floating potential fluctuations from a glow discharge plasma system by the 0–1 test and compared it with the results obtained from Lyapunov exponent.

Cryptanalysis of 'less short' RSA secret exponents

NARCIS (Netherlands)

Verheul, E.R.; Tilborg, van H.C.A.

1997-01-01

In some applications of RSA, it is desirable to have a short secret exponent d. Wiener [6], describes a technique to use continued fractions (CF) in a cryptanalytic attack on an RSA cryptosystem having a ‘short’ secret exponent. Let n=p¿·¿q be the modulus of the system. In the typical case that

Lyapunov stability and thermal stability of partially relaxed fluids and plasmas

International Nuclear Information System (INIS)

Elsaesser, K.; Spiess, P.

1996-01-01

The relation between the Lyapunov stability of a Hamiltonian system and the thermal stability of a fluid whose temperature is controlled from outside is explored: The free energy as a functional of the correct variables (specific volume, local entropy, and some Clebsch potentials of the velocity) may serve as a Lyapunov functional, depending on the open-quote open-quote Casimirs close-quote close-quote as exchanged quantities. For a multi-species plasma one obtains a sufficient condition for stability: γ(v 2 /c 2 s )-1 s the sound speed. Some features of partially relaxed (T=const) cylindrical plasmas are also discussed. copyright 1996 American Institute of Physics

Robust H∞ Control for Singular Time-Delay Systems via Parameterized Lyapunov Functional Approach

Directory of Open Access Journals (Sweden)

Li-li Liu

2014-01-01

Full Text Available A new version of delay-dependent bounded real lemma for singular systems with state delay is established by parameterized Lyapunov-Krasovskii functional approach. In order to avoid generating nonconvex problem formulations in control design, a strategy that introduces slack matrices and decouples the system matrices from the Lyapunov-Krasovskii parameter matrices is used. Examples are provided to demonstrate that the results in this paper are less conservative than the existing corresponding ones in the literature.

Lyapunov Functions and Solutions of the Lyapunov Matrix Equation for Marginally Stable Systems

DEFF Research Database (Denmark)

Kliem, Wolfhard; Pommer, Christian

2000-01-01

We consider linear systems of differential equations $I \\ddot{x}+B \\dot{x}+C{x}={0}$ where $I$ is the identity matrix and $B$ and $C$ are general complex $n$ x $n$ matrices. Our main interest is to determine conditions for complete marginalstability of these systems. To this end we find solutions...... of the Lyapunov matrix equation and characterize the set of matrices $(B, C)$ which guarantees marginal stability. The theory is applied to gyroscopic systems, to indefinite damped systems, and to circulatory systems, showing how to choose certain parameter matrices to get sufficient conditions for marginal...... stability.Comparison is made with some known results for equations with real system matrices.Moreover more general cases are investigated and several examples are given....

Stretching Diagnostics and Mixing Properties In The Stratosphere

Science.gov (United States)

Legras, B.; Shuckburgh, E.

The "finite size Lyapunov exponent" and the "effective diffusivity" are two diagnos- tics of mixing which have been recently introduced to investigate atmospheric flows. Both have been used to successfully identify the barriers to transport, for instance at the edge of the stratospheric polar vortex. Here we compare the two diagnostics in detail. The equivalent length has the advantage of arising as a mixing quantification from a rigid theoretical framework, however it has the disadvantage of being an aver- age quantity (the average around a tracer contour). The finite size Lyapunov exponent may be defined at any point in the flow, and quantifies the stretching properties expe- rienced by a fluid parcel both in its past and future evolution. In particular, the lines of maximum stretching at any time delineate the building blocks of the chaotic stirring. However the interpretation of the finite size Lyapunov exponent as a mixing time is less direct and depends on the alignment of tracer contours with the stretching lines.

Edge state preparation in a one-dimensional lattice by quantum Lyapunov control

International Nuclear Information System (INIS)

Zhao, X L; Shi, Z C; Qin, M; Yi, X X

2017-01-01

Quantum Lyapunov control uses a feedback control methodology to determine control fields applied to control quantum systems in an open-loop way. In this work, we employ two Lyapunov control schemes to prepare an edge state for a fermionic chain consisting of cold atoms loaded in an optical lattice. Such a chain can be described by the Harper model. Corresponding to the two schemes, two types of quantum Lyapunov functions are considered. The results show that both the schemes are effective at preparing the edge state within a wide range of parameters. We found that the edge state can be prepared with high fidelity even if there are moderate fluctuations of on-site or hopping potentials. Both control schemes can be extended to similar chains (3 m + d , d = 2) of different lengths. Since a regular amplitude control field is easier to apply in practice, an amplitude-modulated control field is used to replace the unmodulated one. Such control approaches provide tools to explore the edge states of one-dimensional topological materials. (paper)

What is the cementation exponent? A new differential interpretation

Science.gov (United States)

Glover, P. W. J.

2009-04-01

Between 1950 and 2002 the total volume of reserves discovered has run to over 1500 Bbbl. for oil and 7.5 Tcf. for gas. Over half of these resources has already been produced, and has driven the global economy for the last fifty years. All of the assessments of the volume of hydrocarbon reserves were made using Archie's relationships (1942). It would be difficult, therefore, to overestimate the impact of either the petrophysical techniques or Archie's relationships on the worldwide economy. Archie's laws link the electrical resistivity of a rock to its porosity, to the resistivity of the water that saturates its pores, and to the fractional saturation of the pore space with the water, and are used to calculate the hydrocarbon saturation of the reservoir rock from which the reserves are then calculated. Archie's laws contain two exponents, m and n, which Archie called the cementation exponent and the saturation exponent, respectively. The conductivity of the hydrocarbon saturated rock is highly sensitive to changes in either exponent. However, despite the importance of the cementation exponent, few petrophysicists, commercial or academic, are able to describe its real physical meaning. The purpose of this contribution is to investigate the elusive physical meaning of the cementation exponent. We review the traditional interpretation of the cementation exponent and consider the extension of Archie's first law to two conducting phases. Consequently, we develop a new differential interpretation of the cementation exponent that is based on a new definition for the connectedness of the conducting phases in a porous medium. In this interpretation the connectedness of a porous medium is defined as the availability of pathways for transport, where the connectedness is the inverse of the formation resistivity factor, G = σo σw = 1 F (and may also be called the conductivity formation factor). Porosity is defined as the fractional amount of pore space in the usual manner

First-passage exponents of multiple random walks

International Nuclear Information System (INIS)

Ben-Naim, E; Krapivsky, P L

2010-01-01

We investigate first-passage statistics of an ensemble of N noninteracting random walks on a line. Starting from a configuration in which all particles are located in the positive half-line, we study S n (t), the probability that the nth rightmost particle remains in the positive half-line up to time t. This quantity decays algebraically, S n (t)∼t -β n , in the long-time limit. Interestingly, there is a family of nontrivial first-passage exponents, β 1 2 N-1 ; the only exception is the two-particle case where β 1 = 1/3. In the N → ∞ limit, however, the exponents attain a scaling form, β n (N) → β(z) with z=(n-N/2)/√N. We also demonstrate that the smallest exponent decays exponentially with N. We deduce these results from first-passage kinetics of a random walk in an N-dimensional cone and confirm them using numerical simulations. Additionally, we investigate the family of exponents that characterizes leadership statistics of multiple random walks and find that in this case, the cone provides an excellent approximation.

A new theoretical interpretation of Archie's saturation exponent

Directory of Open Access Journals (Sweden)

P. W. J. Glover

2017-07-01

Full Text Available This paper describes the extension of the concepts of connectedness and conservation of connectedness that underlie the generalized Archie's law for n phases to the interpretation of the saturation exponent. It is shown that the saturation exponent as defined originally by Archie arises naturally from the generalized Archie's law. In the generalized Archie's law the saturation exponent of any given phase can be thought of as formally the same as the phase (i.e. cementation exponent, but with respect to a reference subset of phases in a larger n-phase medium. Furthermore, the connectedness of each of the phases occupying a reference subset of an n-phase medium can be related to the connectedness of the subset itself by Gi = GrefSini. This leads naturally to the idea of the term Sini for each phase i being a fractional connectedness, where the fractional connectednesses of any given reference subset sum to unity in the same way that the connectednesses sum to unity for the whole medium. One of the implications of this theory is that the saturation exponent of any phase can be now be interpreted as the rate of change of the fractional connectedness with saturation and connectivity within the reference subset.

A new exponent in self-avoiding walks

International Nuclear Information System (INIS)

Srivastava, V.

1983-06-01

Existence of a new exponent is reported in the problem of nonintersecting self-avoiding random walks. It is connected with the asymptotic behaviour of the growth of number of such walks of larger and larger length. The value of the exponent is found to be nearly 0.90 for all two-dimensional and nearly 0.96 for all three-dimensional lattices studied here. (author)

How We Tend To Overestimate Powerlaw Tail Exponents

OpenAIRE

Nassim N. Taleb

2012-01-01

In the presence of a layer of metaprobabilities (from uncertainty concerning the parameters), the asymptotic tail exponent corresponds to the lowest possible tail exponent regardless of its probability. The problem explains "Black Swan" effects, i.e., why measurements tend to chronically underestimate tail contributions, rather than merely deliver imprecise but unbiased estimates.

Synchronization Analysis of the Supermarket Refrigeration System

DEFF Research Database (Denmark)

Wisniewski, Rafal; Chen, Liang; Larsen, Lars Finn Sloth

2009-01-01

is analyzed using the bifurcation and chaos theory. It is demonstrated that the system can have a complex chaotic behavior, which is far from the synchronization. This shows that making the system chaotic is a good choice for a de-synchronization strategy. The positive maximum Lyapunov exponent is usually...... taken as an indication of the existence of chaos. It is used in the paper as a measure of performance for the tendency of the system to synchronize, that is, the higher value of the maximum Lyapunov exponent the lower risk for synchronization....

Chaotic dynamics in dense fluids

International Nuclear Information System (INIS)

Posch, H.A.; Hoover, W.G.

1987-09-01

We present calculations of the full spectra of Lyapunov exponents for 8- and 32-particle systems with periodic boundary conditions and interacting with the repulsive part of a Lennard-Jones potential both in equilibrium and nonequilibrium steady states. Lyapunov characteristic exponents λ/sub n/ describe the mean exponential rates of divergence and convergence of neighbouring trajectories in phase-space. They are useful in characterizing the stochastic properties of a dynamical system. A new algorithm for their calculation is presented which incorporates ideas from control theory and constraint nonequilibrium molecular dynamics. 4 refs., 1 fig

The Evolution of the Exponent of Zipf's Law in Language Ontogeny

Science.gov (United States)

Baixeries, Jaume; Elvevåg, Brita; Ferrer-i-Cancho, Ramon

2013-01-01

It is well-known that word frequencies arrange themselves according to Zipf's law. However, little is known about the dependency of the parameters of the law and the complexity of a communication system. Many models of the evolution of language assume that the exponent of the law remains constant as the complexity of a communication systems increases. Using longitudinal studies of child language, we analysed the word rank distribution for the speech of children and adults participating in conversations. The adults typically included family members (e.g., parents) or the investigators conducting the research. Our analysis of the evolution of Zipf's law yields two main unexpected results. First, in children the exponent of the law tends to decrease over time while this tendency is weaker in adults, thus suggesting this is not a mere mirror effect of adult speech. Second, although the exponent of the law is more stable in adults, their exponents fall below 1 which is the typical value of the exponent assumed in both children and adults. Our analysis also shows a tendency of the mean length of utterances (MLU), a simple estimate of syntactic complexity, to increase as the exponent decreases. The parallel evolution of the exponent and a simple indicator of syntactic complexity (MLU) supports the hypothesis that the exponent of Zipf's law and linguistic complexity are inter-related. The assumption that Zipf's law for word ranks is a power-law with a constant exponent of one in both adults and children needs to be revised. PMID:23516390

Quantum synchronization in an optomechanical system based on Lyapunov control.

Science.gov (United States)

Li, Wenlin; Li, Chong; Song, Heshan

2016-06-01

We extend the concepts of quantum complete synchronization and phase synchronization, which were proposed in A. Mari et al., Phys. Rev. Lett. 111, 103605 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.103605, to more widespread quantum generalized synchronization. Generalized synchronization can be considered a necessary condition or a more flexible derivative of complete synchronization, and its criterion and synchronization measure are proposed and analyzed in this paper. As examples, we consider two typical generalized synchronizations in a designed optomechanical system. Unlike the effort to construct a special coupling synchronization system, we purposefully design extra control fields based on Lyapunov control theory. We find that the Lyapunov function can adapt to more flexible control objectives, which is more suitable for generalized synchronization control, and the control fields can be achieved simply with a time-variant voltage. Finally, the existence of quantum entanglement in different generalized synchronizations is also discussed.

Complex Dynamics of an Adnascent-Type Game Model

Directory of Open Access Journals (Sweden)

Baogui Xin

2008-01-01

Full Text Available The paper presents a nonlinear discrete game model for two oligopolistic firms whose products are adnascent. (In biology, the term adnascent has only one sense, “growing to or on something else,” e.g., “moss is an adnascent plant.” See Webster's Revised Unabridged Dictionary published in 1913 by C. & G. Merriam Co., edited by Noah Porter. The bifurcation of its Nash equilibrium is analyzed with Schwarzian derivative and normal form theory. Its complex dynamics is demonstrated by means of the largest Lyapunov exponents, fractal dimensions, bifurcation diagrams, and phase portraits. At last, bifurcation and chaos anticontrol of this system are studied.

Chaotic dynamics of respiratory sounds

International Nuclear Information System (INIS)

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

Chaotic dynamics of respiratory sounds

Energy Technology Data Exchange (ETDEWEB)

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.

Dynamical behavior of RF-biased Josephson junctions (II)

Energy Technology Data Exchange (ETDEWEB)

Xi-Dan, Wang; Xi-Xian, Yao

1985-09-01

Numerical investigations of a differential equation describing a rf-biased Josephson junction, in which the interference term current is included, are carried out in some parameter region. The existence of the intermittant transition to chaos is obtained and the critical exponent of the scaling law is determined in agreement with theoretical predictions. Furthermore, the Lyapunov exponent is calculated for several parameters, then the fractal dimension of strange attractor d/sub L/ is obtained, its dependence on the Lyapunov exponent is defined by Kaplan and Yorke. In addition, the Kolmogorov capacity of strange attractor d/sub c/ is also calculated by box-counting algorithm. Such calculated values of d/sub L/ and d/sub c/ are close to each other as expected.

New zero-input overflow stability proofs based on Lyapunov theory

NARCIS (Netherlands)

Werter, M.J.; Ritzerfeld, J.H.F.

1989-01-01

The authors demonstrate some proofs of zero-input overflow-oscillation suppression in recursive digital filters. The proofs are based on the second method of Lyapunov. For second-order digital filters with complex conjugated poles, the state describes a trajectory in the phase plane, spiraling

A New Approach to the Method of Lyapunov Functionals and Its Applications

Directory of Open Access Journals (Sweden)

Yunguo Jin

2013-01-01

Full Text Available We show some results which can replace the graph theory used to construct global Lyapunov functions in some coupled systems of differential equations. We present an example of an epidemic model with stage structure and latency spreading in a heterogeneous host population and obtain a more general threshold for the extinction and persistence of a disease. Using some results obtained by mathematical induction and suitable Lyapunov functionals, we prove the global stability of the endemic equilibrium. For some coupled systems of differential equations, by a similar approach to the discussion of the epidemic model, the conditions of threshold property or global stability can be established without the assumption that the relative matrix is irreducible.

Lyapunov functions for a dengue disease transmission model

International Nuclear Information System (INIS)

Tewa, Jean Jules; Dimi, Jean Luc; Bowong, Samuel

2009-01-01

In this paper, we study a model for the dynamics of dengue fever when only one type of virus is present. For this model, Lyapunov functions are used to show that when the basic reproduction ratio is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is also globally asymptotically stable.

Lyapunov functions for a dengue disease transmission model

Energy Technology Data Exchange (ETDEWEB)

Tewa, Jean Jules [Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde (Cameroon)], E-mail: tewa@univ-metz.fr; Dimi, Jean Luc [Department of Mathematics, Faculty of Science, University Marien Ngouabi, P.O. Box 69, Brazzaville (Congo, The Democratic Republic of the)], E-mail: jldimi@yahoo.fr; Bowong, Samuel [Department of Mathematics and Computer Science, Faculty of Science, University of Douala, P.O. Box 24157, Douala (Cameroon)], E-mail: samuelbowong@yahoo.fr

2009-01-30

In this paper, we study a model for the dynamics of dengue fever when only one type of virus is present. For this model, Lyapunov functions are used to show that when the basic reproduction ratio is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is also globally asymptotically stable.

Stability of a nonlinear second order equation under parametric bounded noise excitation

International Nuclear Information System (INIS)

Wiebe, Richard; Xie, Wei-Chau

2016-01-01

The motivation for the following work is a structural column under dynamic axial loads with both deterministic (harmonic transmitted forces from the surrounding structure) and random (wind and/or earthquake) loading components. The bounded noise used herein is a sinusoid with an argument composed of a random (Wiener) process deviation about a mean frequency. By this approach, a noise parameter may be used to investigate the behavior through the spectrum from simple harmonic forcing, to a bounded random process with very little harmonic content. The stability of both the trivial and non-trivial stationary solutions of an axially-loaded column (which is modeled as a second order nonlinear equation) under parametric bounded noise excitation is investigated by use of Lyapunov exponents. Specifically the effect of noise magnitude, amplitude of the forcing, and damping on stability of a column is investigated. First order averaging is employed to obtain analytical approximations of the Lyapunov exponents of the trivial solution. For the non-trivial stationary solution however, the Lyapunov exponents are obtained via Monte Carlo simulation as the stability equations become analytically intractable. (paper)

On the Lojasiewicz exponent at infinity of real polynomials

International Nuclear Information System (INIS)

Ha Huy Vui; Pham Tien Son

2007-07-01

Let f : R n → R be a nonconstant polynomial function. In this paper, using the information from 'the curve of tangency' of f, we provide a method to determine the Lojasiewicz exponent at infinity of f. As a corollary, we give a computational criterion to decide if the Lojasiewicz exponent at infinity is finite or not. Then, we obtain a formula to calculate the set of points at which the polynomial f is not proper. Moreover, a relation between the Lojasiewicz exponent at infinity of f with the problem of computing the global optimum of f is also established. (author)

Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation

NARCIS (Netherlands)

Maschke, Bernhard M.J.; Ortega, Romeo; Schaft, Arjan J. van der

1998-01-01

It is well known that the total energy is a suitable Lyapunov function to study the stability of the trivial equilibrium of an isolated standard Hamiltonian system. In many practical instances, however, the system is in interaction with its environment through some constant forcing terms. This gives

Novel stability criteria for uncertain delayed Cohen-Grossberg neural networks using discretized Lyapunov functional

International Nuclear Information System (INIS)

Souza, Fernando O.; Palhares, Reinaldo M.; Ekel, Petr Ya.

2009-01-01

This paper deals with the stability analysis of delayed uncertain Cohen-Grossberg neural networks (CGNN). The proposed methodology consists in obtaining new robust stability criteria formulated as linear matrix inequalities (LMIs) via the Lyapunov-Krasovskii theory. Particularly one stability criterion is derived from the selection of a parameter-dependent Lyapunov-Krasovskii functional, which allied with the Gu's discretization technique and a simple strategy that decouples the system matrices from the functional matrices, assures a less conservative stability condition. Two computer simulations are presented to support the improved theoretical results.

An alternative bifurcation analysis of the Rose-Hindmarsh model

International Nuclear Information System (INIS)

Nikolov, Svetoslav

2005-01-01

The paper presents an alternative study of the bifurcation behavior of the Rose-Hindmarsh model using Lyapunov-Andronov's theory. This is done on the basis of the obtained analytical formula expressing the first Lyapunov's value (this is not Lyapunov exponent) at the boundary of stability. From the obtained results the following new conclusions are made: Transition to chaos and the occurrence of chaotic oscillations in the Rose-Hindmarsh system take place under hard stability loss

Application of Lyapunov's Second Method in the Stability Analysis of ...

African Journals Online (AJOL)

In this paper, Lyapunov's method for determining the stability of non-linear systems under dynamic states is presented. The paper highlights a practical application of the method to investigate the stability of crude oil/natural gas separation process. Mathematical state models for the separation process, used in the ...

Journal of Genetics | Indian Academy of Sciences

Indian Academy of Sciences (India)

Keywords. fitness; invasion exponent; adaptive dynamics; game theory; Lyapunov exponent; invasibility; Malthusian parameter. Abstract. The analysis of evolutionary models requires an appropriate definition for fitness. In this paper, I review such definitions in relation to the five major dimensions by which models may be ...

Design of Connectivity Preserving Flocking Using Control Lyapunov Function

OpenAIRE

Erfianto, Bayu; Bambang, Riyanto T.; Hindersah, Hilwadi; Muchtadi-Alamsyah, Intan

2016-01-01

This paper investigates cooperative flocking control design with connectivity preserving mechanism. During flocking, interagent distance is measured to determine communication topology of the flocks. Then, cooperative flocking motion is built based on cooperative artificial potential field with connectivity preserving mechanism to achieve the common flocking objective. The flocking control input is then obtained by deriving cooperative artificial potential field using control Lyapunov functio...

Generation of hyperchaos from the Chen-Lee system via sinusoidal perturbation

International Nuclear Information System (INIS)

Tam, L.M.; Chen, J.H.; Chen, H.K.; Wai Meng Si Tou

2008-01-01

A system with more than one positive Lyapunov exponent can be classified as a hyperchaotic system. In this study, a sinusoidal perturbation was designed for generating hyperchaos from the Chen-Lee chaotic system. The hyperchaos was identified by the existence of two positive Lyapunov exponents and bifurcation diagrams. The system is hyperchaotic in several different regions of the parameters c, ε, and ω. It was found that this method not only can enhance or suppress chaotic behavior, but also induces chaos in non-chaotic parameter ranges. In addition, two interesting dynamical behaviors, Hopf bifurcation and intermittency, were also found in this study

Chaotic Patterns in Aeroelastic Signals

Directory of Open Access Journals (Sweden)

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.

Generation of hyperchaos from the Chen-Lee system via sinusoidal perturbation

Energy Technology Data Exchange (ETDEWEB)

Tam, L.M. [Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Thomas Pereira S.J., Taipa, Macau (China)], E-mail: fstlmt@umac.mo; Chen, J.H. [Department of Mechanical Engineering, Chung Hua University, Hsinchu, Taiwan (China); Chen, H.K. [Department of Mechanical Engineering, Hsiuping Institute of Technology, Taichung, Taiwan (China); Wai Meng Si Tou [Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Thomas Pereira S.J., Taipa, Macau (China)

2008-11-15

A system with more than one positive Lyapunov exponent can be classified as a hyperchaotic system. In this study, a sinusoidal perturbation was designed for generating hyperchaos from the Chen-Lee chaotic system. The hyperchaos was identified by the existence of two positive Lyapunov exponents and bifurcation diagrams. The system is hyperchaotic in several different regions of the parameters c, {epsilon}, and {omega}. It was found that this method not only can enhance or suppress chaotic behavior, but also induces chaos in non-chaotic parameter ranges. In addition, two interesting dynamical behaviors, Hopf bifurcation and intermittency, were also found in this study.

A family of stadium-like billiards with parabolic boundaries under scaling analysis

International Nuclear Information System (INIS)

Livorati, Andre L P; Loskutov, Alexander; Leonel, Edson D

2011-01-01

Some chaotic properties of a family of stadium-like billiards with parabolic focusing components, which is described by a two-dimensional nonlinear area-preserving map, are studied. Critical values of billiard geometric parameters corresponding to a sudden change of the maximal Lyapunov exponent are found. It is shown that the maximal Lyapunov exponent obtained for chaotic orbits of this family is scaling invariant with respect to the control parameters describing the geometry of the billiard. We also show that this behavior is observed for a generic one-parameter family of mapping with the nonlinearity given by a tangent function.

Hyperchaos Numerical Simulation and Control in a 4D Hyperchaotic System

Directory of Open Access Journals (Sweden)

Junhai Ma

2013-01-01

Full Text Available A hyperchaotic system is introduced, and the complex dynamical behaviors of such system are investigated by means of numerical simulations. The bifurcation diagrams, Lyapunov exponents, hyperchaotic attractors, the power spectrums, and time charts are mapped out through the theory analysis and dynamic simulations. The chaotic and hyper-chaotic attractors exist and alter over a wide range of parameters according to the variety of Lyapunov exponents and bifurcation diagrams. Furthermore, linear feedback controllers are designed for stabilizing the hyperchaos to the unstable equilibrium points; thus, we achieve the goal of a second control which is more useful in application.

Interpolation of polytopic control Lyapunov functions for discrete–time linear systems

NARCIS (Netherlands)

Nguyen, T.T.; Lazar, M.; Spinu, V.; Boje, E.; Xia, X.

2014-01-01

This paper proposes a method for interpolating two (or more) polytopic control Lyapunov functions (CLFs) for discrete--time linear systems subject to polytopic constraints, thereby combining different control objectives. The corresponding interpolated CLF is used for synthesis of a stabilizing

Automated detection of sleep apnea from electrocardiogram signals using nonlinear parameters

International Nuclear Information System (INIS)

Acharya, U Rajendra; Faust, Oliver; Chua, Eric Chern-Pin; Lim, Teik-Cheng; Lim, Liang Feng Benjamin

2011-01-01

Sleep apnoea is a very common sleep disorder which can cause symptoms such as daytime sleepiness, irritability and poor concentration. To monitor patients with this sleeping disorder we measured the electrical activity of the heart. The resulting electrocardiography (ECG) signals are both non-stationary and nonlinear. Therefore, we used nonlinear parameters such as approximate entropy, fractal dimension, correlation dimension, largest Lyapunov exponent and Hurst exponent to extract physiological information. This information was used to train an artificial neural network (ANN) classifier to categorize ECG signal segments into one of the following groups: apnoea, hypopnoea and normal breathing. ANN classification tests produced an average classification accuracy of 90%; specificity and sensitivity were 100% and 95%, respectively. We have also proposed unique recurrence plots for the normal, hypopnea and apnea classes. Detecting sleep apnea with this level of accuracy can potentially reduce the need of polysomnography (PSG). This brings advantages to patients, because the proposed system is less cumbersome when compared to PSG

Hypergeometric continuation of divergent perturbation series: I. Critical exponents of the Bose–Hubbard model

International Nuclear Information System (INIS)

Sanders, Sören; Holthaus, Martin

2017-01-01

We study the connection between the exponent of the order parameter of the Mott insulator-to-superfluid transition occurring in the two-dimensional Bose–Hubbard model, and the divergence exponents of its one- and two-particle correlation functions. We find that at the multicritical points all divergence exponents are related to each other, allowing us to express the critical exponent in terms of one single divergence exponent. This approach correctly reproduces the critical exponent of the three-dimensional XY universality class. Because divergence exponents can be computed in an efficient manner by hypergeometric analytic continuation, our strategy is applicable to a wide class of systems. (paper)

Hypergeometric continuation of divergent perturbation series: I. Critical exponents of the Bose-Hubbard model

Science.gov (United States)

Sanders, Sören; Holthaus, Martin

2017-10-01

We study the connection between the exponent of the order parameter of the Mott insulator-to-superfluid transition occurring in the two-dimensional Bose-Hubbard model, and the divergence exponents of its one- and two-particle correlation functions. We find that at the multicritical points all divergence exponents are related to each other, allowing us to express the critical exponent in terms of one single divergence exponent. This approach correctly reproduces the critical exponent of the three-dimensional XY universality class. Because divergence exponents can be computed in an efficient manner by hypergeometric analytic continuation, our strategy is applicable to a wide class of systems.

Predictability of chaotic dynamics a finite-time Lyapunov exponents approach

CERN Document Server

Vallejo, Juan C

2017-01-01

This book is primarily concerned with the computational aspects of predictability of dynamical systems – in particular those where observation, modeling and computation are strongly interdependent. Unlike with physical systems under control in laboratories, for instance in celestial mechanics, one is confronted with the observation and modeling of systems without the possibility of altering the key parameters of the objects studied. Therefore, the numerical simulations offer an essential tool for analyzing these systems. With the widespread use of computer simulations to solve complex dynamical systems, the reliability of the numerical calculations is of ever-increasing interest and importance. This reliability is directly related to the regularity and instability properties of the modeled flow. In this interdisciplinary scenario, the underlying physics provide the simulated models, nonlinear dynamics provides their chaoticity and instability properties, and the computer sciences provide the actual numerica...

Chaos and its synchronization in two-neuron systems with discrete delays

International Nuclear Information System (INIS)

Zhou Shangbo; Liao Xiaofeng; Yu Juebang; Wong Kwokwo

2004-01-01

It is well known that complex dynamic behaviors exist in time-delayed neural systems. Infinite positive Lyapunov exponents can be found in time-delayed chaotic systems since the dimension of such systems is infinite. However, theoretical and experimental models studied thus far are low dimensional systems with only one positive Lyapunov exponent. Consequently, messages masked by such chaotic systems are shown to be easily extracted in some cases. Therefore, communication system with a higher security level can be design by means of the time-delayed neuron systems. In this paper, we firstly investigate the dynamical behaviors of two-neuron systems with discrete delays. Then, the chaos synchronization in time-delayed neuron system is studied based on the method of designing the coupled system and employing Krasovskii-Lyapunov theory to search the synchronization conditions. Numerical results illustrate the correctness of our theoretical analyses

The hurst exponent and long-time correlation

International Nuclear Information System (INIS)

Wang, G.; Antar, G.; Devynck, P.

1999-10-01

The rescaled range statistics (R/S) method is applied to the ion saturation current fluctuations measured by Langmuir probe at edge on Tore Supra to evaluate the Hurst exponent. Data block randomization is carried out to the data sets in order to investigate the relationship between the Hurst exponent and long time correlation. It is observed that h is well above 0.5 in the long time self-similar range. However, it is found that the information which leads to H > 0.5 is totally contained in the short-time correlation and no link to long times is found. (authors)

Espectro de Lyapunov de un Oscilador Colpitts en Base Común

Directory of Open Access Journals (Sweden)

Camilo Andrés Florez

2013-09-01

Full Text Available En el presente documento se presenta la definición de exponentes de Lyapunov de un sistema autónomo no lineal de tiempo continuo y una técnica recomendada para medir dicho conjunto de exponentes (espectro, con la finalidad de detectar la existencia de ciclos límites o de caos en un circuito oscilador Colpitts implementado con un transistor BJT. A partir del modelo de Ebers-Möll del transistor BJT se derivaron las ecuaciones de estado que rigen al circuito, luego se adoptó un caso numérico de estudio, y mediante el uso de un programa de simulación matemática se aplicó la metodología propuesta para determinar el espectro de Lyapunov del oscilador. Los resultados obtenidos evidencian la existencia de caos para algunos conjuntos de valores de los parámetros del circuito.

A variational approach to Lyapunov type inequalities from ODEs to PDEs

CERN Document Server

Cañada, Antonio

2015-01-01

This book highlights the current state of Lyapunov-type inequalities through a detailed analysis. Aimed toward researchers and students working in differential equations and those interested in the applications of stability theory and resonant systems, the book begins with an overview Lyapunov’s original results and moves forward to include prevalent results obtained in the past ten years. Detailed proofs and an emphasis on basic ideas are provided for different boundary conditions for ordinary differential equations, including Neumann, Dirichlet, periodic, and antiperiodic conditions. Novel results of higher eigenvalues, systems of equations, partial differential equations as well as variational approaches are presented. To this respect, a new and unified variational point of view  is introduced for the treatment of such problems and a systematic discussion of different types of boundary conditions is featured. Various problems make the study of Lyapunov-type inequalities of interest to those in pure and ...

Subdiffusive master equation with space-dependent anomalous exponent and structural instability

Science.gov (United States)

Fedotov, Sergei; Falconer, Steven

2012-03-01

We derive the fractional master equation with space-dependent anomalous exponent. We analyze the asymptotic behavior of the corresponding lattice model both analytically and by Monte Carlo simulation. We show that the subdiffusive fractional equations with constant anomalous exponent μ in a bounded domain [0,L] are not structurally stable with respect to the nonhomogeneous variations of parameter μ. In particular, the Gibbs-Boltzmann distribution is no longer the stationary solution of the fractional Fokker-Planck equation whatever the space variation of the exponent might be. We analyze the random distribution of μ in space and find that in the long-time limit, the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events. We show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a “Black Swan,” the low probability event of the small value of the anomalous exponent that completely dominates the long-time behavior of subdiffusive systems.

Global stability analysis of epidemiological models based on Volterra–Lyapunov stable matrices

International Nuclear Information System (INIS)

Liao Shu; Wang Jin

2012-01-01

Highlights: ► Global dynamics of high dimensional dynamical systems. ► A systematic approach for global stability analysis. ► Epidemiological models of environment-dependent diseases. - Abstract: In this paper, we study the global dynamics of a class of mathematical epidemiological models formulated by systems of differential equations. These models involve both human population and environmental component(s) and constitute high-dimensional nonlinear autonomous systems, for which the global asymptotic stability of the endemic equilibria has been a major challenge in analyzing the dynamics. By incorporating the theory of Volterra–Lyapunov stable matrices into the classical method of Lyapunov functions, we present an approach for global stability analysis and obtain new results on some three- and four-dimensional model systems. In addition, we conduct numerical simulation to verify the analytical results.

Stabilisation of discrete-time polynomial fuzzy systems via a polynomial lyapunov approach

Science.gov (United States)

Nasiri, Alireza; Nguang, Sing Kiong; Swain, Akshya; Almakhles, Dhafer

2018-02-01

This paper deals with the problem of designing a controller for a class of discrete-time nonlinear systems which is represented by discrete-time polynomial fuzzy model. Most of the existing control design methods for discrete-time fuzzy polynomial systems cannot guarantee their Lyapunov function to be a radially unbounded polynomial function, hence the global stability cannot be assured. The proposed control design in this paper guarantees a radially unbounded polynomial Lyapunov functions which ensures global stability. In the proposed design, state feedback structure is considered and non-convexity problem is solved by incorporating an integrator into the controller. Sufficient conditions of stability are derived in terms of polynomial matrix inequalities which are solved via SOSTOOLS in MATLAB. A numerical example is presented to illustrate the effectiveness of the proposed controller.

Chaotic characteristics enhanced by impeller of perturbed six-bent-bladed turbine in stirred tank

Directory of Open Access Journals (Sweden)

Deyu Luan

Full Text Available The fundamental way of improving the mixing efficiency is to induce the chaotic flow in a stirred vessel. The impeller form plays an important role for changing the structure of flow field and realizing chaotic mixing. Based on the velocity time series acquired by the experiment of particle image velocimetry (PIV, with the software Matlab, the macro-instability (MI, largest Lyapunov exponent (LLE, and Kolmogorov entropy in the water stirred tank is investigated respectively with the impeller of perturbed six-bent-bladed turbine (6PBT. The results show that the MI characteristics are obvious and two peak values of MI frequency are observed at the speed N = 60 rpm. With the increasing speed (more than 100 rpm, the peak characteristics of MI frequency disappear and a multi-scale wavelet structure of characterizing the chaotic flow field appears. Moreover, under the speed N = 60 rpm, the LLE is less than 0 and Kolmogorov entropy is 0, which means that the flow field is in the periodic moving state. As the speed is increased to more than 100 rpm, the LLE and Kolmogorov entropy are all more than 0, which indicates that the flow field goes into the chaotic mixing. When the speed reaches up to about 210 rpm, both of the LLE and Kolmogorov entropy achieve the optimum values, which will result in an excellent chaos with the highest mixing efficient. So it is feasible that the MI frequency, the LLE and the Kolmogorov entropy can be used to analyze the flow field characteristics in a stirred tank. The research results promote the understanding of the chaotic mixing mechanism and provide a theoretical reference for the development of new type impeller. Keywords: Macro-instability, The largest Lyapunov exponent, Kolmogorov entropy, The impeller of perturbed six-bent-bladed turbine, Chaotic mixing, PIV

Condensation and critical exponents of an ideal non-Abelian gas

Science.gov (United States)

Talaei, Zahra; Mirza, Behrouz; Mohammadzadeh, Hosein

2017-11-01

We investigate an ideal gas obeying non-Abelian statistics and derive the expressions for some thermodynamic quantities. It is found that thermodynamic quantities are finite at the condensation point where their derivatives diverge and, near this point, they behave as \\vert T-Tc\\vert^{-ρ} in which Tc denotes the condensation temperature and ρ is a critical exponent. The critical exponents related to the heat capacity and compressibility are obtained by fitting numerical results and others are obtained using the scaling law hypothesis for a three-dimensional non-Abelian ideal gas. This set of critical exponents introduces a new universality class.

Detecting Chaos from Agricultural Product Price Time Series

Directory of Open Access Journals (Sweden)

Xin Su

2014-12-01

Full Text Available Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos of the data. The approaches used include the R/S analysis, the BDS test, the power spectra, the recurrence plot, the largest Lyapunov exponent, the Kolmogorov entropy, and the correlation dimension. The results show that there is chaos in agricultural wholesale price data, which provides a good theoretical basis for selecting reasonable forecasting models as prediction techniques based on chaos theory can be applied to forecasting agricultural prices.

Improving the quality of extracting dynamics from interspike intervals via a resampling approach

Science.gov (United States)

Pavlova, O. N.; Pavlov, A. N.

2018-04-01

We address the problem of improving the quality of characterizing chaotic dynamics based on point processes produced by different types of neuron models. Despite the presence of embedding theorems for non-uniformly sampled dynamical systems, the case of short data analysis requires additional attention because the selection of algorithmic parameters may have an essential influence on estimated measures. We consider how the preliminary processing of interspike intervals (ISIs) can increase the precision of computing the largest Lyapunov exponent (LE). We report general features of characterizing chaotic dynamics from point processes and show that independently of the selected mechanism for spike generation, the performed preprocessing reduces computation errors when dealing with a limited amount of data.

Application of chaos theory to the particle dynamics of asymmetry-induced transport

Science.gov (United States)

Eggleston, D. L.

2018-03-01

The techniques of chaos theory are employed in an effort to better understand the complex single-particle dynamics of asymmetry-induced transport in non-neutral plasmas. The dynamical equations are re-conceptualized as describing time-independent trajectories in a four-dimensional space consisting of the radius r, rotating frame angle ψ, axial position z, and axial velocity v. Results include the identification of an integral of the motion, fixed-point analysis of the dynamical equations, the construction and interpretation of Poincaré sections to visualize the dynamics, and, for the case of chaotic motion, numerical calculation of the largest Lyapunov exponent. Chaotic cases are shown to be associated with the overlap of resonance islands formed by the applied asymmetry.

Synchronization of hyperchaotic oscillators via single unidirectional chaotic-coupling

International Nuclear Information System (INIS)

Zou Yanli; Zhu Jie; Chen Guanrong; Luo Xiaoshu

2005-01-01

In this paper, synchronization of two hyperchaotic oscillators via a single variable's unidirectional coupling is studied. First, the synchronizability of the coupled hyperchaotic oscillators is proved mathematically. Then, the convergence speed of this synchronization scheme is analyzed. In order to speed up the response with a relatively large coupling strength, two kinds of chaotic coupling synchronization schemes are proposed. In terms of numerical simulations and the numerical calculation of the largest conditional Lyapunov exponent, it is shown that in a given range of coupling strengths, chaotic-coupling synchronization is quicker than the typical continuous-coupling synchronization. Furthermore, A circuit realization based on the chaotic synchronization scheme is designed and Pspice circuit simulation validates the simulated hyperchaos synchronization mechanism

Short-term data forecasting based on wavelet transformation and chaos theory

Science.gov (United States)

Wang, Yi; Li, Cunbin; Zhang, Liang

2017-09-01

A sketch of wavelet transformation and its application was given. Concerning the characteristics of time sequence, Haar wavelet was used to do data reduction. After processing, the effect of “data nail” on forecasting was reduced. Chaos theory was also introduced, a new chaos time series forecasting flow based on wavelet transformation was proposed. The largest Lyapunov exponent was larger than zero from small data sets, it verified the data change behavior still met chaotic behavior. Based on this, chaos time series to forecast short-term change behavior could be used. At last, the example analysis of the price from a real electricity market showed that the forecasting method increased the precision of the forecasting more effectively and steadily.

Quantifying chaotic dynamics from integrate-and-fire processes

Energy Technology Data Exchange (ETDEWEB)

Pavlov, A. N. [Department of Physics, Saratov State University, Astrakhanskaya Str. 83, 410012 Saratov (Russian Federation); Saratov State Technical University, Politehnicheskaya Str. 77, 410054 Saratov (Russian Federation); Pavlova, O. N. [Department of Physics, Saratov State University, Astrakhanskaya Str. 83, 410012 Saratov (Russian Federation); Mohammad, Y. K. [Department of Physics, Saratov State University, Astrakhanskaya Str. 83, 410012 Saratov (Russian Federation); Tikrit University Salahudin, Tikrit Qadisiyah, University Str. P.O. Box 42, Tikrit (Iraq); Kurths, J. [Potsdam Institute for Climate Impact Research, Telegraphenberg A 31, 14473 Potsdam (Germany); Institute of Physics, Humboldt University Berlin, 12489 Berlin (Germany)

2015-01-15

Characterizing chaotic dynamics from integrate-and-fire (IF) interspike intervals (ISIs) is relatively easy performed at high firing rates. When the firing rate is low, a correct estimation of Lyapunov exponents (LEs) describing dynamical features of complex oscillations reflected in the IF ISI sequences becomes more complicated. In this work we discuss peculiarities and limitations of quantifying chaotic dynamics from IF point processes. We consider main factors leading to underestimated LEs and demonstrate a way of improving numerical determining of LEs from IF ISI sequences. We show that estimations of the two largest LEs can be performed using around 400 mean periods of chaotic oscillations in the regime of phase-coherent chaos. Application to real data is discussed.

Nonlinear dynamics modeling and simulation of two-wheeled self-balancing vehicle

Directory of Open Access Journals (Sweden)

Yunping Liu

2016-11-01

Full Text Available Two-wheeled self-balancing vehicle system is a kind of naturally unstable underactuated system with high-rank unstable multivariable strongly coupling complicated dynamic nonlinear property. Nonlinear dynamics modeling and simulation, as a basis of two-wheeled self-balancing vehicle dynamics research, has the guiding effect for system design of the project demonstration and design phase. Dynamics model of the two-wheeled self-balancing vehicle is established by importing a TSi ProPac package to the Mathematica software (version 8.0, which analyzes the stability and calculates the Lyapunov exponents of the system. The relationship between external force and stability of the system is analyzed by the phase trajectory. Proportional–integral–derivative control is added to the system in order to improve the stability of the two-wheeled self-balancing vehicle. From the research, Lyapunov exponent can be used to research the stability of hyperchaos system. The stability of the two-wheeled self-balancing vehicle is better by inputting the proportional–integral–derivative control. The Lyapunov exponent and phase trajectory can help us analyze the stability of a system better and lay the foundation for the analysis and control of the two-wheeled self-balancing vehicle system.

How complex a dynamical network can be?

International Nuclear Information System (INIS)

Baptista, M.S.; Kakmeni, F. Moukam; Del Magno, Gianluigi; Hussein, M.S.

2011-01-01

Positive Lyapunov exponents measure the asymptotic exponential divergence of nearby trajectories of a dynamical system. Not only they quantify how chaotic a dynamical system is, but since their sum is an upper bound for the rate of information production, they also provide a convenient way to quantify the complexity of a dynamical network. We conjecture based on numerical evidences that for a large class of dynamical networks composed by equal nodes, the sum of the positive Lyapunov exponents is bounded by the sum of all the positive Lyapunov exponents of both the synchronization manifold and its transversal directions, the last quantity being in principle easier to compute than the latter. As applications of our conjecture we: (i) show that a dynamical network composed of equal nodes and whose nodes are fully linearly connected produces more information than similar networks but whose nodes are connected with any other possible connecting topology; (ii) show how one can calculate upper bounds for the information production of realistic networks whose nodes have parameter mismatches, randomly chosen; (iii) discuss how to predict the behavior of a large dynamical network by knowing the information provided by a system composed of only two coupled nodes.

Estabilización del Péndulo Invertido Sobre Dos Ruedas mediante el método de Lyapunov

Directory of Open Access Journals (Sweden)

O. Octavio Gutiérrez Frías

2013-01-01

Full Text Available Resumen: En este trabajo, se presenta un controlador no lineal para estabilizar el sistema Péndulo Invertido Sobre Dos Ruedas. Como primera etapa la estrategia de control, se basa en una linealización parcial por realimentación, para posteriormente proponer una función candidata de Lyapunov en combinación con el principio de invariancia de LaSalle con el fin de obtener el controlador esta- bilizador. El sistema en lazo cerrado obtenido es asintóticamente estable localmente alrededor del punto de equilibrio inestable, con un dominio de atracción calculable. Abstract: In this paper, a nonlinear controller is presented for the stabilization of the two wheels inverted pendulum. The control strategy is based on partial feedback linealization, in first stage and then a suitable function Lyapunov in conjunction with LaSalle's invariance principle is formed to obtain a stabilizing feedback controller. The obtained closed-loop system is locally asymptotically stable around its unstable equilibrium point, with a computable domain of attraction. Palabras clave: Sistema Subactuado, Péndulo Invertido Sobre Dos Ruedas, Método de Lyapunov, Control No Lineal, Keywords: Under Actuated System, Two Wheels Inverted Pendulum, Lyapunov Approach, Non-Linear Control

Controlling chaos in the permanent magnet synchronous motor

International Nuclear Information System (INIS)

Zribi, Mohamed; Oteafy, Ahmed; Smaoui, Nejib

2009-01-01

The Permanent Magnet Synchronous Motor (PMSM) is known to exhibit chaotic behavior under certain conditions. This paper proposes to use an instantaneous Lyapunov exponent control algorithm to control the PMSM. One of the objectives of the control approach is to bring order to the PMSM and to drive it to any user-defined desired state. Simulation results under different operating conditions indicate that the proposed control scheme works well. Moreover, the proposed Lyapunov exponent control scheme is able to induce chaos on the permanent magnet synchronous motor. Simulation results show the effectiveness of the proposed control scheme in chaotifing the response of the motor.

Unstable oscillators based hyperchaotic circuit

DEFF Research Database (Denmark)

Murali, K.; Tamasevicius, A.; G. Mykolaitis, A.

1999-01-01

A simple 4th order hyperchaotic circuit with unstable oscillators is described. The circuit contains two negative impedance converters, two inductors, two capacitors, a linear resistor and a diode. The Lyapunov exponents are presented to confirm hyperchaotic nature of the oscillations in the circ...... in the circuit. The performance of the circuit is investigated by means of numerical integration of appropriate differential equations, PSPICE simulations, and hardware experiment.......A simple 4th order hyperchaotic circuit with unstable oscillators is described. The circuit contains two negative impedance converters, two inductors, two capacitors, a linear resistor and a diode. The Lyapunov exponents are presented to confirm hyperchaotic nature of the oscillations...

Qualitative and numerical study of Bianchi IX Models

International Nuclear Information System (INIS)

Francisco, G.; Matsas, G.E.A.

1987-01-01

The qualitative behaviour of trajectories in the Mixmaster universe is studied. The Lyapunov exponents computed directly from the differential equations and from the Poincare map are shown to be different. A detailed discussion of the role of these exponents in analysing the effect of chaos on trajectories is presented. (Author) [pt

State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach.

Science.gov (United States)

Yu, Jue; Zhuang, Jian; Yu, Dehong

2015-01-01

This paper concerns a state feedback integral control using a Lyapunov function approach for a rotary direct drive servo valve (RDDV) while considering parameter uncertainties. Modeling of this RDDV servovalve reveals that its mechanical performance is deeply influenced by friction torques and flow torques; however, these torques are uncertain and mutable due to the nature of fluid flow. To eliminate load resistance and to achieve satisfactory position responses, this paper develops a state feedback control that integrates an integral action and a Lyapunov function. The integral action is introduced to address the nonzero steady-state error; in particular, the Lyapunov function is employed to improve control robustness by adjusting the varying parameters within their value ranges. This new controller also has the advantages of simple structure and ease of implementation. Simulation and experimental results demonstrate that the proposed controller can achieve higher control accuracy and stronger robustness. Copyright © 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Partial differential equations with variable exponents variational methods and qualitative analysis

CERN Document Server

Radulescu, Vicentiu D

2015-01-01

Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis provides researchers and graduate students with a thorough introduction to the theory of nonlinear partial differential equations (PDEs) with a variable exponent, particularly those of elliptic type. The book presents the most important variational methods for elliptic PDEs described by nonhomogeneous differential operators and containing one or more power-type nonlinearities with a variable exponent. The authors give a systematic treatment of the basic mathematical theory and constructive meth

Generalized direct Lyapunov method for the analysis of stability and attraction in general time systems

International Nuclear Information System (INIS)

Druzhinina, O V; Shestakov, A A

2002-01-01

A generalized direct Lyapunov method is put forward for the study of stability and attraction in general time systems of the following types: the classical dynamical system in the sense of Birkhoff, the general system in the sense of Zubov, the general system in the sense of Seibert, the general system with delay, and the general 'input-output' system. For such systems, with the help of generalized Lyapunov functions with respect to two filters, two quasifilters, or two filter bases, necessary and sufficient conditions for stability and attraction are obtained under minimal assumptions about the mathematical structure of the general system

Lyapunov based control of hybrid energy storage system in electric vehicles

DEFF Research Database (Denmark)

El Fadil, H.; Giri, F.; Guerrero, Josep M.

2012-01-01

This paper deals with a Lyapunov based control principle in a hybrid energy storage system for electric vehicle. The storage system consists on fuel cell (FC) as a main power source and a supercapacitor (SC) as an auxiliary power source. The power stage of energy conversion consists on a boost...

Spacetime dependence of the anomalous exponent of electric transport in the disorder model

International Nuclear Information System (INIS)

Egami, Takeshi; Suzuki, Koshiro; Watanabe, Katsuhiro

2012-01-01

Spacetime dependence of the anomalous exponent of electric transport in the disorder model is investigated. We show that the anomalous exponent evolves with time, according to the time evolution of the number of the effective neighbouring sites. Transition from subdiffusive to normal transport is recovered at macroscopic timescales. Plateaus appear in the history of the anomalous exponent due to the discreteness of the hopping sites, which is compatible with the conventional treatment to regard the anomalous exponent as a constant. We also show that, among various microscopic spatial structures, the number of the effective neighbouring sites is the only element which determines the anomalous exponent. This is compatible with the mesoscopic model of Scher–Montroll. These findings are verified by means of Monte Carlo simulation. The well-known expression of the anomalous exponent in the conventional multiple trapping model is derived by deducing it as a special case of the disorder model. (paper)

Laminar Flame Velocity and Temperature Exponent of Diluted DME-Air Mixture

Science.gov (United States)

Naseer Mohammed, Abdul; Anwar, Muzammil; Juhany, Khalid A.; Mohammad, Akram

2017-03-01

In this paper, the laminar flame velocity and temperature exponent diluted dimethyl ether (DME) air mixtures are reported. Laminar premixed mixture of DME-air with volumetric dilutions of carbon dioxides (CO2) and nitrogen (N2) are considered. Experiments were conducted using a preheated mesoscale high aspect-ratio diverging channel with inlet dimensions of 25 mm × 2 mm. In this method, flame velocities are extracted from planar flames that were stabilized near adiabatic conditions inside the channel. The flame velocities are then plotted against the ratio of mixture temperature and the initial reference temperature. A non-linear power law regression is observed suitable. This regression analysis gives the laminar flame velocity at the initial reference temperature and temperature exponent. Decrease in the laminar flame velocity and increase in temperature exponent is observed for CO2 and N2 diluted mixtures. The addition of CO2 has profound influence when compared to N2 addition on both flame velocity and temperature exponent. Numerical prediction of the similar mixture using a detailed reaction mechanism is obtained. The computational mechanism predicts higher magnitudes for laminar flame velocity and smaller magnitudes of temperature exponent compared to experimental data.

Lyapunov stability robust analysis and robustness design for linear continuous-time systems

NARCIS (Netherlands)

Luo, J.S.; Johnson, A.; Bosch, van den P.P.J.

1995-01-01

The linear continuous-time systems to be discussed are described by state space models with structured time-varying uncertainties. First, the explicit maximal perturbation bound for maintaining quadratic Lyapunov stability of the closed-loop systems is presented. Then, a robust design method is

A comment on measuring the Hurst exponent of financial time series

Science.gov (United States)

Couillard, Michel; Davison, Matt

2005-03-01

A fundamental hypothesis of quantitative finance is that stock price variations are independent and can be modeled using Brownian motion. In recent years, it was proposed to use rescaled range analysis and its characteristic value, the Hurst exponent, to test for independence in financial time series. Theoretically, independent time series should be characterized by a Hurst exponent of 1/2. However, finite Brownian motion data sets will always give a value of the Hurst exponent larger than 1/2 and without an appropriate statistical test such a value can mistakenly be interpreted as evidence of long term memory. We obtain a more precise statistical significance test for the Hurst exponent and apply it to real financial data sets. Our empirical analysis shows no long-term memory in some financial returns, suggesting that Brownian motion cannot be rejected as a model for price dynamics.

Can the bivariate Hurst exponent be higher than an average of the separate Hurst exponents?

Czech Academy of Sciences Publication Activity Database

Krištoufek, Ladislav

2015-01-01

Roč. 431, č. 1 (2015), s. 124-127 ISSN 0378-4371 R&D Projects: GA ČR(CZ) GP14-11402P Institutional support: RVO:67985556 Keywords : Correlations * Power- law cross-correlations * Bivariate Hurst exponent * Spectrum coherence Subject RIV: AH - Economics Impact factor: 1.785, year: 2015 http://library.utia.cas.cz/separaty/2015/E/kristoufek-0452314.pdf

Inverted rank distributions: Macroscopic statistics, universality classes, and critical exponents

Science.gov (United States)

Eliazar, Iddo; Cohen, Morrel H.

2014-01-01

An inverted rank distribution is an infinite sequence of positive sizes ordered in a monotone increasing fashion. Interlacing together Lorenzian and oligarchic asymptotic analyses, we establish a macroscopic classification of inverted rank distributions into five “socioeconomic” universality classes: communism, socialism, criticality, feudalism, and absolute monarchy. We further establish that: (i) communism and socialism are analogous to a “disordered phase”, feudalism and absolute monarchy are analogous to an “ordered phase”, and criticality is the “phase transition” between order and disorder; (ii) the universality classes are characterized by two critical exponents, one governing the ordered phase, and the other governing the disordered phase; (iii) communism, criticality, and absolute monarchy are characterized by sharp exponent values, and are inherently deterministic; (iv) socialism is characterized by a continuous exponent range, is inherently stochastic, and is universally governed by continuous power-law statistics; (v) feudalism is characterized by a continuous exponent range, is inherently stochastic, and is universally governed by discrete exponential statistics. The results presented in this paper yield a universal macroscopic socioeconophysical perspective of inverted rank distributions.

Lyapunov-Based Control Scheme for Single-Phase Grid-Connected PV Central Inverters

NARCIS (Netherlands)

Meza, C.; Biel, D.; Jeltsema, D.; Scherpen, J. M. A.

A Lyapunov-based control scheme for single-phase single-stage grid-connected photovoltaic central inverters is presented. Besides rendering the closed-loop system globally stable, the designed controller is able to deal with the system uncertainty that depends on the solar irradiance. A laboratory

Colored chaos

International Nuclear Information System (INIS)

Mueller, B.

1997-01-01

The report contains viewgraphs on the following: ergodicity and chaos; Hamiltonian dynamics; metric properties; Lyapunov exponents; KS entropy; dynamical realization; lattice formulation; and numerical results

Colored chaos

Energy Technology Data Exchange (ETDEWEB)

Mueller, B.

1997-09-22

The report contains viewgraphs on the following: ergodicity and chaos; Hamiltonian dynamics; metric properties; Lyapunov exponents; KS entropy; dynamical realization; lattice formulation; and numerical results.

Critical exponents from the effective average action

International Nuclear Information System (INIS)

Tetradis, N.; Wetterich, C.

1993-07-01

We compute the critical behaviour of three-dimensional scalar theories using a new exact non-perturbative evolution equation. Our values for the critical exponents agree well with previous precision estimates. (orig.)

Effect of asynchronous updating on the stability of cellular automata

International Nuclear Information System (INIS)

Baetens, J.M.; Van der Weeën, P.; De Baets, B.

2012-01-01

Highlights: ► An upper bound on the Lyapunov exponent of asynchronously updated CA is established. ► The employed update method has repercussions on the stability of CAs. ► A decision on the employed update method should be taken with care. ► Substantial discrepancies arise between synchronously and asynchronously updated CA. ► Discrepancies between different asynchronous update schemes are less pronounced. - Abstract: Although cellular automata (CAs) were conceptualized as utter discrete mathematical models in which the states of all their spatial entities are updated simultaneously at every consecutive time step, i.e. synchronously, various CA-based models that rely on so-called asynchronous update methods have been constructed in order to overcome the limitations that are tied up with the classical way of evolving CAs. So far, only a few researchers have addressed the consequences of this way of updating on the evolved spatio-temporal patterns, and the reachable stationary states. In this paper, we exploit Lyapunov exponents to determine to what extent the stability of the rules within a family of totalistic CAs is affected by the underlying update method. For that purpose, we derive an upper bound on the maximum Lyapunov exponent of asynchronously iterated CAs, and show its validity, after which we present a comparative study between the Lyapunov exponents obtained for five different update methods, namely one synchronous method and four well-established asynchronous methods. It is found that the stability of CAs is seriously affected if one of the latter methods is employed, whereas the discrepancies arising between the different asynchronous methods are far less pronounced and, finally, we discuss the repercussions of our findings on the development of CA-based models.

On the Kalman Filter error covariance collapse into the unstable subspace

Directory of Open Access Journals (Sweden)

A. Trevisan

2011-03-01

Full Text Available When the Extended Kalman Filter is applied to a chaotic system, the rank of the error covariance matrices, after a sufficiently large number of iterations, reduces to N⁺ + N⁰ where N⁺ and N⁰ are the number of positive and null Lyapunov exponents. This is due to the collapse into the unstable and neutral tangent subspace of the solution of the full Extended Kalman Filter. Therefore the solution is the same as the solution obtained by confining the assimilation to the space spanned by the Lyapunov vectors with non-negative Lyapunov exponents. Theoretical arguments and numerical verification are provided to show that the asymptotic state and covariance estimates of the full EKF and of its reduced form, with assimilation in the unstable and neutral subspace (EKF-AUS are the same. The consequences of these findings on applications of Kalman type Filters to chaotic models are discussed.

On the Topological Changes of Local Hurst Exponent in Polar Regions

Science.gov (United States)

Consolini, G.; De Michelis, P.

2014-12-01

Geomagnetic activity during magnetic substorms and storms is related to the dinamical and topological changes of the current systems flowing in the Earth's magnetosphere-ionosphere. This is particularly true in the case of polar regions where the enhancement of auroral electrojet current system is responsible for the observed geomagnetic perturbations. Here, using the DMA-technique we evaluate the local Hurst exponent (H"older exponent) for a set of 46 geomagnetic observatories, widely distributed in the northern hemisphere, during one of the most famous and strong geomagnetic storm, the Bastille event, and reconstruct a sequence of polar maps showing the dinamical changes of the topology of the local Hurst exponent with the geomagnetic activity level. The topological evolution of local Hurst exponent maps is discussed in relation to the dinamical changes of the current systems flowing in the polar ionosphere. G. Consolini has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under Grant agreement no. 313038/STORM for this research.

Variation of CRE with exponents of time and number of fractions

International Nuclear Information System (INIS)

Supe, S.J.; Rao, S.M.; Sawant, S.G.; Bisht, J.S.

1976-01-01

The concept of NSD has been modified into TDF's by Orton and Ellis and CRE's by Kirk et al. It was aimed to study the variability of these new concepts on the exponents of time and number of fractions. It was found that TDF has larger variation with the exponents compared to that of CRE. The use of CRE and NSD for solving the treatment scheduling problems or for intercomparison of various regimes has been simplified by providing readymade estimation of CRE for various doses/fraction with increasing number of fractions. As there is increasing evidence for the change of exponents J and H, nomograms are presented to determine the CRE for various values of J and H. The variation of decay correction factors with the exponent H is also evaluated and is presented. This will help various radiotherapists to use CRE and the decay correction factors consistent with their clinical findings. (orig.) [de

Gross-Pitaevski map as a chaotic dynamical system.

Science.gov (United States)

Guarneri, Italo

2017-03-01

The Gross-Pitaevski map is a discrete time, split-operator version of the Gross-Pitaevski dynamics in the circle, for which exponential instability has been recently reported. Here it is studied as a classical dynamical system in its own right. A systematic analysis of Lyapunov exponents exposes strongly chaotic behavior. Exponential growth of energy is then shown to be a direct consequence of rotational invariance and for stationary solutions the full spectrum of Lyapunov exponents is analytically computed. The present analysis includes the "resonant" case, when the free rotation period is commensurate to 2π, and the map has countably many constants of the motion. Except for lowest-order resonances, this case exhibits an integrable-chaotic transition.

Abstraction of continuous dynamical systems utilizing lyapunov functions

DEFF Research Database (Denmark)

Sloth, Christoffer; Wisniewski, Rafael

2010-01-01

This paper considers the development of a method for abstracting continuous dynamical systems by timed automata. The method is based on partitioning the state space of dynamical systems with invariant sets, which form cells representing locations of the timed automata. To enable verification...... of the dynamical system based on the abstraction, conditions for obtaining sound, complete, and refinable abstractions are set up. It is proposed to partition the state space utilizing sub-level sets of Lyapunov functions, since they are positive invariant sets. The existence of sound abstractions for Morse......-Smale systems and complete and refinable abstractions for linear systems are shown....

Pramana – Journal of Physics | Indian Academy of Sciences

Indian Academy of Sciences (India)

An attractor widening and merging crisis is seen in the phase space in the aerosol case. Crisis-induced intermittency is seen in the time series and the laminar length distribution of times before bursts give rise to a power law with the exponent = -1/3. The maximum Lyapunov exponent near the crisis fluctuates around zero ...

Chaos optimization algorithms based on chaotic maps with different probability distribution and search speed for global optimization

Science.gov (United States)

Yang, Dixiong; Liu, Zhenjun; Zhou, Jilei

2014-04-01

Chaos optimization algorithms (COAs) usually utilize the chaotic map like Logistic map to generate the pseudo-random numbers mapped as the design variables for global optimization. Many existing researches indicated that COA can more easily escape from the local minima than classical stochastic optimization algorithms. This paper reveals the inherent mechanism of high efficiency and superior performance of COA, from a new perspective of both the probability distribution property and search speed of chaotic sequences generated by different chaotic maps. The statistical property and search speed of chaotic sequences are represented by the probability density function (PDF) and the Lyapunov exponent, respectively. Meanwhile, the computational performances of hybrid chaos-BFGS algorithms based on eight one-dimensional chaotic maps with different PDF and Lyapunov exponents are compared, in which BFGS is a quasi-Newton method for local optimization. Moreover, several multimodal benchmark examples illustrate that, the probability distribution property and search speed of chaotic sequences from different chaotic maps significantly affect the global searching capability and optimization efficiency of COA. To achieve the high efficiency of COA, it is recommended to adopt the appropriate chaotic map generating the desired chaotic sequences with uniform or nearly uniform probability distribution and large Lyapunov exponent.

Four-loop critical exponents for the Gross-Neveu-Yukawa models

International Nuclear Information System (INIS)

Zerf, Nikolai; Mihaila, Luminita N.; Herbut, Igor F.; Scherer, Michael M.

2017-09-01

We study the chiral Ising, the chiral XY and the chiral Heisenberg models at four-loop order with the perturbative renormalization group in 4-ε dimensions and compute critical exponents for the Gross-Neveu-Yukawa fixed points to order O(ε 4 ). Further, we provide Pade estimates for the correlation length exponent, the boson and fermion anomalous dimension as well as the leading correction to scaling exponent in 2+1 dimensions. We also confirm the emergence of supersymmetric field theories at four loops for the chiral Ising and the chiral XY models with N=1/4 and N=1/2 fermions, respectively. Furthermore, applications of our results relevant to various quantum transitions in the context of Dirac and Weyl semimetals are discussed, including interaction-induced transitions in graphene and surface states of topological insulators.

Four-loop critical exponents for the Gross-Neveu-Yukawa models

Energy Technology Data Exchange (ETDEWEB)

Zerf, Nikolai; Mihaila, Luminita N. [Heidelberg Univ. (Germany). Inst. fuer Theoretische Physik; Marquard, Peter [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Herbut, Igor F. [Simon Fraser Univ., Burnaby, BC (Canada). Dept. of Physics; Scherer, Michael M. [Koeln Univ. (Germany). Inst. for Theoretical Physics

2017-09-15

We study the chiral Ising, the chiral XY and the chiral Heisenberg models at four-loop order with the perturbative renormalization group in 4-ε dimensions and compute critical exponents for the Gross-Neveu-Yukawa fixed points to order O(ε{sup 4}). Further, we provide Pade estimates for the correlation length exponent, the boson and fermion anomalous dimension as well as the leading correction to scaling exponent in 2+1 dimensions. We also confirm the emergence of supersymmetric field theories at four loops for the chiral Ising and the chiral XY models with N=1/4 and N=1/2 fermions, respectively. Furthermore, applications of our results relevant to various quantum transitions in the context of Dirac and Weyl semimetals are discussed, including interaction-induced transitions in graphene and surface states of topological insulators.

Fujita Exponent for a Nonlinear Degenerate Parabolic Equation with Localized Source

Directory of Open Access Journals (Sweden)

Yulan Wang

2014-01-01

Full Text Available This paper is devoted to understand the blow-up properties of reaction-diffusion equations which combine a localized reaction term with nonlinear diffusion. In particular, we study the critical exponent of a p-Laplacian equation with a localized reaction. We obtain the Fujita exponent qc of the equation.

High-accuracy critical exponents for O(N) hierarchical 3D sigma models

International Nuclear Information System (INIS)

Godina, J. J.; Li, L.; Meurice, Y.; Oktay, M. B.

2006-01-01

The critical exponent γ and its subleading exponent Δ in the 3D O(N) Dyson's hierarchical model for N up to 20 are calculated with high accuracy. We calculate the critical temperatures for the measure δ(φ-vector.φ-vector-1). We extract the first coefficients of the 1/N expansion from our numerical data. We show that the leading and subleading exponents agree with Polchinski equation and the equivalent Litim equation, in the local potential approximation, with at least 4 significant digits

Analysis of Real Ship Rolling Dynamics under Wave Excitement Force Composed of Sums of Cosine Functions

International Nuclear Information System (INIS)

Zhang, Y. S.; Cai, F.; Xu, W. M.

2011-01-01

The ship motion equation with a cosine wave excitement force describes the slip moments in regular waves. A new kind of wave excitement force model, with the form as sums of cosine functions was proposed to describe ship rolling in irregular waves. Ship rolling time series were obtained by solving the ship motion equation with the fourth-order-Runger-Kutta method. These rolling time series were synthetically analyzed with methods of phase-space track, power spectrum, primary component analysis, and the largest Lyapunove exponent. Simulation results show that ship rolling presents some chaotic characteristic when the wave excitement force was applied by sums of cosine functions. The result well explains the course of ship rolling's chaotic mechanism and is useful for ship hydrodynamic study.

Mode locking and quasiperiodicity in a discrete-time Chialvo neuron model

Science.gov (United States)

Wang, Fengjuan; Cao, Hongjun

2018-03-01

The two-dimensional parameter spaces of a discrete-time Chialvo neuron model are investigated. Our studies demonstrate that for all our choice of two parameters (i) the fixed point is destabilized via Neimark-Sacker bifurcation; (ii) there exist mode locking structures like Arnold tongues and shrimps, with periods organized in a Farey tree sequence, embedded in quasiperiodic/chaotic region. We determine analytically the location of the parameter sets where Neimark-Sacker bifurcation occurs, and the location on this curve where Arnold tongues of arbitrary period are born. Properties of the transition that follows the so-called two-torus from quasiperiodicity to chaos are presented clearly and proved strictly by using numerical simulations such as bifurcation diagrams, the largest Lyapunov exponent diagram on MATLAB and C++.

? and ? nonquadratic stabilisation of discrete-time Takagi-Sugeno systems based on multi-instant fuzzy Lyapunov functions

Science.gov (United States)

Tognetti, Eduardo S.; Oliveira, Ricardo C. L. F.; Peres, Pedro L. D.

2015-01-01

The problem of state feedback control design for discrete-time Takagi-Sugeno (TS) (T-S) fuzzy systems is investigated in this paper. A Lyapunov function, which is quadratic in the state and presents a multi-polynomial dependence on the fuzzy weighting functions at the current and past instants of time, is proposed.This function contains, as particular cases, other previous Lyapunov functions already used in the literature, being able to provide less conservative conditions of control design for TS fuzzy systems. The structure of the proposed Lyapunov function also motivates the design of a new stabilising compensator for Takagi-Sugeno fuzzy systems. The main novelty of the proposed state feedback control law is that the gain is composed of matrices with multi-polynomial dependence on the fuzzy weighting functions at a set of past instants of time, including the current one. The conditions for the existence of a stabilising state feedback control law that minimises an upper bound to the ? or ? norms are given in terms of linear matrix inequalities. Numerical examples show that the approach can be less conservative and more efficient than other methods available in the literature.

Bilinear Approximate Model-Based Robust Lyapunov Control for Parabolic Distributed Collectors

KAUST Repository

Elmetennani, Shahrazed

2016-11-09

This brief addresses the control problem of distributed parabolic solar collectors in order to maintain the field outlet temperature around a desired level. The objective is to design an efficient controller to force the outlet fluid temperature to track a set reference despite the unpredictable varying working conditions. In this brief, a bilinear model-based robust Lyapunov control is proposed to achieve the control objectives with robustness to the environmental changes. The bilinear model is a reduced order approximate representation of the solar collector, which is derived from the hyperbolic distributed equation describing the heat transport dynamics by means of a dynamical Gaussian interpolation. Using the bilinear approximate model, a robust control strategy is designed applying Lyapunov stability theory combined with a phenomenological representation of the system in order to stabilize the tracking error. On the basis of the error analysis, simulation results show good performance of the proposed controller, in terms of tracking accuracy and convergence time, with limited measurement even under unfavorable working conditions. Furthermore, the presented work is of interest for a large category of dynamical systems knowing that the solar collector is representative of physical systems involving transport phenomena constrained by unknown external disturbances.

Quantum Butterfly Effect in Weakly Interacting Diffusive Metals

Directory of Open Access Journals (Sweden)

Aavishkar A. Patel

2017-09-01

Full Text Available We study scrambling, an avatar of chaos, in a weakly interacting metal in the presence of random potential disorder. It is well known that charge and heat spread via diffusion in such an interacting disordered metal. In contrast, we show within perturbation theory that chaos spreads in a ballistic fashion. The squared anticommutator of the electron-field operators inherits a light-cone-like growth, arising from an interplay of a growth (Lyapunov exponent that scales as the inelastic electron scattering rate and a diffusive piece due to the presence of disorder. In two spatial dimensions, the Lyapunov exponent is universally related at weak coupling to the sheet resistivity. We are able to define an effective temperature-dependent butterfly velocity, a speed limit for the propagation of quantum information that is much slower than microscopic velocities such as the Fermi velocity and that is qualitatively similar to that of a quantum critical system with a dynamical critical exponent z>1.

Dispersive Sachdev-Ye-Kitaev model: Band structure and quantum chaos

Science.gov (United States)

Zhang, Pengfei

2017-11-01

The Sachdev-Ye-Kitaev (SYK) model is a concrete model for a non-Fermi liquid with maximally chaotic behavior in (0 +1 ) dimensions. In order to gain some insights into real materials in higher dimensions where fermions could hop between different sites, here we consider coupling a SYK lattice by constant hopping. We call this the dispersive SYK model. Focusing on (1 +1 ) -dimensional homogeneous hopping, by either tuning the temperature or the relative strength of the random interaction (hopping) and constant hopping, we find a crossover between a dispersive metal to an incoherent metal, where the dynamic exponent z changes from 1 to ∞ . We study the crossover by calculating the spectral function, charge density correlator, and the Lyapunov exponent. We further find the Lyapunov exponent becomes larger when the chemical potential is tuned to approach a van Hove singularity because of the large density of states near the Fermi surface. The effect of the topological nontrivial bands is also discussed.

Design of Connectivity Preserving Flocking Using Control Lyapunov Function

Directory of Open Access Journals (Sweden)

Bayu Erfianto

2016-01-01

Full Text Available This paper investigates cooperative flocking control design with connectivity preserving mechanism. During flocking, interagent distance is measured to determine communication topology of the flocks. Then, cooperative flocking motion is built based on cooperative artificial potential field with connectivity preserving mechanism to achieve the common flocking objective. The flocking control input is then obtained by deriving cooperative artificial potential field using control Lyapunov function. As a result, we prove that our flocking protocol establishes group stabilization and the communication topology of multiagent flocking is always connected.

Nonlinear Lyapunov-based boundary control of distributed heat transfer mechanisms in membrane distillation plant

KAUST Repository

Eleiwi, Fadi; Laleg-Kirati, Taous-Meriem

2015-01-01

This paper presents a nonlinear Lyapunov-based boundary control for the temperature difference of a membrane distillation boundary layers. The heat transfer mechanisms inside the process are modeled with a 2D advection-diffusion equation. The model

Lyapunov stability and its application to systems of ordinary differential equations

Science.gov (United States)

Kennedy, E. W.

1979-01-01

An outline and a brief introduction to some of the concepts and implications of Lyapunov stability theory are presented. Various aspects of the theory are illustrated by the inclusion of eight examples, including the Cartesian coordinate equations of the two-body problem, linear and nonlinear (Van der Pol's equation) oscillatory systems, and the linearized Kustaanheimo-Stiefel element equations for the unperturbed two-body problem.

A new wind power prediction method based on chaotic theory and Bernstein Neural Network

International Nuclear Information System (INIS)

Wang, Cong; Zhang, Hongli; Fan, Wenhui; Fan, Xiaochao

2016-01-01

The accuracy of wind power prediction is important for assessing the security and economy of the system operation when wind power connects to the grids. However, multiple factors cause a long delay and large errors in wind power prediction. Hence, efficient wind power forecasting approaches are still required for practical applications. In this paper, a new wind power forecasting method based on Chaos Theory and Bernstein Neural Network (BNN) is proposed. Firstly, the largest Lyapunov exponent as a judgment for wind power system's chaotic behavior is made. Secondly, Phase Space Reconstruction (PSR) is used to reconstruct the wind power series' phase space. Thirdly, the prediction model is constructed using the Bernstein polynomial and neural network. Finally, the weights and thresholds of the model are optimized by Primal Dual State Transition Algorithm (PDSTA). The practical hourly data of wind power generation in Xinjiang is used to test this forecaster. The proposed forecaster is compared with several current prominent research findings. Analytical results indicate that the forecasting error of PDSTA + BNN is 3.893% for 24 look-ahead hours, and has lower errors obtained compared with the other forecast methods discussed in this paper. The results of all cases studying confirm the validity of the new forecast method. - Highlights: • Lyapunov exponent is used to verify chaotic behavior of wind power series. • Phase Space Reconstruction is used to reconstruct chaotic wind power series. • A new Bernstein Neural Network to predict wind power series is proposed. • Primal dual state transition algorithm is chosen as the training strategy of BNN.

Hurst exponent and prediction based on weak-form efficient market hypothesis of stock markets

Science.gov (United States)

Eom, Cheoljun; Choi, Sunghoon; Oh, Gabjin; Jung, Woo-Sung

2008-07-01

We empirically investigated the relationships between the degree of efficiency and the predictability in financial time-series data. The Hurst exponent was used as the measurement of the degree of efficiency, and the hit rate calculated from the nearest-neighbor prediction method was used for the prediction of the directions of future price changes. We used 60 market indexes of various countries. We empirically discovered that the relationship between the degree of efficiency (the Hurst exponent) and the predictability (the hit rate) is strongly positive. That is, a market index with a higher Hurst exponent tends to have a higher hit rate. These results suggested that the Hurst exponent is useful for predicting future price changes. Furthermore, we also discovered that the Hurst exponent and the hit rate are useful as standards that can distinguish emerging capital markets from mature capital markets.

Intermittency exponent of the turbulent energy cascade

International Nuclear Information System (INIS)

Cleve, J.; Greiner, M.; Pearson, B.R.; Sreenivasan, K.R.

2006-12-01

We consider the turbulent energy dissipation from one-dimensional records in experiments using air and gaseous helium at cryogenic temperatures, and obtain the intermittency exponent via the two-point correlation function of the energy dissipation. The air data are obtained in a number of flows in a wind tunnel and the atmospheric boundary layer at a height of about 35 m above the ground. The helium data correspond to the centerline of a jet exhausting into a container. The air data on the intermittency exponent are consistent with each other and with a trend that increases with the Taylor microscale Reynolds number, R λ , of up to about 1000 and saturates thereafter. On the other hand, the helium data cluster around a constant value at nearly all R λ , this being about half of the asymptotic value for the air data. Some possible explanation is offered for this anomaly. (author)

Study of chaos in chaotic satellite systems

Science.gov (United States)

Khan, Ayub; Kumar, Sanjay

2018-01-01

In this paper, we study the qualitative behaviour of satellite systems using bifurcation diagrams, Poincaré section, Lyapunov exponents, dissipation, equilibrium points, Kaplan-Yorke dimension etc. Bifurcation diagrams with respect to the known parameters of satellite systems are analysed. Poincaré sections with different sowing axes of the satellite are drawn. Eigenvalues of Jacobian matrices for the satellite system at different equilibrium points are calculated to justify the unstable regions. Lyapunov exponents are estimated. From these studies, chaos in satellite system has been established. Solution of equations of motion of the satellite system are drawn in the form of three-dimensional, two-dimensional and time series phase portraits. Phase portraits and time series display the chaotic nature of the considered system.

The anomalous scaling exponents of turbulence in general dimension from random geometry

Energy Technology Data Exchange (ETDEWEB)

Eling, Christopher [Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP (United Kingdom); Oz, Yaron [Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978 (Israel)

2015-09-22

We propose an analytical formula for the anomalous scaling exponents of inertial range structure functions in incompressible fluid turbulence. The formula is a Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation and is valid in any number of space dimensions. It incorporates intermittency in a novel way by dressing the Kolmogorov linear scaling via a coupling to a lognormal random geometry. The formula has one real parameter γ that depends on the number of space dimensions. The scaling exponents satisfy the convexity inequality, and the supersonic bound constraint. They agree with the experimental and numerical data in two and three space dimensions, and with numerical data in four space dimensions. Intermittency increases with γ, and in the infinite γ limit the scaling exponents approach the value one, as in Burgers turbulence. At large n the nth order exponent scales as √n. We discuss the relation between fluid flows and black hole geometry that inspired our proposal.

Nature of exponents found in the critical regime of YBCO

International Nuclear Information System (INIS)

Marhas, Manmeet Kaur; Saravanan, P.; Balakrishnan, K.; Srinivasan, R.; Kanjilal, D.; Metha, G.K.; Pai, S.P.; Pinto, R.; Vedvyas, M.; Ogale, S.B.; Mohan Rao, G.; Nathan, Senthil; Mohan, S.

1997-01-01

Full text: Fluctuation effects in electrical conductivity near T c is an important tool for studying the nature of phase transition in high T c ceramics. Probing critical regime by way of experiments demand data of good precision. Measurements were carried out on well characterised high T c films prepared by laser ablation and high pressure oxygen sputtering. High energy ion irradiation carried out to see the effect of disorder. Precise electrical resistivity measurements were carried out near T c with a temperature control accuracy better than 10 mK and large number of data points were collected in this regime. 100 MeV oxygen and 200 MeV Ag ions were used with varying fluences for irradiation at 77K. The data was analysed using existing models of critical fluctuation effects. The exponent of electrical conductivity in laser ablated thin films whose transition widths are less than 1 K was 1.33 and is independent of disorder caused by high energy ion irradiation and this could be identified as the exponent for excess conductivity in the critical intermediate charged fluctuation regime as proposed by Fisher. The exponent is around 2.7 in those films whose transition widths are greater than 1 K and also was independent of disorder and this could be identified as exponent in the para coherence regime

Scaling exponents for fracture surfaces in opal glass

Energy Technology Data Exchange (ETDEWEB)

Chavez-Guerrero, L., E-mail: guerreroleo@hotmail.com [Facultad de Ingenieria Mecanica y Electrica. Cd. Universitaria s/n, C.P. 66450, Universidad Autonoma de Nuevo Leon, Nuevo Leon (Mexico); Center of Innovation, Research and Development on Engineering and Technology, Universidad Autonoma de Nuevo Leon Monterrey, C.P. 66600, Apodaca, Nuevo Leon (Mexico); Garza, F.J., E-mail: fjgarza@gama.fime.uanl.mx [Facultad de Ciencias Quimicas, Cd. Universitaria s/n, C.P. 66450, Universidad Autonoma de Nuevo Leon, Nuevo Leon (Mexico); Hinojosa, M., E-mail: hinojosa@gama.fime.uanl.mx [Facultad de Ingenieria Mecanica y Electrica. Cd. Universitaria s/n, C.P. 66450, Universidad Autonoma de Nuevo Leon, Nuevo Leon (Mexico); Center of Innovation, Research and Development on Engineering and Technology, Universidad Autonoma de Nuevo Leon Monterrey, C.P. 66600, Apodaca, Nuevo Leon (Mexico)

2010-09-25

We have investigated the scaling properties of fracture surfaces in opal glass. Specimens with two different opacifying particle sizes (1 {mu}m and 0.4 {mu}m) were broken by three-point bending test and the resulting fracture surfaces were analyzed using Atomic Force Microscopy. The analysis of the self-affine behavior was performed using the Variable Bandwidth and Height-Height Correlation Methods, and both the roughness exponent, {zeta}, and the correlation length, {xi}, were determined. It was found that the roughness exponent obtained in both samples is {zeta} {approx} 0.8; whereas the correlation length in both fractures is of the order of the particle size, demonstrating the dependence of this self-affine parameter on the microstructure of opal glass.

Scaling exponents for fracture surfaces in opal glass

International Nuclear Information System (INIS)

Chavez-Guerrero, L.; Garza, F.J.; Hinojosa, M.

2010-01-01

We have investigated the scaling properties of fracture surfaces in opal glass. Specimens with two different opacifying particle sizes (1 μm and 0.4 μm) were broken by three-point bending test and the resulting fracture surfaces were analyzed using Atomic Force Microscopy. The analysis of the self-affine behavior was performed using the Variable Bandwidth and Height-Height Correlation Methods, and both the roughness exponent, ζ, and the correlation length, ξ, were determined. It was found that the roughness exponent obtained in both samples is ζ ∼ 0.8; whereas the correlation length in both fractures is of the order of the particle size, demonstrating the dependence of this self-affine parameter on the microstructure of opal glass.

Determination of critical exponents of inhomogeneous Gd films

Energy Technology Data Exchange (ETDEWEB)

Rosales-Rivera, A., E-mail: arosalesr@unal.edu.co [Laboratorio de Magnetismo y Materiales Avanzados, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Colombia, Sede Manizales, Manizales (Colombia); Salazar, N.A. [Laboratorio de Magnetismo y Materiales Avanzados, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Colombia, Sede Manizales, Manizales (Colombia); Hovorka, O.; Idigoras, O.; Berger, A. [CIC nanoGUNE Consolider, Tolosa Hiribidea 76, E-20018 Donostia-San Sebastian (Spain)

2012-08-15

The role of inhomogeneity on the critical behavior is studied for non-epitaxial Gd films. For this purpose, the film inhomogeneity was varied experimentally by annealing otherwise identical samples at different temperatures T{sub AN}=200, 400, and 500 Degree-Sign C. Vibrating sample magnetometry (VSM) was used for magnetization M vs. T measurements at different external fields H. A method based upon the linear superposition of different sample parts having different Curie temperatures T{sub C} was used to extract the critical exponents and the intrinsic distribution of Curie temperatures. We found that this method allows extracting reliable values of the critical exponents for all annealing temperatures, which enabled us to study the effects of disorder onto the universality class of Gd films.

Determination of critical exponents of inhomogeneous Gd films

International Nuclear Information System (INIS)

Rosales-Rivera, A.; Salazar, N.A.; Hovorka, O.; Idigoras, O.; Berger, A.

2012-01-01

The role of inhomogeneity on the critical behavior is studied for non-epitaxial Gd films. For this purpose, the film inhomogeneity was varied experimentally by annealing otherwise identical samples at different temperatures T AN =200, 400, and 500 °C. Vibrating sample magnetometry (VSM) was used for magnetization M vs. T measurements at different external fields H. A method based upon the linear superposition of different sample parts having different Curie temperatures T C was used to extract the critical exponents and the intrinsic distribution of Curie temperatures. We found that this method allows extracting reliable values of the critical exponents for all annealing temperatures, which enabled us to study the effects of disorder onto the universality class of Gd films.

PHYSIOLOGICAL RESPONSES DURING MATCHES AND PROFILE OF ELITE PENCAK SILAT EXPONENTS

Directory of Open Access Journals (Sweden)

Benedict Tan

2002-12-01

Full Text Available This is a descriptive, cross-sectional study describing the physiological responses during competitive matches and profile of elite exponents of an emerging martial art sport, pencak silat. Thirty exponents (21 males and 9 females were involved in the study. Match responses (i.e. heart rate (HR throughout match and capillary blood lactate concentration, [La], at pre-match and at the end of every round were obtained during actual competitive duels. Elite silat exponents' physiological attributes were assessed via anthropometry, vertical jump, isometric grip strength, maximal oxygen uptake, and the Wingate 30 s anaerobic test of the upper and lower body, in the laboratory. The match response data showed that silat competitors' mean HR was > 84% of estimated HR maximum and levels of [La] ranged from 6.7 - 18.7 mMol-1 during matches. This suggests that competitive silat matches are characterised by high aerobic and anaerobic responses. In comparison to elite taekwondo and judo athletes' physiological characteristics, elite silat exponents have lower aerobic fitness and grip strength, but greater explosive leg power (vertical jump. Generally, they also possessed a similar anaerobic capability in the lower but markedly inferior anaerobic capability in the upper body

Quantum computation of multifractal exponents through the quantum wavelet transform

International Nuclear Information System (INIS)

Garcia-Mata, Ignacio; Giraud, Olivier; Georgeot, Bertrand

2009-01-01

We study the use of the quantum wavelet transform to extract efficiently information about the multifractal exponents for multifractal quantum states. We show that, combined with quantum simulation algorithms, it enables to build quantum algorithms for multifractal exponents with a polynomial gain compared to classical simulations. Numerical results indicate that a rough estimate of fractality could be obtained exponentially fast. Our findings are relevant, e.g., for quantum simulations of multifractal quantum maps and of the Anderson model at the metal-insulator transition.

Dynamics of quantum-classical differences for chaotic systems

International Nuclear Information System (INIS)

Ballentine, L.E.

2002-01-01

The differences between quantum and classical dynamics can be studied through the moments and correlations of the position and momentum variables in corresponding quantum and classical statistical states. In chaotic states the quantum-classical differences grow exponentially with an exponent that exceeds the classical Lyapunov exponent. It is shown analytically that the quantum-classical differences scale as (ℎ/2π) 2 , and that the exponent for the growth of these differences is independent of (ℎ/2π). The quantum-classical difference exponent is studied for two quartic potential models, and the results are compared with previous work on the Henon-Heiles model

Thickness dependence of effective critical exponents in three-dimensional Ising plates

International Nuclear Information System (INIS)

Marques, M.I.; Gonzalo, J.A.

2000-01-01

Phase transitions in ising plates of equal area and different thickness have been studied by the Monte Carlo approach. The evolution of the critical temperature and of the effective critical exponents with the thickness of the lattice has been numerically determined. The thickness dependence of the maximum value of the effective critical exponents is well described by an exponential decay towards the respective three-dimensional value. (author)

Taxonomía de asteroides y cometas basada en los espectros de Lyapunov

Science.gov (United States)

Tancredi, G.; Motta, V.; Froeschlé, C.

Estudiaremos dos familias de objetos que sufren encuentros cercanos con planetas, a saber: la familia de cometas de Júpiter (JF) y los asteroides cercanos a la Tierra (NEAs). El movimiento de estos objetos es caótico en una escala de tiempo corta. Más aún, debido a los cambios erráticos en los elementos orbitales, la comparación de los valores actuales da poca información acerca de la posible vinculación dinámica entre los objetos de una misma familia. Calculamos una estimación finita de los Exponentes Característicos de Lyapunov (LCE), los llamamos Indicadores Característicos de Lyapunov (LCI) para ambas familias y analizamos las características del espacio de fase donde tiene lugar el movimiento de estos objetos. Integrando en un período suficientemente largo (e.g. 20000 años), encontramos que el LCI alcanza un valor cuasi-constante. La mayoría de los miembros de ambas familias muestran una concentración de los tiempos de Lyapunov (inverso del LCI) de alrededor de 50-100 años (Tancredi, 1995, Astron & Astrop., 299, 288). La concentración de los tiempos de Lyapunov es mayor para la familia de Júpiter que para los NEAs. Entre estos últimos, la menor dispersión se da para aquellos que cruzan la órbita de la Tierra. Se demostró que el espectro de los `indicadores locales' (Froeschlé et. al., 1990, Cel. Mec. 56, 307) o ``números de estiramiento'' (Voglis and Contopoulos, 1994, J. Phys. A 26, 4899) (relacionados con el LCI) son invariantes y nos dan una información más completa sobre el comportamiento caótico. Mediante la comparación de espectros discutimos la similitud entre los objetos de una misma familia y analizamos las diferentes posibles rutas al caos. Los espectros se clasifican mediante la comparación de los momentos de las distribuciones de los `números de estiramiento'. Aplicamos un método de agrupamiento jerárquico (Zappala et. al., 1990, Astron. J. 100, 2030) para identificar ``familias'' de espectros (grupos de espectros

New relation for critical exponents in the Ising model

International Nuclear Information System (INIS)

Pishtshev, A.

2007-01-01

The Ising model in a transverse field is considered at T=0. From the analysis of the power low behaviors of the energy gap and the order parameter as functions of the field a new relation between the respective critical exponents, β>=1/(8s 2 ), is derived. By using the Suzuki equivalence from this inequality a new relation for critical exponents in the Ising model, β>=1/(8ν 2 ), is obtained. A number of numerical examples for different cases illustrates the generality and validity of the relation. By applying this relation the estimation ν=(1/4) 1/3 ∼0.62996 for the 3D-Ising model is proposed

Lyapunov Orbits in the Jupiter System Using Electrodynamic Tethers

Science.gov (United States)

Bokelmann, Kevin; Russell, Ryan P.; Lantoine, Gregory

2013-01-01

Various researchers have proposed the use of electrodynamic tethers for power generation and capture from interplanetary transfers. The effect of tether forces on periodic orbits in Jupiter-satellite systems are investigated. A perturbation force is added to the restricted three-body problem model and a series of simplifications allows development of a conservative system that retains the Jacobi integral. Expressions are developed to find modified locations of equilibrium positions. Modified families of Lyapunov orbits are generated as functions of tether size and Jacobi integral. Zero velocity curves and stability analyses are used to evaluate the dynamical properties of tether-modified orbits.

Critical exponents for the Reggeon quantum spin model

International Nuclear Information System (INIS)

Brower, R.C.; Furman, M.A.

1978-01-01

The Reggeon quantum spin (RQS) model on the transverse lattice in D dimensional impact parameter space has been conjectured to have the same critical behaviour as the Reggeon field theory (RFT). Thus from a high 'temperature' series of ten (D=2) and twenty (D=1) terms for the RQS model the authors extrapolate to the critical temperature T=Tsub(c) by Pade approximants to obtain the exponents eta=0.238 +- 0.008, z=1.16 +- 0.01, γ=1.271 +- 0.007 for D=2 and eta=0.317 +- 0.002, z=1.272 +- 0.007, γ=1.736 +- 0.001, lambda=0.57 +- 0.03 for D=1. These exponents naturally interpolate between the D=0 and D=4-epsilon results for RFT as expected on the basis of the universality conjecture. (Auth.)

Monte Carlo-Based Tail Exponent Estimator

Czech Academy of Sciences Publication Activity Database

Baruník, Jozef; Vácha, Lukáš

2010-01-01

Roč. 2010, č. 6 (2010), s. 1-26 R&D Projects: GA ČR GA402/09/0965; GA ČR GD402/09/H045; GA ČR GP402/08/P207 Institutional research plan: CEZ:AV0Z10750506 Keywords : Hill estimator * α-stable distributions * tail exponent estimation Subject RIV: AH - Economics http://library.utia.cas.cz/separaty/2010/E/barunik-0342493.pdf

Dynamic Analysis and Circuit Design of a Novel Hyperchaotic System with Fractional-Order Terms

Directory of Open Access Journals (Sweden)

Abir Lassoued

2017-01-01

Full Text Available A novel hyperchaotic system with fractional-order (FO terms is designed. Its highly complex dynamics are investigated in terms of equilibrium points, Lyapunov spectrum, and attractor forms. It will be shown that the proposed system exhibits larger Lyapunov exponents than related hyperchaotic systems. Finally, to enhance its potential application, a related circuit is designed by using the MultiSIM Software. Simulation results verify the effectiveness of the suggested circuit.

A new combined approach on Hurst exponent estimate and its applications in realized volatility

Science.gov (United States)

Luo, Yi; Huang, Yirong

2018-02-01

The purpose of this paper is to propose a new estimator of Hurst exponent based on the combined information of the conventional rescaled range methods. We demonstrate the superiority of the proposed estimator by Monte Carlo simulations, and the applications in estimating the Hurst exponent of daily volatility series in Chinese stock market. Moreover, we indicate the impact of the type of estimator and structural break on the estimating results of Hurst exponent.

Pattern formations in chaotic spatio-temporal systems

Indian Academy of Sciences (India)

5Beijing–Hong Kong–Singapore Joint Center of Nonlinear and Complex Systems, ... For theoretical studies most previous work has focused on pattern .... Lyapunov exponents from arbitrary initial conditions, and the plots look rather.

A simple chaotic delay differential equation

International Nuclear Information System (INIS)

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

Scilab software package for the study of dynamical systems

Science.gov (United States)

Bordeianu, C. C.; Beşliu, C.; Jipa, Al.; Felea, D.; Grossu, I. V.

2008-05-01

This work presents a new software package for the study of chaotic flows and maps. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropy. Various well known examples are implemented, with the capability of the users inserting their own ODE. Program summaryProgram title: Chaos Catalogue identifier: AEAP_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAP_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 885 No. of bytes in distributed program, including test data, etc.: 5925 Distribution format: tar.gz Programming language: Scilab 3.1.1 Computer: PC-compatible running Scilab on MS Windows or Linux Operating system: Windows XP, Linux RAM: below 100 Megabytes Classification: 6.2 Nature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE). Solution method: Numerical solving of ordinary differential equations. The chaotic behavior of the nonlinear dynamical system is analyzed using Poincaré sections, phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropies. Restrictions: The package routines are normally able to handle ODE systems of high orders (up to order twelve and possibly higher), depending on the nature of the problem. Running time: 10 to 20 seconds for problems that do not

Nonlinear dynamics of regenerative cutting processes-Comparison of two models

International Nuclear Information System (INIS)

Wang, X.S.; Hu, J.; Gao, J.B.

2006-01-01

Understanding the nonlinear dynamics of cutting processes is essential for the improvement of machining technology. We study machine cutting processes by two different models, one has been recently introduced by Litak [Litak G. Chaotic vibrations in a regenerative cutting process. Chaos, Solitons and Fractals 2002;13:1531-5] and the other is the classic delay differential equation model. Although chaotic solutions have been found in both models, well known routes to chaos, such as period-doubling or quasi-periodic motion to chaos are not observed in either model. Careful analysis shows that the chaotic motion from the Litak's model has sharper spectral peaks, a smaller correlation dimension and a smaller value for the largest positive Lyapunov exponent. Implications to the control of chaos in cutting processes are discussed

Studies on Five Senses Treatment

Science.gov (United States)

Sato, Sadaka; Miao, Tiejun; Oyama-Higa, Mayumi

2011-06-01

This study proposed a therapy from complementary and alternative medicine to treat mental disorder by through interactions of five senses between therapist and patient. In this method sounding a certain six voices play an important role in healing and recovery. First, we studied effects of speaking using scalp- EEG measurement. Chaos analysis of EEG showed a largely enhanced largest Lyapunov exponent (LLE) during the speaking. In addition, EEG power spectrum showed an increase over most frequencies. Second, we performed case studies on mental disorder using the therapy. Running power spectrum of EEG of patients indicated decreasing power at end of treatment, implying five senses therapy induced relaxed and lowered energy in central neural system. The results agreed with patient's reports that there were considerable decline in anxiety and improvements in mood.

Achieving Synchronization in Arrays of Coupled Differential Systems with Time-Varying Couplings

Directory of Open Access Journals (Sweden)

Xinlei Yi

2013-01-01

Full Text Available We study complete synchronization of the complex dynamical networks described by linearly coupled ordinary differential equation systems (LCODEs. Here, the coupling is timevarying in both network structure and reaction dynamics. Inspired by our previous paper (Lu et al. (2007-2008, the extended Hajnal diameter is introduced and used to measure the synchronization in a general differential system. Then we find that the Hajnal diameter of the linear system induced by the time-varying coupling matrix and the largest Lyapunov exponent of the synchronized system play the key roles in synchronization analysis of LCODEs with identity inner coupling matrix. As an application, we obtain a general sufficient condition guaranteeing directed time-varying graph to reach consensus. Example with numerical simulation is provided to show the effectiveness of the theoretical results.

Gestures recognition based on wavelet and LLE

International Nuclear Information System (INIS)

Ai, Qingsong; Liu, Quan; Lu, Ying; Yuan, Tingting

2013-01-01

Wavelet analysis is a time–frequency, non-stationary method while the largest Lyapunov exponent (LLE) is used to judge the non-linear characteristic of systems. Because surface electromyography signal (SEMGS) is a complex signal that is characterized by non-stationary and non-linear properties. This paper combines wavelet coefficient and LLE together as the new feature of SEMGS. The proposed method not only reflects the non-stationary and non-linear characteristics of SEMGS, but also is suitable for its classification. Then, the BP (back propagation) neural network is employed to implement the identification of six gestures (fist clench, fist extension, wrist extension, wrist flexion, radial deviation, ulnar deviation). The experimental results indicate that based on the proposed method, the identification of these six gestures can reach an average rate of 97.71 %.

A mathematical analysis of the effects of Hebbian learning rules on the dynamics and structure of discrete-time random recurrent neural networks.

Science.gov (United States)

Siri, Benoît; Berry, Hugues; Cessac, Bruno; Delord, Bruno; Quoy, Mathias

2008-12-01

We present a mathematical analysis of the effects of Hebbian learning in random recurrent neural networks, with a generic Hebbian learning rule, including passive forgetting and different timescales, for neuronal activity and learning dynamics. Previous numerical work has reported that Hebbian learning drives the system from chaos to a steady state through a sequence of bifurcations. Here, we interpret these results mathematically and show that these effects, involving a complex coupling between neuronal dynamics and synaptic graph structure, can be analyzed using Jacobian matrices, which introduce both a structural and a dynamical point of view on neural network evolution. Furthermore, we show that sensitivity to a learned pattern is maximal when the largest Lyapunov exponent is close to 0. We discuss how neural networks may take advantage of this regime of high functional interest.

Lojasiewicz exponents and Newton polyhedra

International Nuclear Information System (INIS)

Pham Tien Son

2006-07-01

In this paper we obtain the exact value of the Lojasiewicz exponent at the origin of analytic map germs on K n (K = R or C under the Newton non-degeneracy condition, using information from their Newton polyhedra. We also give some conclusions on Newton non-degenerate analytic map germs. As a consequence, we obtain a link between Newton non-degenerate ideals and their integral closures, thus leading to a simple proof of a result of Saia. Similar results are also considered to polynomial maps which are Newton non-degenerate at infinity. (author)

Caos, statistica e metodi di ricampionamento

Directory of Open Access Journals (Sweden)

Simone Giannerini

2007-10-01

Full Text Available Chaos theory offers to the statistician new perspectives for time series analysis as well as concepts and ideas that have a through contribution to statistics. On the other hand, statistical methodology has shown to play a crucial role for the comprehension of nonlinear and chaotic phenomena. From this standpoint we present some essential notions for the analysis of chaotic time series. Particular attention is given to the problem of estimating Lyapunov exponents, together with the derivation of confidence intervals for estimates. For this latter problem we propose a solution based on resampling by means of spline interpolation. We show from simulations that the distribution of the maximal Lyapunov exponent obtained by way of resampling a single series with our method, agrees with the true distribution obtained from series with different initial conditions.

Chaotic weak chimeras and their persistence in coupled populations of phase oscillators

International Nuclear Information System (INIS)

Bick, Christian; Ashwin, Peter

2016-01-01

Nontrivial collective behavior may emerge from the interactive dynamics of many oscillatory units. Chimera states are chaotic patterns of spatially localized coherent and incoherent oscillations. The recently-introduced notion of a weak chimera gives a rigorously testable characterization of chimera states for finite-dimensional phase oscillator networks. In this paper we give some persistence results for dynamically invariant sets under perturbations and apply them to coupled populations of phase oscillators with generalized coupling. In contrast to the weak chimeras with nonpositive maximal Lyapunov exponents constructed so far, we show that weak chimeras that are chaotic can exist in the limit of vanishing coupling between coupled populations of phase oscillators. We present numerical evidence that positive Lyapunov exponents can persist for a positive measure set of this inter-population coupling strength. (paper)

Computational analysis of kidney scintigrams

Energy Technology Data Exchange (ETDEWEB)

Vrincianu, D.; Puscasu, E.; Creanga, D. [University Al. I. Cuza, Faculty of Physics, 11 Blvd. Carol I, 700506, Iasi (Romania); Stefanescu, C. [University of Medicine and Pharmacy Gr. T. Popa, Iasi (Romania)

2013-11-13

The scintigraphic investigation of normal and pathological kidneys was carried out using specialized gamma-camera device from nuclear medicine hospital department. Technetium 90m isotope with gamma radiation emission, coupled with vector molecules for kidney tissues was introduced into the subject body, its dynamics being recorded as data source for kidney clearance capacity. Two representative data series were investigated, corresponding to healthy and pathological organs respectively. The semi-quantitative tests applied for the comparison of the two distinct medical situations were: the shape of probability distribution histogram, the power spectrum, the auto-correlation function and the Lyapunov exponent. While power spectrum led to similar results in both cases, significant differences were revealed by means of distribution probability, Lyapunov exponent and correlation time, recommending these numerical tests as possible complementary tools in clinical diagnosis.

Nonlinear analysis of RED - a comparative study

International Nuclear Information System (INIS)

Jiang Kai; Wang Xiaofan; Xi Yugeng

2004-01-01

Random Early Detection (RED) is an active queue management (AQM) mechanism for routers on the Internet. In this paper, performance of RED and Adaptive RED are compared from the viewpoint of nonlinear dynamics. In particular, we reveal the relationship between the performance of the network and its nonlinear dynamical behavior. We measure the maximal Lyapunov exponent and Hurst parameter of the average queue length of RED and Adaptive RED, as well as the throughput and packet loss rate of the aggregate traffic on the bottleneck link. Our simulation scenarios include FTP flows and Web flows, one-way and two-way traffic. In most situations, Adaptive RED has smaller maximal Lyapunov exponents, lower Hurst parameters, higher throughput and lower packet loss rate than that of RED. This confirms that Adaptive RED has better performance than RED

Nonlinear analysis of RED - a comparative study

Energy Technology Data Exchange (ETDEWEB)

Jiang Kai; Wang Xiaofan E-mail: xfwang@sjtu.edu.cn; Xi Yugeng

2004-09-01

Random Early Detection (RED) is an active queue management (AQM) mechanism for routers on the Internet. In this paper, performance of RED and Adaptive RED are compared from the viewpoint of nonlinear dynamics. In particular, we reveal the relationship between the performance of the network and its nonlinear dynamical behavior. We measure the maximal Lyapunov exponent and Hurst parameter of the average queue length of RED and Adaptive RED, as well as the throughput and packet loss rate of the aggregate traffic on the bottleneck link. Our simulation scenarios include FTP flows and Web flows, one-way and two-way traffic. In most situations, Adaptive RED has smaller maximal Lyapunov exponents, lower Hurst parameters, higher throughput and lower packet loss rate than that of RED. This confirms that Adaptive RED has better performance than RED.

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

OpenAIRE

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...

Critical exponents predicted by grouping of Feynman diagrams in φ4 model

International Nuclear Information System (INIS)

Kaupuzs, J.

2001-01-01

Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ 4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments. (orig.)

Inter-relationship between scaling exponents for describing self-similar river networks

Science.gov (United States)

Yang, Soohyun; Paik, Kyungrock

2015-04-01

Natural river networks show well-known self-similar characteristics. Such characteristics are represented by various power-law relationships, e.g., between upstream length and drainage area (exponent h) (Hack, 1957), and in the exceedance probability distribution of upstream area (exponent ɛ) (Rodriguez-Iturbe et al., 1992). It is empirically revealed that these power-law exponents are within narrow ranges. Power-law is also found in the relationship between drainage density (the total stream length divided by the total basin area) and specified source area (the minimum drainage area to form a stream head) (exponent η) (Moussa and Bocquillon, 1996). Considering that above three scaling relationships all refer to fundamental measures of 'length' and 'area' of a given drainage basin, it is natural to hypothesize plausible inter-relationship between these three scaling exponents. Indeed, Rigon et al. (1996) demonstrated the relationship between ɛ and h. In this study, we expand this to a more general ɛ-η-h relationship. We approach ɛ-η relationship in an analytical manner while η-h relationship is demonstrated for six study basins in Korea. Detailed analysis and implications will be presented. References Hack, J. T. (1957). Studies of longitudinal river profiles in Virginia and Maryland. US, Geological Survey Professional Paper, 294. Moussa, R., & Bocquillon, C. (1996). Fractal analyses of tree-like channel networks from digital elevation model data. Journal of Hydrology, 187(1), 157-172. Rigon, R., Rodriguez-Iturbe, I., Maritan, A., Giacometti. A., Tarboton, D. G., & Rinaldo, A. (1996). On Hack's Law. Water Resources Research, 32(11), 3367-3374. Rodríguez-Iturbe, I., Ijjasz-Vasquez, E. J., Bras, R. L., & Tarboton, D. G. (1992). Power law distributions of discharge mass and energy in river basins. Water Resources Research, 28(4), 1089-1093.

Lyapunov, singular and bred vectors in a multi-scale system: an empirical exploration of vectors related to instabilities

International Nuclear Information System (INIS)

Norwood, Adrienne; Kalnay, Eugenia; Ide, Kayo; Yang, Shu-Chih; Wolfe, Christopher

2013-01-01

We compute and compare the three types of vectors frequently used to explore the instability properties of dynamical models, namely Lyapunov vectors (LVs), singular vectors (SVs) and bred vectors (BVs) in two systems, using the Wolfe–Samelson (2007 Tellus A 59 355–66) algorithm to compute all of the Lyapunov vectors. The first system is the Lorenz (1963 J. Atmos. Sci. 20 130–41) three-variable model. Although the leading Lyapunov vector, LV1, grows fastest globally, the second Lyapunov vector, LV2, which has zero growth globally, often grows faster than LV1 locally. Whenever this happens, BVs grow closer to LV2, suggesting that in larger atmospheric or oceanic models where several instabilities can grow in different areas of the world, BVs will grow toward the fastest growing local unstable mode. A comparison of their growth rates at different times shows that all three types of dynamical vectors have the ability to predict regime changes and the duration of the new regime based on their growth rates in the last orbit of the old regime, as shown for BVs by Evans et al (2004 Bull. Am. Meteorol. Soc. 520–4). LV1 and BVs have similar predictive skill, LV2 has a tendency to produce false alarms, and even LV3 shows that maximum decay is also associated with regime change. Initial and final SVs grow much faster and are the most accurate predictors of regime change, although the characteristics of the initial SVs are strongly dependent on the length of the optimization window. The second system is the toy ‘ocean-atmosphere’ model developed by Peña and Kalnay (2004 Nonlinear Process. Geophys. 11 319–27) coupling three Lorenz (1963 J. Atmos. Sci. 20 130–41) systems with different time scales, in order to test the effects of fast and slow modes of growth on the dynamical vectors. A fast ‘extratropical atmosphere’ is weakly coupled to a fast ‘tropical atmosphere’ which is, in turn, strongly coupled to a slow ‘ocean’ system, the latter coupling

On identifying relationships between the flood scaling exponent and basin attributes.

Science.gov (United States)

Medhi, Hemanta; Tripathi, Shivam

2015-07-01

Floods are known to exhibit self-similarity and follow scaling laws that form the basis of regional flood frequency analysis. However, the relationship between basin attributes and the scaling behavior of floods is still not fully understood. Identifying these relationships is essential for drawing connections between hydrological processes in a basin and the flood response of the basin. The existing studies mostly rely on simulation models to draw these connections. This paper proposes a new methodology that draws connections between basin attributes and the flood scaling exponents by using observed data. In the proposed methodology, region-of-influence approach is used to delineate homogeneous regions for each gaging station. Ordinary least squares regression is then applied to estimate flood scaling exponents for each homogeneous region, and finally stepwise regression is used to identify basin attributes that affect flood scaling exponents. The effectiveness of the proposed methodology is tested by applying it to data from river basins in the United States. The results suggest that flood scaling exponent is small for regions having (i) large abstractions from precipitation in the form of large soil moisture storages and high evapotranspiration losses, and (ii) large fractions of overland flow compared to base flow, i.e., regions having fast-responding basins. Analysis of simple scaling and multiscaling of floods showed evidence of simple scaling for regions in which the snowfall dominates the total precipitation.

Chaos synchronization and parameter identification of three time scales brushless DC motor system

International Nuclear Information System (INIS)

Ge, Z.-M.; Cheng, J.-W.

2005-01-01

Chaotic anticontrol and chaos synchronization of brushless DC motor system are studied in this paper. Nondimensional dynamic equations of three time scale brushless DC motor system are presented. Using numerical results, such as phase diagram, bifurcation diagram, and Lyapunov exponent, periodic and chaotic motions can be observed. Then, chaos synchronization of two identical systems via additional inputs and Lyapunov stability theory are studied. And further, the parameter of the system is traced via adaptive control and random optimization method

Stochastic model of Zipf's law and the universality of the power-law exponent.

Science.gov (United States)

Yamamoto, Ken

2014-04-01

We propose a stochastic model of Zipf's law, namely a power-law relation between rank and size, and clarify as to why a specific value of its power-law exponent is quite universal. We focus on the successive total of a multiplicative stochastic process. By employing properties of a well-known stochastic process, we concisely show that the successive total follows a stationary power-law distribution, which is directly related to Zipf's law. The formula of the power-law exponent is also derived. Finally, we conclude that the universality of the rank-size exponent is brought about by symmetry between an increase and a decrease in the random growth rate.

Fermat's Last Theorem for Factional and Irrational Exponents

Science.gov (United States)

Morgan, Frank

2010-01-01

Fermat's Last Theorem says that for integers n greater than 2, there are no solutions to x[superscript n] + y[superscript n] = z[superscript n] among positive integers. What about rational exponents? Irrational n? Negative n? See what an undergraduate senior seminar discovered.

Pesin’s entropy formula for stochastic flows of diffeomorphisms

Institute of Scientific and Technical Information of China (English)

刘培东

1996-01-01

Pesin’s entropy formula relating entropy and Lyapunov exponents within the context of random dynamical systems generated by (discrete or continuous) stochastic flows of diffeomorphisms (including solution flows of stochastic differential equations on manifolds) is proved.

On generalized scaling laws with continuously varying exponents

International Nuclear Information System (INIS)

Sittler, Lionel; Hinrichsen, Haye

2002-01-01

Many physical systems share the property of scale invariance. Most of them show ordinary power-law scaling, where quantities can be expressed as a leading power law times a scaling function which depends on scaling-invariant ratios of the parameters. However, some systems do not obey power-law scaling, instead there is numerical evidence for a logarithmic scaling form, in which the scaling function depends on ratios of the logarithms of the parameters. Based on previous ideas by Tang we propose that this type of logarithmic scaling can be explained by a concept of local scaling invariance with continuously varying exponents. The functional dependence of the exponents is constrained by a homomorphism which can be expressed as a set of partial differential equations. Solving these equations we obtain logarithmic scaling as a special case. The other solutions lead to scaling forms where logarithmic and power-law scaling are mixed

Weak and strong chaos in Fermi-Pasta-Ulam models and beyond

Science.gov (United States)

Pettini, Marco; Casetti, Lapo; Cerruti-Sola, Monica; Franzosi, Roberto; Cohen, E. G. D.

2005-03-01

We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. (ii) A strong stochasticity threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, the third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions.

Neural chaos and schizophrenia

Czech Academy of Sciences Publication Activity Database

Bob, P.; Chládek, Jan; Šusta, M.; Glaslová, K.; Jagla, F.; Kukleta, M.

2007-01-01

Roč. 26, č. 4 (2007), s. 298-305 ISSN 0231-5882 Institutional research plan: CEZ:AV0Z20650511 Keywords : EDA * Lyapunov exponent * schizophrenia * chaos Subject RIV: FL - Psychiatry, Sexuology Impact factor: 1.286, year: 2007

Variation of Zipf's exponent in one hundred live languages: A study of the Holy Bible translations

Science.gov (United States)

Mehri, Ali; Jamaati, Maryam

2017-08-01

Zipf's law, as a power-law regularity, confirms long-range correlations between the elements in natural and artificial systems. In this article, this law is evaluated for one hundred live languages. We calculate Zipf's exponent for translations of the holy Bible to several languages, for this purpose. The results show that, the average of Zipf's exponent in studied texts is slightly above unity. All studied languages in some families have Zipf's exponent lower/higher than unity. It seems that geographical distribution impresses the communication between speakers of different languages in a language family, and affect similarity between their Zipf's exponent. The Bible has unique concept regardless of its language, but the discrepancy in grammatical rules and syntactic regularities in applying stop words to make sentences and imply a certain concept, lead to difference in Zipf's exponent for various languages.

Identification of exponent from load-deformation relation for soft materials from impact tests

Science.gov (United States)

Ciornei, F. C.; Alaci, S.; Romanu, I. C.; Ciornei, M. C.; Sopon, G.

2018-01-01

When two bodies are brought into contact, the magnitude of occurring reaction forces increase together with the amplitude of deformations. The load-deformation dependency of two contacting bodies is described by a function having the form F = Cxα . An accurate illustration of this relationship assumes finding the precise coefficient C and exponent α. This representation proved to be very useful in hardness tests, in dynamic systems modelling or in considerations upon the elastic-plastic ratio concerning a Hertzian contact. The classical method for identification of the exponent consists in finding it from quasi-static tests. The drawback of the method is the fact that the accurate estimation of the exponent supposes precise identification of the instant of contact initiation. To overcome this aspect, the following observation is exploited: during an impact process, the dissipated energy is converted into heat released by internal friction in the materials and energy for plastic deformations. The paper is based on the remark that for soft materials the hysteresis curves obtained for a static case are similar to the ones obtained for medium velocities. Furthermore, utilizing the fact that for the restitution phase the load-deformation dependency is elastic, a method for finding the α exponent for compression phase is proposed. The maximum depth of the plastic deformations obtained for a series of collisions, by launching, from different heights, a steel ball in free falling on an immobile prism made of soft material, is evaluated by laser profilometry method. The condition that the area of the hysteresis loop equals the variation of kinetical energy of the ball is imposed and two tests are required for finding the exponent. Five collisions from different launching heights of the ball were taken into account. For all the possible impact-pair cases, the values of the exponent were found and close values were obtained.

Effect of density of state on isotope effect exponent of two-band superconductors

International Nuclear Information System (INIS)

Udomsamuthirun, P.; Kumvongsa, C.; Burakorn, A.; Changkanarth, P.; Yoksan, S.

2005-01-01

The exact formula of T c 's equation and the isotope effect exponent of two-band s-wave superconductors in weak-coupling limit are derived by considering the influence of two kinds of density of state: constant and van Hove singularity. The paring interaction in each band consisted of two parts: the electron-phonon interaction and non-electron-phonon interaction are included in our model. We find that the interband interaction of electron-phonon show more effect on isotope exponent than the intraband interaction and the isotope effect exponent with constant density of state can fit to experimental data, MgB 2 and high-T c superconductor, better than van Hove singularity density of state

Effect of noise on estimation of Lyapunov exponents from a time series

Energy Technology Data Exchange (ETDEWEB)

Serletis, Apostolos [Department of Economics, University of Calgary, Calgary, Alta., T2N 1N4 (Canada)]. E-mail: http://econ.ucalgary.ca/serletis.htm; Shahmoradi, Asghar [Department of Economics, University of Calgary, Calgary, Alta., T2N 1N4 (Canada); Serletis, Demitre [Division of Neurosurgery, University of Toronto, Toronto, ON, M5G 1L5 (Canada)

2007-04-15

We argue that dynamical noise can dramatically change the dynamics of low-dimensional chaotic systems. Moreover, we show that chaos tests are highly sensitive to dynamical noise and this becomes worse when the intensity of the noise increases.

Continuation of probability density functions using a generalized Lyapunov approach

Energy Technology Data Exchange (ETDEWEB)

Baars, S., E-mail: s.baars@rug.nl [Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen (Netherlands); Viebahn, J.P., E-mail: viebahn@cwi.nl [Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB, Amsterdam (Netherlands); Mulder, T.E., E-mail: t.e.mulder@uu.nl [Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht (Netherlands); Kuehn, C., E-mail: ckuehn@ma.tum.de [Technical University of Munich, Faculty of Mathematics, Boltzmannstr. 3, 85748 Garching bei München (Germany); Wubs, F.W., E-mail: f.w.wubs@rug.nl [Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen (Netherlands); Dijkstra, H.A., E-mail: h.a.dijkstra@uu.nl [Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht University, Princetonplein 5, 3584 CC Utrecht (Netherlands); School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY (United States)

2017-05-01

Techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved. Here, we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial differential equations near fixed points, under a small noise approximation. Key innovation is the efficient solution of a generalized Lyapunov equation using an iterative method involving low-rank approximations. We apply and illustrate the capabilities of the method using a problem in physical oceanography, i.e. the occurrence of multiple steady states of the Atlantic Ocean circulation.

Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

Science.gov (United States)

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.

Estimation of Hurst Exponent for the Financial Time Series

Science.gov (United States)

Kumar, J.; Manchanda, P.

2009-07-01

Till recently statistical methods and Fourier analysis were employed to study fluctuations in stock markets in general and Indian stock market in particular. However current trend is to apply the concepts of wavelet methodology and Hurst exponent, see for example the work of Manchanda, J. Kumar and Siddiqi, Journal of the Frankline Institute 144 (2007), 613-636 and paper of Cajueiro and B. M. Tabak. Cajueiro and Tabak, Physica A, 2003, have checked the efficiency of emerging markets by computing Hurst component over a time window of 4 years of data. Our goal in the present paper is to understand the dynamics of the Indian stock market. We look for the persistency in the stock market through Hurst exponent and fractal dimension of time series data of BSE 100 and NIFTY 50.

Power-law Exponent in Multiplicative Langevin Equation with Temporally Correlated Noise

Science.gov (United States)

Morita, Satoru

2018-05-01

Power-law distributions are ubiquitous in nature. Random multiplicative processes are a basic model for the generation of power-law distributions. For discrete-time systems, the power-law exponent is known to decrease as the autocorrelation time of the multiplier increases. However, for continuous-time systems, it is not yet clear how the temporal correlation affects the power-law behavior. Herein, we analytically investigated a multiplicative Langevin equation with colored noise. We show that the power-law exponent depends on the details of the multiplicative noise, in contrast to the case of discrete-time systems.

Dynamics of the stochastic low concentration trimolecular oscillatory chemical system with jumps

Science.gov (United States)

Wei, Yongchang; Yang, Qigui

2018-06-01

This paper is devoted to discern long time dynamics through the stochastic low concentration trimolecular oscillatory chemical system with jumps. By Lyapunov technique, this system is proved to have a unique global positive solution, and the asymptotic stability in mean square of such model is further established. Moreover, the existence of random attractor and Lyapunov exponents are obtained for the stochastic homeomorphism flow generated by the corresponding global positive solution. And some numerical simulations are given to illustrate the presented results.

Controversy in the allometric application of fixed- versus varying-exponent models: a statistical and mathematical perspective.

Science.gov (United States)

Tang, Huadong; Hussain, Azher; Leal, Mauricio; Fluhler, Eric; Mayersohn, Michael

2011-02-01

This commentary is a reply to a recent article by Mahmood commenting on the authors' article on the use of fixed-exponent allometry in predicting human clearance. The commentary discusses eight issues that are related to criticisms made in Mahmood's article and examines the controversies (fixed-exponent vs. varying-exponent allometry) from the perspective of statistics and mathematics. The key conclusion is that any allometric method, which is to establish a power function based on a limited number of animal species and to extrapolate the resulting power function to human values (varying-exponent allometry), is infused with fundamental statistical errors. Copyright © 2010 Wiley-Liss, Inc.

Landau-like theory for universality of critical exponents in quasistationary states of isolated mean-field systems.

Science.gov (United States)

Ogawa, Shun; Yamaguchi, Yoshiyuki Y

2015-06-01

An external force dynamically drives an isolated mean-field Hamiltonian system to a long-lasting quasistationary state, whose lifetime increases with population of the system. For second order phase transitions in quasistationary states, two nonclassical critical exponents have been reported individually by using a linear and a nonlinear response theories in a toy model. We provide a simple way to compute the critical exponents all at once, which is an analog of the Landau theory. The present theory extends the universality class of the nonclassical exponents to spatially periodic one-dimensional systems and shows that the exponents satisfy a classical scaling relation inevitably by using a key scaling of momentum.

LYAPUNOV-Based Sensor Failure Detection and Recovery for the Reverse Water Gas Shift Process

Science.gov (United States)

Haralambous, Michael G.

2002-01-01

Livingstone, a model-based AI software system, is planned for use in the autonomous fault diagnosis, reconfiguration, and control of the oxygen-producing reverse water gas shift (RWGS) process test-bed located in the Applied Chemistry Laboratory at KSC. In this report the RWGS process is first briefly described and an overview of Livingstone is given. Next, a Lyapunov-based approach for detecting and recovering from sensor failures, differing significantly from that used by Livingstone, is presented. In this new method, models used are in t e m of the defining differential equations of system components, thus differing from the qualitative, static models used by Livingstone. An easily computed scalar inequality constraint, expressed in terms of sensed system variables, is used to determine the existence of sensor failures. In the event of sensor failure, an observer/estimator is used for determining which sensors have failed. The theory underlying the new approach is developed. Finally, a recommendation is made to use the Lyapunov-based approach to complement the capability of Livingstone and to use this combination in the RWGS process.

Multivalued synchronization by Poincaré coupling

Science.gov (United States)

Ontañón-García, L. J.; Campos-Cantón, E.; Femat, R.; Campos-Cantón, I.; Bonilla-Marín, M.

2013-10-01

This work presents multivalued chaotic synchronization via coupling based on the Poincaré plane. The coupling is carried out by an underdamped signal, triggered every crossing event of the trajectory of the master system through a previously defined Poincaré plane. A master-slave system is explored, and the synchronization between the systems is detected via the auxiliary system approach and the maximum conditional Lyapunov exponent. Due to the response to specific conditions two phenomena may be obtained: univalued and multivalued synchronization. Since the Lyapunov exponent is not enough to detect these two phenomena, the distance between the pieces of trajectories of the slave and auxiliary systems with different initial conditions is also used as a tool for the detection of multivalued synchronization. Computer simulations using the benchmark chaotic systems of Lorenz and Rössler are used to exemplify the approach proposed.

Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay

International Nuclear Information System (INIS)

Pyragas, V.; Pyragas, K.

2011-01-01

We propose a simple adaptive delayed feedback control algorithm for stabilization of unstable periodic orbits with unknown periods. The state dependent time delay is varied continuously towards the period of controlled orbit according to a gradient-descent method realized through three simple ordinary differential equations. We demonstrate the efficiency of the algorithm with the Roessler and Mackey-Glass chaotic systems. The stability of the controlled orbits is proven by computation of the Lyapunov exponents of linearized equations. -- Highlights: → A simple adaptive modification of the delayed feedback control algorithm is proposed. → It enables the control of unstable periodic orbits with unknown periods. → The delay time is varied continuously according to a gradient descend method. → The algorithm is embodied by three simple ordinary differential equations. → The validity of the algorithm is proven by computation of the Lyapunov exponents.

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

Directory of Open Access Journals (Sweden)

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.

Chaos of the Relativistic Forced van der Pol Oscillator

International Nuclear Information System (INIS)

Ashkenazya, Y.; Gorma, C; Horwitz, L. P.

1998-01-01

A manifestly relativistically covariant form of the van der Pol oscillator in 1 + 1 dimensions is studied. We show that the driven relativistic equations, for which z and t are coupled, relax very quickly to a pair of identical decoupled equations, due to a rapid vanishing of the angular momentum (the boost in 1 + 1 dimensions). A similar effect occurs in the damped driven covariant Duffing oscillator previously treated. This effect is an example of entrainment, or synchronization (phase locking) , of coupled chaotic systems. The Lyapunov exponents are calculated using the very efficient method of Habib and Ryne. We show a Poincare map that demonstrates this effect and maintains remarkable stability in spite of the inevitable accumulation of computer error in the chaotic region. For our choice of parameters, the positive Lyapunov exponent is about 0.242 almost independently of the integration method

Critical exponents of extremal Kerr perturbations

Science.gov (United States)

Gralla, Samuel E.; Zimmerman, Peter

2018-05-01

We show that scalar, electromagnetic, and gravitational perturbations of extremal Kerr black holes are asymptotically self-similar under the near-horizon, late-time scaling symmetry of the background metric. This accounts for the Aretakis instability (growth of transverse derivatives) as a critical phenomenon associated with the emergent symmetry. We compute the critical exponent of each mode, which is equivalent to its decay rate. It follows from symmetry arguments that, despite the growth of transverse derivatives, all generally covariant scalar quantities decay to zero.

Structured Lyapunov functions for synchronization of identical affine-in-control agents-Unified approach

Czech Academy of Sciences Publication Activity Database

Hengster-Movric, K.; Šebek, M.; Čelikovský, Sergej

2016-01-01

Roč. 353, č. 14 (2016), s. 3457-3486 ISSN 0016-0032 R&D Projects: GA ČR GA13-20433S Grant - others:GA ČR(CZ) GJ16-25493Y Institutional support: RVO:67985556 Keywords : Multi-agent nonlinear systems * structured Lyapunov functions Subject RIV: BC - Control Systems Theory Impact factor: 3.139, year: 2016 http://library.utia.cas.cz/separaty/2016/TR/celikovsky-0462691.pdf

Scaling exponent and dispersity of polymers in solution by diffusion NMR.

Science.gov (United States)

Williamson, Nathan H; Röding, Magnus; Miklavcic, Stanley J; Nydén, Magnus

2017-05-01

Molecular mass distribution measurements by pulsed gradient spin echo nuclear magnetic resonance (PGSE NMR) spectroscopy currently require prior knowledge of scaling parameters to convert from polymer self-diffusion coefficient to molecular mass. Reversing the problem, we utilize the scaling relation as prior knowledge to uncover the scaling exponent from within the PGSE data. Thus, the scaling exponent-a measure of polymer conformation and solvent quality-and the dispersity (M w /M n ) are obtainable from one simple PGSE experiment. The method utilizes constraints and parametric distribution models in a two-step fitting routine involving first the mass-weighted signal and second the number-weighted signal. The method is developed using lognormal and gamma distribution models and tested on experimental PGSE attenuation of the terminal methylene signal and on the sum of all methylene signals of polyethylene glycol in D 2 O. Scaling exponent and dispersity estimates agree with known values in the majority of instances, leading to the potential application of the method to polymers for which characterization is not possible with alternative techniques. Copyright © 2017 Elsevier Inc. All rights reserved.

Distribution of the largest aftershocks in branching models of triggered seismicity: Theory of the universal Baath law

International Nuclear Information System (INIS)

Saichev, A.; Sornette, D.

2005-01-01

Using the epidemic-type aftershock sequence (ETAS) branching model of triggered seismicity, we apply the formalism of generating probability functions to calculate exactly the average difference between the magnitude of a mainshock and the magnitude of its largest aftershock over all generations. This average magnitude difference is found empirically to be independent of the mainshock magnitude and equal to 1.2, a universal behavior known as Baath's law. Our theory shows that Baath's law holds only sufficiently close to the critical regime of the ETAS branching process. Allowing for error bars ±0.1 for Baath's constant value around 1.2, our exact analytical treatment of Baath's law provides new constraints on the productivity exponent α and the branching ratio n: 0.9 < or approx. α≤1 and 0.8 < or approx. n≤1. We propose a method for measuring α based on the predicted renormalization of the Gutenberg-Richter distribution of the magnitudes of the largest aftershock. We also introduce the 'second Baath law for foreshocks': the probability that a main earthquake turns out to be the foreshock does not depend on its magnitude ρ

High-resolution satellite image segmentation using Hölder exponents

Indian Academy of Sciences (India)

Keywords. High resolution image; texture analysis; segmentation; IKONOS; Hölder exponent; cluster. ... are that. • it can be used as a tool to measure the roughness ... uses reinforcement learning to learn the reward values of ..... The numerical.

Estimation of time-dependent Hurst exponents with variational smoothing and application to forecasting foreign exchange rates

Science.gov (United States)

Garcin, Matthieu

2017-10-01

Hurst exponents depict the long memory of a time series. For human-dependent phenomena, as in finance, this feature may vary in the time. It justifies modelling dynamics by multifractional Brownian motions, which are consistent with time-dependent Hurst exponents. We improve the existing literature on estimating time-dependent Hurst exponents by proposing a smooth estimate obtained by variational calculus. This method is very general and not restricted to the sole Hurst framework. It is globally more accurate and easier than other existing non-parametric estimation techniques. Besides, in the field of Hurst exponents, it makes it possible to make forecasts based on the estimated multifractional Brownian motion. The application to high-frequency foreign exchange markets (GBP, CHF, SEK, USD, CAD, AUD, JPY, CNY and SGD, all against EUR) shows significantly good forecasts. When the Hurst exponent is higher than 0.5, what depicts a long-memory feature, the accuracy is higher.

Non Lyapunov stability of a constant spatially developing 2-D gas flow

Science.gov (United States)

Balint, Agneta M.; Balint, Stefan; Tanasie, Loredana

2017-01-01

Different types of stabilities (global, local) and instabilities (global absolute, local convective) of the constant spatially developing 2-D gas flow are analyzed in a particular phase space of continuously differentiable functions, endowed with the usual algebraic operations and the topology generated by the uniform convergence on the plane. For this purpose the Euler equations linearized at the constant flow are used. The Lyapunov stability analysis was presented in [1] and this paper is a continuation of [1].

Nonlinear dynamics analysis of the human balance control subjected to physical and sensory perturbations.

Science.gov (United States)

Ashtiani, Mohammed N; Mahmood-Reza, Azghani

2017-01-01

Postural control after applying perturbation involves neural and muscular efforts to limit the center of mass (CoM) motion. Linear dynamical approaches may not unveil all complexities of body efforts. This study was aimed at determining two nonlinear dynamics parameters (fractal dimension (FD) and largest Lyapunov exponent (LLE)) in addition to the linear standing metrics of balance in perturbed stance. Sixteen healthy young males were subjected to sudden rotations of the standing platform. The vision and cognition during the standing were also interfered. Motion capturing was used to measure the lower limb joints and the CoM displacements. The CoM path length as a linear parameter was increased by elimination of vision (pnonlinear metric FD was decreased due to the cognitive loads (pnonlinear metrics of the perturbed stance showed that a combination of them may properly represent the body behavior.

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

Directory of Open Access Journals (Sweden)

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.

Economy, Movement Dynamics, and Muscle Activity of Human Walking at Different Speeds

DEFF Research Database (Denmark)

Raffalt, Peter Christian; Guul, Martin Kjær; Nielsen, A. N.

2017-01-01

The complex behaviour of human walking with respect to movement variability, economy and muscle activity is speed dependent. It is well known that a U-shaped relationship between walking speed and economy exists. However, it is an open question if the movement dynamics of joint angles and centre...... of mass and muscle activation strategy also exhibit a U-shaped relationship with walking speed. We investigated the dynamics of joint angle trajectories and the centre of mass accelerations at five different speeds ranging from 20 to 180% of the predicted preferred speed (based on Froude speed) in twelve...... healthy males. The muscle activation strategy and walking economy were also assessed. The movement dynamics was investigated using a combination of the largest Lyapunov exponent and correlation dimension. We observed an intermediate stage of the movement dynamics of the knee joint angle and the anterior...

Randomness confidence bands of fractal scaling exponents for financial price returns

International Nuclear Information System (INIS)

Ibarra-Valdez, C.; Alvarez, J.; Alvarez-Ramirez, J.

2016-01-01

Highlights: • A robust test for randomness of price returns is proposed. • The DFA scaling exponent is contrasted against confidence bands for random sequences. • The size of the band depends of the sequence length. • Crude oil and USA stock markets have been rarely inefficient. - Abstract: The weak-form of the efficient market hypothesis (EMH) establishes that price returns behave as a pure random process and so their outcomes cannot be forecasted. The detrended fluctuation analysis (DFA) has been widely used to test the weak-form of the EMH by showing that time series of price returns are serially uncorrelated. In this case, the DFA scaling exponent exhibits deviations from the theoretical value of 0.5. This work considers the test of the EMH for DFA implementation on a sliding window, which is an approach that is intended to monitor the evolution of markets. Under these conditions, the scaling exponent exhibits important variations over the scrutinized period that can offer valuable insights in the behavior of the market provided the estimated scaling value is kept within strict statistical tests to verify the presence or not of serial correlations in the price returns. In this work, the statistical tests are based on comparing the estimated scaling exponent with the values obtained from pure Gaussian sequences with the length of the real time series. In this way, the presence of serial correlations can be guaranteed only in terms of the confidence bands of a pure Gaussian process. The crude oil (WTI) and the USA stock (DJIA) markets are used to illustrate the methodology.

Beyond Critical Exponents in Neuronal Avalanches

Science.gov (United States)

Friedman, Nir; Butler, Tom; Deville, Robert; Beggs, John; Dahmen, Karin

2011-03-01

Neurons form a complex network in the brain, where they interact with one another by firing electrical signals. Neurons firing can trigger other neurons to fire, potentially causing avalanches of activity in the network. In many cases these avalanches have been found to be scale independent, similar to critical phenomena in diverse systems such as magnets and earthquakes. We discuss models for neuronal activity that allow for the extraction of testable, statistical predictions. We compare these models to experimental results, and go beyond critical exponents.

Fractal dimension of cantori

International Nuclear Information System (INIS)

Li, W.; Bak, P.

1986-01-01

At a critical point the golden-mean Kolmogorov-Arnol'd-Moser trajectory of Chirikov's standard map breaks up into a fractal orbit called a cantorus. The transition describes a pinning of the incommensurate phase of the Frenkel-Kontorowa model. We find that the fractal dimension of the cantorus is D = 0 and that the transition from the Kolmogorov-Arnol'd-Moser trajectory with dimension D = 1 to the cantorus is governed by an exponent ν = 0.98. . . and a universal scaling function. It is argued that the exponent is equal to that of the Lyapunov exponent

Two (multi point nonlinear Lyapunov systems associated with an n th order nonlinear system of differential equations – existence and uniqueness

Directory of Open Access Journals (Sweden)

Murty K. N.

2000-01-01

Full Text Available This paper presents a criterion for the existence and uniqueness of solutions to two and multipoint boundary value problems associated with an n th order nonlinear Lyapunov system. A variation of parameters formula is developed and used as a tool to obtain existence and uniqueness. We discuss solution of the second order problem by the ADI method and develop a fixed point method to find the general solution of the n th order Lyapunov system. The results of Barnett (SIAM J. Appl. Anal. 24(1, 1973 are a particular case.

Magnetic entropy change and critical exponents in double perovskite Y2NiMnO6

Science.gov (United States)

Sharma, G.; Tripathi, T. S.; Saha, J.; Patnaik, S.

2014-11-01

We report the magnetic entropy change (ΔSM) and the critical exponents in the double perovskite manganite Y2NiMnO6 with a ferromagnetic to paramagnetic transition TC~85 K. For a magnetic field change ΔH=80 kOe, a maximum magnetic entropy change ΔSM=-6.57 J/kg K is recorded around TC. The critical exponents β=0.363±0.05 and γ=1.331±0.09 obtained from power law fitting to spontaneous magnetization MS(T) and the inverse initial susceptibility χ0-1(T) satisfy well to values derived for a 3D-Heisenberg ferromagnet. The critical exponent δ=4.761±0.129 is determined from the isothermal magnetization at TC. The scaling exponents corresponding to second order phase transition are consistent with the exponents from Kouvel-Fisher analysis and satisfy Widom's scaling relation δ=1+(γ/β). Additionally, they also satisfy the single scaling equation M(H,ɛ)=ɛβf±(H/ɛ) according to which the magnetization-field-temperature data around TC should collapse into two curves for temperatures below and above TC.

Un algoritmo de replanificación en tiempo real basado en un índice de estabilidad de Lyapunov para líneas de metro

Directory of Open Access Journals (Sweden)

A. Berbey

2014-04-01

Full Text Available Resumen: En este trabajo, se propone un nuevo índice basado en el método directo de Lyapunov para el diseño de un algoritmo de reprogramación en tiempo real para líneas de metro. En este estudio se utiliza una versión modificada de un modelo de espacio de estados en tiempo real discreto, que considera los efectos de saturación en la línea de metro. Una vez que el modelo de espacio de estados se ha obtenido, el método directo de Lyapunov se aplica con el fin de analizar la estabilidad del sistema de la línea de metro. Como resultado de este análisis no sólo se propone un nuevo índice de estabilidad, sino también la creación de tres zonas de estabilidad para indicar el estado actual del sistema. Finalmente, se presenta un nuevo algoritmo que permite la reprogramación del calendario de los trenes en tiempo real en presencia de perturbaciones medianas. Abstract: A new Lyapunov-based index for designing a rescheduling algorithm in real time for metro lines has been proposed in this paper. A modified real time discrete space state model which considers saturation effects in the metro line has been utilized in this study. Once the space state model has been obtained, the direct method of Lyapunov is applied in order to analyze the stability of the metro line system. As a result of this analysis not only a new stability index is proposed, but also the establishment of three stability zones to indicate the current state of the system. Finally, a new algorithm which allows the rescheduling of the timetable in the real time of the trains under presence of medium disturbances has been presented. Palabras clave: Sistema de metro, estabilidad de Lyapunov, planificación en tiempo real, Keywords: Metro system, Lyapunov stability, real time planning, traffic regulation

Nonlinear chaos-dynamical approach to analysis of atmospheric ...

Indian Academy of Sciences (India)

false nearest neighbors, Lyapunov's exponents, surrogate data, nonlinear prediction ... Chaotic dynamics; time series of the 222Rn concentration; universal complex ... tems is due to a number of applications, including the ..... Computer Engineering. ... Ternovsky,Quantum Systems in Physics, Chemistry, and. Biology, pp.

Random number generation based on digital differential chaos

KAUST Repository

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

2012-01-01

In this paper, we present a fully digital differential chaos based random number generator. The output of the digital circuit is proved to be chaotic by calculating the output time series maximum Lyapunov exponent. We introduce a new post processing

Resonance – Journal of Science Education | Indian Academy of ...

Indian Academy of Sciences (India)

Home; Journals; Resonance – Journal of Science Education; Volume 20; Issue 3 ... Lorenz system; deterministic chaos; unpredictability; Lyapunov exponent; fractals. ... Professor of Physics Dean Graduate Studies Indian Institute of Science Education & Research Dr Homi Bhabha Road Pashan, Pune 411008, India ...

A MONTE-CARLO METHOD FOR ESTIMATING THE CORRELATION EXPONENT

NARCIS (Netherlands)

MIKOSCH, T; WANG, QA

We propose a Monte Carlo method for estimating the correlation exponent of a stationary ergodic sequence. The estimator can be considered as a bootstrap version of the classical Hill estimator. A simulation study shows that the method yields reasonable estimates.

Adaptive Fuzzy-Lyapunov Controller Using Biologically Inspired Swarm Intelligence

Directory of Open Access Journals (Sweden)

Alejandro Carrasco Elizalde

2008-01-01

Full Text Available The collective behaviour of swarms produces smarter actions than those achieved by a single individual. Colonies of ants, flocks of birds and fish schools are examples of swarms interacting with their environment to achieve a common goal. This cooperative biological intelligence is the inspiration for an adaptive fuzzy controller developed in this paper. Swarm intelligence is used to adjust the parameters of the membership functions used in the adaptive fuzzy controller. The rules of the controller are designed using a computing-with-words approach called Fuzzy-Lyapunov synthesis to improve the stability and robustness of an adaptive fuzzy controller. Computing-with-words provides a powerful tool to manipulate numbers and symbols, like words in a natural language.

Fractal characters and hurst exponent of radon exhalation rate from uranium Tailings

International Nuclear Information System (INIS)

Hu Hanqiao; Tan Kaixuan; Li Chunguang; Lv Junwen; Liu Dong

2010-01-01

The uranium tailings radon exhalation is an important environmental problem. The change of the radon exhalation rate of uranium tailings with the time through laboratory experiments is measured, and the results show that the radon exhalation rate of the tailings change obviously with time in non-periodic oscillations. Applying fractal analysis to the radon exhalation rate time-series data by R/S method, the Hurst exponent of the entire time series data is 0.83, the fractal dimension is 1.17. Mobile Hurst exponent is between 0.5 and 0.8 in most cases. The Hurst exponent of the experiments in the later part are below 0.5. The exhalation rate of uranium tailings radon does not meet the long-term trend of random walk theory, the radon exhalation rate has long-term memory, but the short-term memory is not distinct. The radon exhalation from uranium tailings is a deterministic chaotic dynamics. (authors)

Explanation of the values of Hack's drainage basin, river length scaling exponent

Science.gov (United States)

Hunt, A. G.

2015-08-01

Percolation theory can be used to find water flow paths of least resistance. The application of percolation theory to drainage networks allows identification of the range of exponent values that describe the tortuosity of rivers in real river networks, which is then used to generate the observed scaling between drainage basin area and channel length, a relationship known as Hack's law. Such a theoretical basis for Hack's law allows interpretation of the range of exponent values based on an assessment of the heterogeneity of the substrate.

Density-scaling exponents and virial potential-energy correlation ...

Indian Academy of Sciences (India)

This paper investigates the relation between the density-scaling exponent γ and the virial potential energy correlation coefficient R at several thermodynamic state points in three dimensions for the generalized (2n, n) Lennard-Jones (LJ) system for n = 4, 9, 12, 18, as well as for the standard n = 6 LJ system in two,three, and ...

Spectrum-based estimators of the bivariate Hurst exponent

Czech Academy of Sciences Publication Activity Database

Krištoufek, Ladislav

2014-01-01

Roč. 90, č. 6 (2014), art. 062802 ISSN 1539-3755 R&D Projects: GA ČR(CZ) GP14-11402P Institutional support: RVO:67985556 Keywords : bivariate Hurst exponent * power- law cross-correlations * estimation Subject RIV: AH - Economics Impact factor: 2.288, year: 2014 http://library.utia.cas.cz/separaty/2014/E/kristoufek-0436818.pdf

Quasiperiodic transition to chaos in a plasma

International Nuclear Information System (INIS)

Weixing, D.; Huang Wei; Wang Xiaodong; Yu, C.X.

1993-01-01

The quasiperiodic transition to chaos in an undriven discharge plasma has been investigated. Results from the power spectrum and Lyapunov exponents quantitatively confirm the transition to chaos through quasiperiodicity. A low-dimension strange attractor has been found for this kind of plasma chaos

Chaos in schizophrenia associations, reality or metaphor?

Czech Academy of Sciences Publication Activity Database

Bob, P.; Šusta, M.; Chládek, Jan; Glaslová, K.; Paluš, Milan

2009-01-01

Roč. 73, č. 3 (2009), s. 179-185 ISSN 0167-8760 Institutional research plan: CEZ:AV0Z20650511; CEZ:AV0Z10300504 Keywords : Chaos * Schizophrenia * Associations * Electrodermal activity * Lyapunov exponent Subject RIV: FH - Neurology Impact factor: 3.045, year: 2009

The use of Lyapunov differential inequalities for estimating the transients of mechanical systems

Science.gov (United States)

Alyshev, A. S.; Dudarenko, N. A.; Melnikov, V. G.; Melnikov, G. I.

2018-05-01

In this paper we consider an autonomous mechanical system in a finite neighborhood of the zero of the phase space of states. The system is given as a matrix differential equation in the Cauchy form with the right-hand side of the polynomial structure. We propose a method for constructing a sequence of linear inhomogeneous differential inequalities for Lyapunov functions. As a result, we obtain estimates of transient processes in the form of functional inequalities.

A fast image encryption algorithm based on chaotic map

Science.gov (United States)

Liu, Wenhao; Sun, Kehui; Zhu, Congxu

2016-09-01

Derived from Sine map and iterative chaotic map with infinite collapse (ICMIC), a new two-dimensional Sine ICMIC modulation map (2D-SIMM) is proposed based on a close-loop modulation coupling (CMC) model, and its chaotic performance is analyzed by means of phase diagram, Lyapunov exponent spectrum and complexity. It shows that this map has good ergodicity, hyperchaotic behavior, large maximum Lyapunov exponent and high complexity. Based on this map, a fast image encryption algorithm is proposed. In this algorithm, the confusion and diffusion processes are combined for one stage. Chaotic shift transform (CST) is proposed to efficiently change the image pixel positions, and the row and column substitutions are applied to scramble the pixel values simultaneously. The simulation and analysis results show that this algorithm has high security, low time complexity, and the abilities of resisting statistical analysis, differential, brute-force, known-plaintext and chosen-plaintext attacks.

Polyhedral Lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the Jacobian is non-singular

NARCIS (Netherlands)

Blanchini, Franco; Giordano, G.

2017-01-01

For a vast class of dynamical networks, including chemical reaction networks (CRNs) with monotonic reaction rates, the existence of a polyhedral Lyapunov function (PLF) implies structural (i.e., parameter-free) local stability. Global structural stability is ensured under the additional

A Globally Stable Lyapunov Pointing and Rate Controller for the Magnetospheric MultiScale Mission (MMS)

Science.gov (United States)

Shah, Neerav

2011-01-01

The Magnetospheric MultiScale Mission (MMS) is scheduled to launch in late 2014. Its primary goal is to discover the fundamental plasma physics processes of reconnection in the Earth's magnetosphere. Each of the four MMS spacecraft is spin-stabilized at a nominal rate of 3 RPM. Traditional spin-stabilized spacecraft have used a number of separate modes to control nutation, spin rate, and precession. To reduce the number of modes and simplify operations, the Delta-H control mode is designed to accomplish nutation control, spin rate control, and precession control simultaneously. A nonlinear design technique, Lyapunov's method, is used to design the Delta-H control mode. A global spin rate controller selected as the baseline controller for MMS, proved to be insufficient due to an ambiguity in the attitude. Lyapunov's design method was used to solve this ambiguity, resulting in a controller that meets the design goals. Simulation results show the advantage of the pointing and rate controller for maneuvers larger than 90 deg and provide insight into the performance of this controller.

Order and chaos in the one-dimensional ϕ4 model: N-dependence and the Second Law of Thermodynamics

Science.gov (United States)

Hoover, William Graham; Aoki, Kenichiro

2017-08-01

We revisit the equilibrium one-dimensional ϕ4 model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These ;computationally stable; sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-numbered and even-numbered particles' coordinates and momenta are either identical or differ only in sign. ;Positive Lyapunov exponents; can and do result if an infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is introduced so that the motion expands into a four-dimensional phase space. In that extended space a positive exponent results. We formulate a standard initial condition for the investigation of the microcanonical chaotic number dependence of the model. We speculate on the uniqueness of the model's chaotic sea and on the connection of such collections of deterministic and time-reversible states to the Second Law of Thermodynamics.

Local Dynamic Stability Assessment of Motion Impaired Elderly Using Electronic Textile Pants.

Science.gov (United States)

Liu, Jian; Lockhart, Thurmon E; Jones, Mark; Martin, Tom

2008-10-01

A clear association has been demonstrated between gait stability and falls in the elderly. Integration of wearable computing and human dynamic stability measures into home automation systems may help differentiate fall-prone individuals in a residential environment. The objective of the current study was to evaluate the capability of a pair of electronic textile (e-textile) pants system to assess local dynamic stability and to differentiate motion-impaired elderly from their healthy counterparts. A pair of e-textile pants comprised of numerous e-TAGs at locations corresponding to lower extremity joints was developed to collect acceleration, angular velocity and piezoelectric data. Four motion-impaired elderly together with nine healthy individuals (both young and old) participated in treadmill walking with a motion capture system simultaneously collecting kinematic data. Local dynamic stability, characterized by maximum Lyapunov exponent, was computed based on vertical acceleration and angular velocity at lower extremity joints for the measurements from both e-textile and motion capture systems. Results indicated that the motion-impaired elderly had significantly higher maximum Lyapunov exponents (computed from vertical acceleration data) than healthy individuals at the right ankle and hip joints. In addition, maximum Lyapunov exponents assessed by the motion capture system were found to be significantly higher than those assessed by the e-textile system. Despite the difference between these measurement techniques, attaching accelerometers at the ankle and hip joints was shown to be an effective sensor configuration. It was concluded that the e-textile pants system, via dynamic stability assessment, has the potential to identify motion-impaired elderly.

a Comparison of Three Hurst Exponent Approaches to Predict Nascent Bubbles in S&P500 Stocks

Science.gov (United States)

Fernández-Martínez, M.; Sánchez-Granero, M. A.; Muñoz Torrecillas, M. J.; McKelvey, Bill

Since the pioneer contributions due to Vandewalle and Ausloos, the Hurst exponent has been applied by econophysicists as a useful indicator to deal with investment strategies when such a value is above or below 0.5, the Hurst exponent of a Brownian motion. In this paper, we hypothesize that the self-similarity exponent of financial time series provides a reliable indicator for herding behavior (HB) in the following sense: if there is HB, then the higher the price, the more the people will buy. This will generate persistence in the stocks which we shall measure by their self-similarity exponents. Along this work, we shall explore whether there is some connections between the self-similarity exponent of a stock (as a HB indicator) and the stock’s future performance under the assumption that the HB will last for some time. With this aim, three approaches to calculate the self-similarity exponent of a time series are compared in order to determine which performs best to identify the transition from random efficient market behavior to HB and hence, to detect the beginning of a bubble. Generalized Hurst Exponent, Detrended Fluctuation Analysis, and GM2 algorithms have been tested. Traditionally, researchers have focused on identifying the beginning of a crash. We study the beginning of the transition from efficient market behavior to a market bubble, instead. Our empirical results support that the higher (respectively the lower) the self-similarity index, the higher (respectively the lower) the mean of the price change, and hence, the better (respectively the worse) the performance of the corresponding stock. This would imply, as a consequence, that the transition process from random efficient market to HB has started. For experimentation purposes, S&P500 stock Index constituted our main data source.

ACCURATE ESTIMATES OF CHARACTERISTIC EXPONENTS FOR SECOND ORDER DIFFERENTIAL EQUATION

Institute of Scientific and Technical Information of China (English)

无

2009-01-01

In this paper, a second order linear differential equation is considered, and an accurate estimate method of characteristic exponent for it is presented. Finally, we give some examples to verify the feasibility of our result.

Large $N$ critical exponents for the chiral Heisenberg Gross-Neveu universality class

OpenAIRE

Gracey, J. A.

2018-01-01

We compute the large N critical exponents η, ηϕ and 1/ν in d dimensions in the chiral Heisenberg Gross-Neveu model to several orders in powers of 1/N. For instance, the large N conformal bootstrap method is used to determine η at O(1/N3) while the other exponents are computed to O(1/N2). Estimates of the exponents for a phase transition in graphene are given which are shown to be commensurate with other approaches. In particular the behavior of the exponents in 2

A semi-analytical treatment of xenon oscillations

International Nuclear Information System (INIS)

Zarei, M.; Minuchehr, A.; Ghaderi, R.

2017-01-01

Highlights: • A two-group two region kinetic core model is developed employing the eigenvalues separation index. • Poison dynamics are investigated within an adiabatic approach. • The overall nonlinear reactor model is recast into a linear time varying framework incorporating the matrix exponential numerical scheme. • The largest Lyapunov exponent is employed to analytically verify model stability. - Abstract: A novel approach is developed to investigate xenon oscillations within a two-group two-region coupled core reactor model incorporating thermal feedback and poison effects. Group-wise neutronic coupling coefficients between the core regions are calculated applying the associated fundamental and first mode eigenvalue separation values. The resultant nonlinear state space representation of the core behavior is quite suitable for evaluation of reactivity induced power transients such as load following operation. The model however comprises a multi-physics coupling of sub-systems with extremely distant relaxation times whose stiffness treatment inquire costly multistep implicit numerical methods. An adiabatic treatment of the sluggish poison dynamics is therefore proposed as a way out. The approach helps further investigate the nonlinear system within a linear time varying (LTV) framework whereby a semi-analytical framework is established. This scheme incorporates a matrix exponential analytical solution of the perturbed system as a quite efficient tool to study load following operation and control purposes. Poison dynamics are updated within larger intervals which exclude the need for specific numerical schemes of stiff systems. Simulation results of the axial offset conducted on a VVER-1000 reactor at the beginning (BOC) and the end of cycle (EOC) display quite acceptable results compared with available benchmarks. The LTV reactor model is further investigated within a stability analysis of the associated time varying systems at these two stages

An analysis of the financial crisis in the KOSPI market using Hurst exponents

Science.gov (United States)

Yim, Kyubin; Oh, Gabjin; Kim, Seunghwan

2014-09-01

Recently, the study of the financial crisis has progressed to include the concept of the complex system, thereby improving the understanding of this extreme event from a neoclassical economic perspective. To determine which variables are related to the financial event caused by the 2008 US subprime crisis using temporal correlations, we investigate the diverse variables that may explain the financial system. These variables include return, volatility, trading volume and inter-trade duration data sets within the TAQ data for 27 highly capitalized individual companies listed on the KOSPI stock market. During 2008 and 2009, the Hurst exponent for the return time series over the whole period was less than 0.5, and the Hurst exponents for other variables, such as the volatility, trading volume and inter-trade duration, were greater than 0.5. Additionally, we analyze the relationships between the variation of temporal correlation and market instability based on these Hurst exponents and the degree of multifractality. We find that for the data related to trading volume, the Hurst exponents do not allow us to detect changes in market status, such as changes from normal to abnormal status, whereas other variables, including the return, volatility and weekly inter-trade duration, indicate a significant change in market status after the Lehman Brothers' bankruptcy. In addition, the multifractality and the measurement defined by subtracting the Hurst exponent of the return time series from that of the volatility time series decrease sharply after the US subprime event and recover approximately 50 days after the Lehman Brothers' collapse. Our findings suggest that the temporal features of financial quantities in the TAQ data set and the market complexity perform very well at diagnosing financial market stability.

An accurate algorithm to calculate the Hurst exponent of self-similar processes

International Nuclear Information System (INIS)

Fernández-Martínez, M.; Sánchez-Granero, M.A.; Trinidad Segovia, J.E.; Román-Sánchez, I.M.

2014-01-01

In this paper, we introduce a new approach which generalizes the GM2 algorithm (introduced in Sánchez-Granero et al. (2008) [52]) as well as fractal dimension algorithms (FD1, FD2 and FD3) (first appeared in Sánchez-Granero et al. (2012) [51]), providing an accurate algorithm to calculate the Hurst exponent of self-similar processes. We prove that this algorithm performs properly in the case of short time series when fractional Brownian motions and Lévy stable motions are considered. We conclude the paper with a dynamic study of the Hurst exponent evolution in the S and P500 index stocks. - Highlights: • We provide a new approach to properly calculate the Hurst exponent. • This generalizes FD algorithms and GM2, introduced previously by the authors. • This method (FD4) results especially appropriate for short time series. • FD4 may be used in both unifractal and multifractal contexts. • As an empirical application, we show that S and P500 stocks improved their efficiency

An accurate algorithm to calculate the Hurst exponent of self-similar processes

Energy Technology Data Exchange (ETDEWEB)

Fernández-Martínez, M., E-mail: fmm124@ual.es [Department of Mathematics, Faculty of Science, Universidad de Almería, 04120 Almería (Spain); Sánchez-Granero, M.A., E-mail: misanche@ual.es [Department of Mathematics, Faculty of Science, Universidad de Almería, 04120 Almería (Spain); Trinidad Segovia, J.E., E-mail: jetrini@ual.es [Department of Accounting and Finance, Faculty of Economics and Business, Universidad de Almería, 04120 Almería (Spain); Román-Sánchez, I.M., E-mail: iroman@ual.es [Department of Accounting and Finance, Faculty of Economics and Business, Universidad de Almería, 04120 Almería (Spain)

2014-06-27

In this paper, we introduce a new approach which generalizes the GM2 algorithm (introduced in Sánchez-Granero et al. (2008) [52]) as well as fractal dimension algorithms (FD1, FD2 and FD3) (first appeared in Sánchez-Granero et al. (2012) [51]), providing an accurate algorithm to calculate the Hurst exponent of self-similar processes. We prove that this algorithm performs properly in the case of short time series when fractional Brownian motions and Lévy stable motions are considered. We conclude the paper with a dynamic study of the Hurst exponent evolution in the S and P500 index stocks. - Highlights: • We provide a new approach to properly calculate the Hurst exponent. • This generalizes FD algorithms and GM2, introduced previously by the authors. • This method (FD4) results especially appropriate for short time series. • FD4 may be used in both unifractal and multifractal contexts. • As an empirical application, we show that S and P500 stocks improved their efficiency.

Global chaos synchronization of new chaotic systems via nonlinear control

International Nuclear Information System (INIS)

Chen, H.-K.

2005-01-01

Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V (e) = 1/2e T e. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach

Parameter-dependent PWQ Lyapunov function stability criteria for uncertain piecewise linear systems

Directory of Open Access Journals (Sweden)

Morten Hovd

2018-01-01

Full Text Available The calculation of piecewise quadratic (PWQ Lyapunov functions is addressed in view of stability analysis of uncertain piecewise linear dynamics. As main contribution, the linear matrix inequality (LMI approach proposed in (Johansson and Rantzer, 1998 for the stability analysis of PWL and PWA dynamics is extended to account for parametric uncertainty based on a improved relaxation technique. The results are applied for the analysis of a Phase Locked Loop (PLL benchmark and the ability to guarantee a stability region in the parameter space well beyond the state of the art is demonstrated.

Zipf's law and influential factors of the Pareto exponent of the city size distribution: Evidence from China

OpenAIRE

GAO Hongying; WU Kangping

2007-01-01

This paper estimates the Pareto exponent of the city size (population size and economy size) distribution, all provinces, and three regions in China in 1997, 2000 and 2003 by OLS, comparatively analyzes the Pareto exponent cross section and times, and empirically analyzes the factors which impacts on the Pareto exponents of provinces. Our analyses show that the size distributions of cities in China follow the Pareto distribution and are of structural features. Variations in the value of the P...

Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement

Science.gov (United States)

Sikora, Grzegorz; Teuerle, Marek; Wyłomańska, Agnieszka; Grebenkov, Denis

2017-08-01

The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.

Numerical simulation of torus breakdown to chaos in an atmospheric-pressure dielectric barrier discharge

International Nuclear Information System (INIS)

Zhang, J.; Wang, Y. H.; Wang, D. Z.

2013-01-01

Understanding the routes to chaos occurring in atmospheric-pressure dielectric barrier discharge systems by changing controlling parameters is very important to predict and control the dynamical behaviors. In this paper, a route of a quasiperiodic torus to chaos via the strange nonchaotic attractor is observed in an atmospheric-pressure dielectric barrier discharge driven by triangle-wave voltage. By increasing the driving frequency, the discharge system first bifurcates to a quasiperiodic torus from a stable single periodic state, and then torus and phase-locking periodic state appear and disappear alternately. In the meantime, the torus becomes increasingly wrinkling and stretching, and gradually approaches a fractal structure with the nonpositive largest Lyapunov exponent, i.e., a strange nonchaotic attractor. After that, the discharge system enters into chaotic state. If the driving frequency is further increased, another well known route of period-doubling bifurcation to chaos is also observed

Numerical simulation of torus breakdown to chaos in an atmospheric-pressure dielectric barrier discharge

Energy Technology Data Exchange (ETDEWEB)

Zhang, J.; Wang, Y. H.; Wang, D. Z. [Key Laboratory of Materials Modification by Laser, Electron, and Ion Beams, School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024 (China)

2013-08-15

Understanding the routes to chaos occurring in atmospheric-pressure dielectric barrier discharge systems by changing controlling parameters is very important to predict and control the dynamical behaviors. In this paper, a route of a quasiperiodic torus to chaos via the strange nonchaotic attractor is observed in an atmospheric-pressure dielectric barrier discharge driven by triangle-wave voltage. By increasing the driving frequency, the discharge system first bifurcates to a quasiperiodic torus from a stable single periodic state, and then torus and phase-locking periodic state appear and disappear alternately. In the meantime, the torus becomes increasingly wrinkling and stretching, and gradually approaches a fractal structure with the nonpositive largest Lyapunov exponent, i.e., a strange nonchaotic attractor. After that, the discharge system enters into chaotic state. If the driving frequency is further increased, another well known route of period-doubling bifurcation to chaos is also observed.

Deterministic nonlinear characteristics of in vivo blood flow velocity and arteriolar diameter fluctuations

International Nuclear Information System (INIS)

Parthimos, D; Osterloh, K; Pries, A R; Griffith, T M

2004-01-01

We have performed a nonlinear analysis of fluctuations in red cell velocity and arteriolar calibre in the mesenteric bed of the anaesthetized rat. Measurements were obtained under control conditions and during local superfusion with N G -nitro-L-arginine (L-NNA, 30 μM) and tetrabutylammonium (TBA, 0.1 mM), which suppress NO synthesis and block Ca 2+ activated K + channels (K Ca ), respectively. Time series were analysed by calculating correlation dimensions and largest Lyapunov exponents. Both statistics were higher for red cell velocity than diameter fluctuations, thereby potentially differentiating between global and local mechanisms that regulate microvascular flow. Evidence for underlying nonlinear structure was provided by analysis of surrogate time series generated from the experimental data following randomization of Fourier phase. Complexity indices characterizing time series under control conditions were in general higher than those derived from data obtained during superfusion with L-NNA and TBA

Complexity Analysis of a Master-Slave Oligopoly Model and Chaos Control

Directory of Open Access Journals (Sweden)

Junhai Ma

2014-01-01

Full Text Available We establish a master-slave oligopoly game model with an upstream monopoly whose output is considered and two downstream oligopolies whose prices are considered. The existence and the local stable region of the Nash equilibrium point are investigated. The complex dynamic properties, such as bifurcation and chaos, are analyzed using bifurcation diagrams, the largest Lyapunov exponent diagrams, and the strange attractor graph. We further analyze the long-run average profit of the three firms and find that they are all optimal in the stable region. In addition, delay feedback control method and limiter control method are used in nondelayed model to control chaos. Furthermore, a delayed master-slave oligopoly game model is considered, and the three firms’ profit in various conditions is analyzed. We find that suitable delayed parameters are important for eliminating chaos and maximizing the profit of the players.

Change of State of a Dynamical Unit in the Transition of Coherence

International Nuclear Information System (INIS)

Yang Yan-Jin; Du Ru-Hai; Wang Sheng-Jun; Jin Tao; Qu Shi-Xian

2015-01-01

The change of state of one map in the network of nonlocal coupled logistic maps at the transition of coherence is studied. With the increase of coupling strength, the network dynamics transits from the incoherent state into the coherent state. In the process, the iteration of the map first changes from chaos to period state, then from periodic to chaotic state again. For the periodic doubling bifurcations, similar to an isolated map, the largest Lyapunov exponent tends to zero from a negative value. However, the states of coupled maps exhibit complex behavior rather than converge to a few fixed values. The behavior brings a new chimera state of coupled logistic maps. The bifurcation diagram is identical to the phase order of maps iterations. For the bifurcation between 1-band and multi-band chaos, the symmetry of chaotic bands emerges and the transition of the order of iteration direction occurs

Hurst's Exponent Determination for Radial Distribution Functions of In, Sn and In-40 wt%Sn Melt

International Nuclear Information System (INIS)

Zhou Yong-Zhi; Li Mei; Geng Hao-Ran; Yang Zhong-Xi; Sun Chun-Jing

2011-01-01

Hurst's exponent of radial distribution functions (RDFs) within the short-range scope of In, Sn and In-40 wt % Sn melts are determined by the rescaled range analysis method. Hurst's exponents H are between 0.94 and 0.97, which display long-range dependence. Within short-range scope, the number of particles from a reference particle belongs to fractional Brownian motion. After RDF serials are randomly scrambled, Hurst's exponents all dramatically dropped, which proves long-range dependence. H irregularly varies as the temperature rises, but the change tendency is not consistent with the correlation radius r c . (general)

Bayesian Estimation of the Logistic Positive Exponent IRT Model

Science.gov (United States)

Bolfarine, Heleno; Bazan, Jorge Luis

2010-01-01

A Bayesian inference approach using Markov Chain Monte Carlo (MCMC) is developed for the logistic positive exponent (LPE) model proposed by Samejima and for a new skewed Logistic Item Response Theory (IRT) model, named Reflection LPE model. Both models lead to asymmetric item characteristic curves (ICC) and can be appropriate because a symmetric…

Quantitative hyperbolicity estimates in one-dimensional dynamics

International Nuclear Information System (INIS)

Day, S; Kokubu, H; Pilarczyk, P; Luzzatto, S; Mischaikow, K; Oka, H

2008-01-01

We develop a rigorous computational method for estimating the Lyapunov exponents in uniformly expanding regions of the phase space for one-dimensional maps. Our method uses rigorous numerics and graph algorithms to provide results that are mathematically meaningful and can be achieved in an efficient way

A new chaotic Hopfield network with piecewise linear activation function

International Nuclear Information System (INIS)

Peng-Sheng, Zheng; Wan-Sheng, Tang; Jian-Xiong, Zhang

2010-01-01

This paper presents a new chaotic Hopfield network with a piecewise linear activation function. The dynamic of the network is studied by virtue of the bifurcation diagram, Lyapunov exponents spectrum and power spectrum. Numerical simulations show that the network displays chaotic behaviours for some well selected parameters

Non-universality of critical exponents in the paraconductivity of short-coherence-length superconductors

International Nuclear Information System (INIS)

Gauzzi, A.

1993-01-01

The Aslamazov-Larkin paraconductivity term is calculated in the case of sufficiently small superconducting coherence length. It is found that the critical exponent of paraconductivity depends on the short-wavelength cut-off of the fluctuation spectrum in the whole Ginzburg-Landau mean-field region. Hence, it is predicted that the Aslamazov-Larkin universal relation between the critical exponent of paraconductivity and the dimensionality of the superconducting state is no longer valid in short-coherence-length superconductors. This prediction is confirmed by paraconductivity measurements on cuprate superconductors. (orig.)

Synchronization of a new fractional-order hyperchaotic system

International Nuclear Information System (INIS)

Wu Xiangjun; Lu Hongtao; Shen Shilei

2009-01-01

In this letter, a new fractional-order hyperchaotic system is proposed. By utilizing the fractional calculus theory and computer simulations, it is found that hyperchaos exists in the new fractional-order four-dimensional system with order less than 4. The lowest order to have hyperchaos in this system is 2.88. The results are validated by the existence of two positive Lyapunov exponents. Using the pole placement technique, a nonlinear state observer is designed to synchronize a class of nonlinear fractional-order systems. The observer method is used to synchronize two identical fractional-order hyperchaotic systems. In addition, the active control technique is applied to synchronize the new fractional-order hyperchaotic system and the fractional-order Chen hyperchaotic system. The two schemes, based on the stability theory of the fractional-order system, are rather simple, theoretically rigorous and convenient to realize synchronization. They do not require the computation of the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the proposed synchronization schemes.

Stochastic bifurcation in a model of love with colored noise

Science.gov (United States)

Yue, Xiaokui; Dai, Honghua; Yuan, Jianping

2015-07-01

In this paper, we wish to examine the stochastic bifurcation induced by multiplicative Gaussian colored noise in a dynamical model of love where the random factor is used to describe the complexity and unpredictability of psychological systems. First, the dynamics in deterministic love-triangle model are considered briefly including equilibrium points and their stability, chaotic behaviors and chaotic attractors. Then, the influences of Gaussian colored noise with different parameters are explored such as the phase plots, top Lyapunov exponents, stationary probability density function (PDF) and stochastic bifurcation. The stochastic P-bifurcation through a qualitative change of the stationary PDF will be observed and bifurcation diagram on parameter plane of correlation time and noise intensity is presented to find the bifurcation behaviors in detail. Finally, the top Lyapunov exponent is computed to determine the D-bifurcation when the noise intensity achieves to a critical value. By comparison, we find there is no connection between two kinds of stochastic bifurcation.

Bubbling route to strange nonchaotic attractor in a nonlinear series LCR circuit with a nonsinusoidal force.

Science.gov (United States)

Senthilkumar, D V; Srinivasan, K; Thamilmaran, K; Lakshmanan, M

2008-12-01

We identify an unconventional route to the creation of a strange nonchaotic attractor (SNA) in a quasiperiodically forced electronic circuit with a nonsinusoidal (square wave) force as one of the quasiperiodic forces through numerical and experimental studies. We find that bubbles appear in the strands of the quasiperiodic attractor due to the instability induced by the additional square-wave-type force. The bubbles then enlarge and get increasingly wrinkled as a function of the control parameter. Finally, the bubbles get extremely wrinkled (while the remaining parts of the strands of the torus remain largely unaffected) resulting in the creation of the SNA; we term this the bubbling route to the SNA. We characterize and confirm this creation from both experimental and numerical data using maximal Lyapunov exponents and their variance, Poincaré maps, Fourier amplitude spectra, and spectral distribution functions. We also strongly confirm the creation of a SNA via the bubbling route by the distribution of the finite-time Lyapunov exponents.

Invariant graphs of a family of non-uniformly expanding skew products over Markov maps

Science.gov (United States)

Walkden, C. P.; Withers, T.

2018-06-01

We consider a family of skew-products of the form where T is a continuous, expanding, locally eventually onto Markov map and is a family of homeomorphisms of . A function is said to be an invariant graph if is an invariant set for the skew-product; equivalently, u(T(x))  =  g x (u(x)). A well-studied problem is to consider the existence, regularity and dimension-theoretic properties of such functions, usually under strong contraction or expansion conditions (in terms of Lyapunov exponents or partial hyperbolicity) in the fibre direction. Here we consider such problems in a setting where the Lyapunov exponent in the fibre direction is zero on a set of periodic orbits but expands except on a neighbourhood of these periodic orbits. We prove that u either has the structure of a ‘quasi-graph’ (or ‘bony graph’) or is as smooth as the dynamics, and we give a criteria for this to happen.

Introduction to linear systems of differential equations

CERN Document Server

Adrianova, L Ya

1995-01-01

The theory of linear systems of differential equations is one of the cornerstones of the whole theory of differential equations. At its root is the concept of the Lyapunov characteristic exponent. In this book, Adrianova presents introductory material and further detailed discussions of Lyapunov exponents. She also discusses the structure of the space of solutions of linear systems. Classes of linear systems examined are from the narrowest to widest: 1)�autonomous, 2)�periodic, 3)�reducible to autonomous, 4)�nearly reducible to autonomous, 5)�regular. In addition, Adrianova considers the following: stability of linear systems and the influence of perturbations of the coefficients on the stability the criteria of uniform stability and of uniform asymptotic stability in terms of properties of the solutions several estimates of the growth rate of solutions of a linear system in terms of its coefficients How perturbations of the coefficients change all the elements of the spectrum of the system is defin...

Lyapunov-based Stability of Feedback Interconnections of Negative Imaginary Systems

KAUST Repository

Ghallab, Ahmed G.

2017-10-19

Feedback control systems using sensors and actuators such as piezoelectric sensors and actuators, micro-electro-mechanical systems (MEMS) sensors and opto-mechanical sensors, are allowing new advances in designing such high precision technologies. The negative imaginary control systems framework allows for robust control design for such high precision systems in the face of uncertainties due to unmodelled dynamics. The stability of the feedback interconnection of negative imaginary systems has been well established in the literature. However, the proofs of stability feedback interconnection which are used in some previous papers have a shortcoming due to a matrix inevitability issue. In this paper, we provide a new and correct Lyapunov-based proof of one such result and show that the result is still true.

Lyapunov-based Stability of Feedback Interconnections of Negative Imaginary Systems

KAUST Repository

Ghallab, Ahmed G.; Mabrok, Mohamed; Petersen, Ian R.

2017-01-01

Feedback control systems using sensors and actuators such as piezoelectric sensors and actuators, micro-electro-mechanical systems (MEMS) sensors and opto-mechanical sensors, are allowing new advances in designing such high precision technologies. The negative imaginary control systems framework allows for robust control design for such high precision systems in the face of uncertainties due to unmodelled dynamics. The stability of the feedback interconnection of negative imaginary systems has been well established in the literature. However, the proofs of stability feedback interconnection which are used in some previous papers have a shortcoming due to a matrix inevitability issue. In this paper, we provide a new and correct Lyapunov-based proof of one such result and show that the result is still true.

On fractality and chaos in Moroccan family business stock returns and volatility

Science.gov (United States)

Lahmiri, Salim

2017-05-01

The purpose of this study is to examine existence of fractality and chaos in returns and volatilities of family business companies listed on the Casablanca Stock Exchange (CSE) in Morocco, and also in returns and volatility of the CSE market index. Detrended fluctuation analysis based Hurst exponent and fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) model are used to quantify fractality in returns and volatility time series respectively. Besides, the largest Lyapunov exponent is employed to quantify chaos in both time series. The empirical results from sixteen family business companies follow. For return series, fractality analysis show that most of family business returns listed on CSE exhibit anti-persistent dynamics, whilst market returns have persistent dynamics. Besides, chaos tests show that business family stock returns are not chaotic while market returns exhibit evidence of chaotic behaviour. For volatility series, fractality analysis shows that most of family business stocks and market index exhibit long memory in volatility. Furthermore, results from chaos tests show that volatility of family business returns is not chaotic, whilst volatility of market index is chaotic. These results may help understanding irregularities patterns in Moroccan family business stock returns and volatility, and how they are different from market dynamics.

On Hurst exponent estimation under heavy-tailed distributions

Czech Academy of Sciences Publication Activity Database

Baruník, Jozef; Krištoufek, Ladislav

2010-01-01

Roč. 389, č. 18 (2010), s. 3844-3855 ISSN 0378-4371 R&D Projects: GA ČR GA402/09/0965 Grant - others:GA UK(CZ) 118310; GA UK(CZ) 46108 Institutional research plan: CEZ:AV0Z10750506 Keywords : high frequency data analysis * heavy tails * Hurst exponent Subject RIV: AH - Economics Impact factor: 1.521, year: 2010 http://library.utia.cas.cz/separaty/2010/E/barunik-0343525.pdf

Large-Signal Lyapunov-Based Stability Analysis of DC/AC Inverters and Inverter-Based Microgrids

Science.gov (United States)

Kabalan, Mahmoud

Microgrid stability studies have been largely based on small-signal linearization techniques. However, the validity and magnitude of the linearization domain is limited to small perturbations. Thus, there is a need to examine microgrids with large-signal nonlinear techniques to fully understand and examine their stability. Large-signal stability analysis can be accomplished by Lyapunov-based mathematical methods. These Lyapunov methods estimate the domain of asymptotic stability of the studied system. A survey of Lyapunov-based large-signal stability studies showed that few large-signal studies have been completed on either individual systems (dc/ac inverters, dc/dc rectifiers, etc.) or microgrids. The research presented in this thesis addresses the large-signal stability of droop-controlled dc/ac inverters and inverter-based microgrids. Dc/ac power electronic inverters allow microgrids to be technically feasible. Thus, as a prelude to examining the stability of microgrids, the research presented in Chapter 3 analyzes the stability of inverters. First, the 13 th order large-signal nonlinear model of a droop-controlled dc/ac inverter connected to an infinite bus is presented. The singular perturbation method is used to decompose the nonlinear model into 11th, 9th, 7th, 5th, 3rd and 1st order models. Each model ignores certain control or structural components of the full order model. The aim of the study is to understand the accuracy and validity of the reduced order models in replicating the performance of the full order nonlinear model. The performance of each model is studied in three different areas: time domain simulations, Lyapunov's indirect method and domain of attraction estimation. The work aims to present the best model to use in each of the three domains of study. Results show that certain reduced order models are capable of accurately reproducing the performance of the full order model while others can be used to gain insights into those three areas of

A modified approach to predict pore pressure using the D exponent method: An example from the Carbonera Formation, Colombia

International Nuclear Information System (INIS)

Solano, Yully P; Uribe, Rodolfo; Frydman, Marcelo; Saavedra, Nestor F; Calderon, Zuly H

2007-01-01

The methodology for the pore pressure prediction known as an exponent is o function of an exponent of adjustment that was originally defined for the Gulf of Mexico (Jorden and Shirley, 1966; Eaton, 1972). A limiting factor of this methodology is the definition of the normal compaction trend (NCT), which needs to be interpreted from the data (Mouchet and Mitchell, 1989). In this study, the D exponent methodology was modified to make it applicable to the Oligocene Carbonera Formation in an oil field of the llanos foothills Colombia. The approach consisted of calculating the ratio between affective stress and the D exponent of each wall, in order to find a robust NCT for the entire field, thus reducing subjectivity in the traditional d exponent methodology. Pore pressure determinations from Measured Direct Tests (MDT) at one wall confirm the predictive capability of our approach

Subgeometric Ergodicity Analysis of Continuous-Time Markov Chains under Random-Time State-Dependent Lyapunov Drift Conditions

Directory of Open Access Journals (Sweden)

Mokaedi V. Lekgari

2014-01-01

Full Text Available We investigate random-time state-dependent Foster-Lyapunov analysis on subgeometric rate ergodicity of continuous-time Markov chains (CTMCs. We are mainly concerned with making use of the available results on deterministic state-dependent drift conditions for CTMCs and on random-time state-dependent drift conditions for discrete-time Markov chains and transferring them to CTMCs.

Fast and unbiased estimator of the time-dependent Hurst exponent

Science.gov (United States)

Pianese, Augusto; Bianchi, Sergio; Palazzo, Anna Maria

2018-03-01

We combine two existing estimators of the local Hurst exponent to improve both the goodness of fit and the computational speed of the algorithm. An application with simulated time series is implemented, and a Monte Carlo simulation is performed to provide evidence of the improvement.

Directional maximum likelihood self-estimation of the path-loss exponent

NARCIS (Netherlands)

Hu, Y.; Leus, G.J.T.; Dong, Min; Zheng, Thomas Fang

2016-01-01

The path-loss exponent (PLE) is a key parameter in wireless propagation channels. Therefore, obtaining the knowledge of the PLE is rather significant for assisting wireless communications and networking to achieve a better performance. Most existing methods for estimating the PLE not only require

Modelling chaotic Hamiltonian systems as a Markov Chain ...

African Journals Online (AJOL)

The behaviour of chaotic Hamiltonian system has been characterised qualitatively in recent times by its appearance on the Poincaré section and quantitatively by the Lyapunov exponent. Studying the dynamics of the two chaotic Hamiltonian systems: the Henon-Heiles system and non-linearly coupled oscillators as their ...

Study of chaos in chaotic satellite systems

Indian Academy of Sciences (India)

Lyapunov exponents are estimated. From these studies, chaosin satellite system has been established. Solution of equations of motion of the satellite system are drawn in the form of three-dimensional, two-dimensional and time series phase portraits. Phase portraits and time series display the chaotic nature of the ...

Dynamic behaviours and control of fractional-order memristor-based ...

Indian Academy of Sciences (India)

Dynamics of fractional-order memristor circuit system and its control are investigated in this paper. With the help of stability theory of fractional-order systems, stability of its equilibrium points is analysed. Then, the chaotic behaviours are validated using phase portraits, the Lyapunov exponents and bifurcation diagrams with ...

Pramana – Journal of Physics | Indian Academy of Sciences

Indian Academy of Sciences (India)

Volume 90 Issue 1 January 2018 pp 13 Research Article. Study of chaos in chaotic satellite systems · AYUB KHAN SANJAY KUMAR · More Details Abstract Fulltext PDF. In this paper,we study the qualitative behaviour of satellite systems using bifurcation diagrams, Poincaré section, Lyapunov exponents, dissipation, ...

Fractal tracer distributions in turbulent field theories

DEFF Research Database (Denmark)

Hansen, J. Lundbek; Bohr, Tomas

1998-01-01

We study the motion of passive tracers in a two-dimensional turbulent velocity field generated by the Kuramoto-Sivashinsky equation. By varying the direction of the velocity-vector with respect to the field-gradient we can continuously vary the two Lyapunov exponents for the particle motion and t...

Indian Academy of Sciences Conference Series | Indian Academy of ...

Indian Academy of Sciences (India)

Also, circuit simulation of the electronic generator is provided using the NI Multisim environment. Portraits of attractors, waveforms of generated oscillations, Lyapunov exponents, and spectra are considered and found to be in good correspondence for the dynamics on the attractive sets of the self-oscillatory systems and for ...

Chaos in the delay logistic equation with discontinuous delays

International Nuclear Information System (INIS)

Sen, Ayan; Mukherjee, Debasis

2009-01-01

This paper analyzes a delay logistic equation which models a feedback control problem. Interval map associated to the system is derived. By calculating Lyapunov exponent, we indicate stable orbit and chaotic phenomenon respectively. The results are verified through computer simulation. We identify the parameter which controls the dynamics.

Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent

International Nuclear Information System (INIS)

Zhikov, Vasilii V; Pastukhova, Svetlana E

2008-01-01

Elliptic equations of p(x)-Laplacian type are investigated. There is a well-known logarithmic condition on the modulus of continuity of the nonlinearity exponent p(x), which ensures that a Laplacian with variable order of nonlinearity inherits many properties of the usual p-Laplacian of constant order. One of these is the so-called improved integrability of the gradient of the solution. It is proved in this paper that this property holds also under a slightly more general condition on the exponent p(x), although then the improvement of integrability is logarithmic rather than power-like. The method put forward is based on a new generalization of Gehring's lemma, which relies upon the reverse Hoelder inequality 'with increased support and exponent on the right-hand side'. A counterexample is constructed that reveals the extent to which the condition on the modulus of continuity obtained is sharp. Bibliography: 28 titles.

The Exponent of High-frequency Source Spectral Falloff and Contribution to Source Parameter Estimates

Science.gov (United States)

Kiuchi, R.; Mori, J. J.

2015-12-01

As a way to understand the characteristics of the earthquake source, studies of source parameters (such as radiated energy and stress drop) and their scaling are important. In order to estimate source parameters reliably, often we must use appropriate source spectrum models and the omega-square model is most frequently used. In this model, the spectrum is flat in lower frequencies and the falloff is proportional to the angular frequency squared. However, Some studies (e.g. Allmann and Shearer, 2009; Yagi et al., 2012) reported that the exponent of the high frequency falloff is other than -2. Therefore, in this study we estimate the source parameters using a spectral model for which the falloff exponent is not fixed. We analyze the mainshock and larger aftershocks of the 2008 Iwate-Miyagi Nairiku earthquake. Firstly, we calculate the P wave and SH wave spectra using empirical Green functions (EGF) to remove the path effect (such as attenuation) and site effect. For the EGF event, we select a smaller earthquake that is highly-correlated with the target event. In order to obtain the stable results, we calculate the spectral ratios using a multitaper spectrum analysis (Prieto et al., 2009). Then we take a geometric mean from multiple stations. Finally, using the obtained spectra ratios, we perform a grid search to determine the high frequency falloffs, as well as corner frequency of both of events. Our results indicate the high frequency falloff exponent is often less than 2.0. We do not observe any regional, focal mechanism, or depth dependencies for the falloff exponent. In addition, our estimated corner frequencies and falloff exponents are consistent between the P wave and SH wave analysis. In our presentation, we show differences in estimated source parameters using a fixed omega-square model and a model allowing variable high-frequency falloff.

Chaotic Bohmian trajectories for stationary states

International Nuclear Information System (INIS)

Cesa, Alexandre; Martin, John; Struyve, Ward

2016-01-01

In Bohmian mechanics, the nodes of the wave function play an important role in the generation of chaos. However, so far, most of the attention has been on moving nodes; little is known about the possibility of chaos in the case of stationary nodes. We address this question by considering stationary states, which provide the simplest examples of wave functions with stationary nodes. We provide examples of stationary wave functions for which there is chaos, as demonstrated by numerical computations, for one particle moving in three spatial dimensions and for two and three entangled particles in two dimensions. Our conclusion is that the motion of the nodes is not necessary for the generation of chaos. What is important is the overall complexity of the wave function. That is, if the wave function, or rather its phase, has a complex spatial variation, it will lead to complex Bohmian trajectories and hence to chaos. Another aspect of our work concerns the average Lyapunov exponent, which quantifies the overall amount of chaos. Since it is very hard to evaluate the average Lyapunov exponent analytically, which is often computed numerically, it is useful to have simple quantities that agree well with the average Lyapunov exponent. We investigate possible correlations with quantities such as the participation ratio and different measures of entanglement, for different systems and different families of stationary wave functions. We find that these quantities often tend to correlate to the amount of chaos. However, the correlation is not perfect, because, in particular, these measures do not depend on the form of the basis states used to expand the wave function, while the amount of chaos does. (paper)

Numerical difficulties to obtain 3-d critical exponents from platonic solids

International Nuclear Information System (INIS)

Alcaraz, F.C.; Herrmann, H.J.

1985-01-01

The possibility to extract critical exponents of 3-d systems exploring the mass gap amplitudes of platonic solids is tested. For the Ising model the proposed method does not work for numerical reasons. (Author) [pt

A development of the direct Lyapunov method for the analysis of transient stability of a system of synchronous generators based on the determination of non- stable equilibria on a multidimensional sphere

Directory of Open Access Journals (Sweden)

A. V. Stepanov

2014-01-01

Full Text Available A development of the direct Lyapunov method for the analysis of transient stability of a system of synchronous generators based on the determination of non- stable equilibria on a multidimensional sphere.We consider the problem of transient stability analysis for a system of synchronous generators under the action of strong perturbations. The aim of our work is to develop methods to analyze a transient stability of the system of synchronous generators, which allow getting trustworthy results on reserve transient stability under different perturbations. For the analysis of transient stability, we use the direct Lyapunov method.One of the problems for this method application is to find the Lypunov function that well reflects the properties of a parallel system of synchronous generators. The most reliable results were obtained when the analysis of transient stability was performed with a Lyapunov function of energy type. Another problem for application of the direct Lyapunov method is to determine the critical value of the Lyapunov function, which requires finding the non-stable equilibria of the system. Determination of the non-stable equilibria requires studying the Lyapunov function in a multidimensional space in a neighborhood of a stable equilibrium for the post-breakdown system; this is a complicated non-linear problem.In the paper, we propose a method for determination of the non-stable equilibria on a multidimensional sphere. The method is based on a search of a minimum of the Lyapunov function on a multidimensional sphere the center of which is a stable equilibrium. Our method allows, comparing with the other, e.g., gradient methods, reliable finding a non-stable equilibrium and calculating the critical value. The reliability of our method is proved by numerical experiments. The developed methods and a program realized in a MATLAB package can be recommended for design of a post-breakdown control system of synchronous generators or as a

Investigating textural controls on Archie's porosity exponent using process-based, pore-scale modelling

Science.gov (United States)

Niu, Q.; Zhang, C.

2017-12-01

Archie's law is an important empirical relationship linking the electrical resistivity of geological materials to their porosity. It has been found experimentally that the porosity exponent m in Archie's law in sedimentary rocks might be related to the degree of cementation, and therefore m is termed as "cementation factor" in most literatures. Despite it has been known for many years, there is lack of well-accepted physical interpretations of the porosity exponent. Some theoretical and experimental evidences have also shown that m may be controlled by the particle and/or pore shape. In this study, we conduct a pore-scale modeling of the porosity exponent that incorporates different geological processes. The evolution of m of eight synthetic samples with different particle sizes and shapes are calculated during two geological processes, i.e., compaction and cementation. The numerical results show that in dilute conditions, m is controlled by the particle shape. As the samples deviate from dilute conditions, m increases gradually due to the strong interaction between particles. When the samples are at static equilibrium, m is noticeably larger than its values at dilution condition. The numerical simulation results also show that both geological compaction and cementation induce a significant increase in m. In addition, the geometric characteristics of these samples (e.g., pore space/throat size, and their distributions) during compaction and cementation are also calculated. Preliminary analysis shows a unique correlation between the pore size broadness and porosity exponent for all eight samples. However, such a correlation is not found between m and other geometric characteristics.

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

Science.gov (United States)

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.

Lyapunov-based decentralized control of a rougher flotation phenomenological simulator

International Nuclear Information System (INIS)

Benaskeur, A.R.; Desbiens, A.

1999-01-01

In this paper a new approach to decentralized control of linear two-by-two plants is presented. The novelty lies in the use of a modified control function of Lyapunov and the introduction of an integral action in each manipulated variable, to ensure zero tracking errors. An appropriate choice of the regulated errors, allows the elimination of the cross terms in the obtained backstepping-based multivariable controller. It will be proven that if the H ∞ -norm of the plant interaction quotient is less than one, the centralized controller can be split up into two independent scalar output feedback regulators. Under these conditions, the global stability and zero tracking errors will still be guaranteed. The developed scheme is successfully applied to the control of a rougher flotation phenomenological simulator. (author)

Quality of Gaussian basis sets: direct optimization of orbital exponents by the method of conjugate gradients

International Nuclear Information System (INIS)

Kari, R.E.; Mezey, P.G.; Csizmadia, I.G.

1975-01-01

Expressions are given for calculating the energy gradient vector in the exponent space of Gaussian basis sets and a technique to optimize orbital exponents using the method of conjugate gradients is described. The method is tested on the (9/sups/5/supp/) Gaussian basis space and optimum exponents are determined for the carbon atom. The analysis of the results shows that the calculated one-electron properties converge more slowly to their optimum values than the total energy converges to its optimum value. In addition, basis sets approximating the optimum total energy very well can still be markedly improved for the prediction of one-electron properties. For smaller basis sets, this improvement does not warrant the necessary expense

Critical exponents in the transition to chaos in one-dimensional

Indian Academy of Sciences (India)

We report the numerically evaluated critical exponents associated with the scaling of generalized fractal dimensions during the transition from order to chaos. The analysis is carried out in detail in the context of unimodal and bimodal maps representing typical one-dimensional discrete dynamical systems. The behavior of ...

Dynamic behavior and chaos control in a complex Riccati-type map ...

African Journals Online (AJOL)

This paper is devoted to analyze the dynamic behavior of a Riccati- type map with complex variables and complex parameters. Fixed points and their asymptotic stability are studied. Lyapunov exponent is computed to indicate chaos. Bifurcation and chaos are discussed. Chaotic behavior of the map has been controlled by ...

Stability analysis and quasinormal modes of Reissner–Nordstrøm ...

Indian Academy of Sciences (India)

2016-06-09

Jun 9, 2016 ... They also determine important features of the space-time and give important information on the background geometry. The Lyapunov exponent (λ) has been used to probe the instability of circular null geodesics and in terms of the quasinormal modes (QNMs) for spherically symmetric space-time of arbitrary ...

Synchronization of delayed systems in the presence of delay time modulation

International Nuclear Information System (INIS)

Kye, Won-Ho; Choi, Muhan; Kim, Myung-Woon; Lee, Soo-Young; Rim, Sunghwan; Kim, Chil-Min; Park, Young-Jai

2004-01-01

We investigate synchronization in the presence of delay time modulation for application to communication. We have observed that the robust synchronization is established by a common delay signal and its threshold is presented using Lyapunov exponents analysis. The influence of the delay time modulation in chaotic oscillators is also discussed

Dynamical generalized Hurst exponent as a tool to monitor unstable periods in financial time series

Science.gov (United States)

Morales, Raffaello; Di Matteo, T.; Gramatica, Ruggero; Aste, Tomaso

2012-06-01

We investigate the use of the Hurst exponent, dynamically computed over a weighted moving time-window, to evaluate the level of stability/instability of financial firms. Financial firms bailed-out as a consequence of the 2007-2008 credit crisis show a neat increase with time of the generalized Hurst exponent in the period preceding the unfolding of the crisis. Conversely, firms belonging to other market sectors, which suffered the least throughout the crisis, show opposite behaviors. We find that the multifractality of the bailed-out firms increase at the crisis suggesting that the multi fractal properties of the time series are changing. These findings suggest the possibility of using the scaling behavior as a tool to track the level of stability of a firm. In this paper, we introduce a method to compute the generalized Hurst exponent which assigns larger weights to more recent events with respect to older ones. In this way large fluctuations in the remote past are less likely to influence the recent past. We also investigate the scaling associated with the tails of the log-returns distributions and compare this scaling with the scaling associated with the Hurst exponent, observing that the processes underlying the price dynamics of these firms are truly multi-scaling.

Synchronising chaotic Chua's circuit using switching feedback control based on piecewise quadratic Lyapunov functions

International Nuclear Information System (INIS)

Hong-Bin, Zhang; Jian-Wei, Xia; Yong-Bin, Yu; Chuang-Yin, Dang

2010-01-01

This paper investigates the chaos synchronisation between two coupled chaotic Chua's circuits. The sufficient condition presented by linear matrix inequalities (LMIs) of global asymptotic synchronisation is attained based on piecewise quadratic Lyapunov functions. First, we obtain the piecewise linear differential inclusions (pwLDIs) model of synchronisation error dynamics, then we design a switching (piecewise-linear) feedback control law to stabilise it based on the piecewise quadratic Laypunov functions. Then we give some numerical simulations to demonstrate the effectiveness of our theoretical results

A new version of Scilab software package for the study of dynamical systems

Science.gov (United States)

Bordeianu, C. C.; Felea, D.; Beşliu, C.; Jipa, Al.; Grossu, I. V.

2009-11-01

This work presents a new version of a software package for the study of chaotic flows, maps and fractals [1]. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropy. Various well-known examples are implemented, with the capability of the users inserting their own ODE or iterative equations. New version program summaryProgram title: Chaos v2.0 Catalogue identifier: AEAP_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAP_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1275 No. of bytes in distributed program, including test data, etc.: 7135 Distribution format: tar.gz Programming language: Scilab 5.1.1. Scilab 5.1.1 should be installed before running the program. Information about the installation can be found at scilab.org/howto/install/windows" xlink:type="simple">http://wiki.scilab.org/howto/install/windows. Computer: PC-compatible running Scilab on MS Windows or Linux Operating system: Windows XP, Linux RAM: below 150 Megabytes Classification: 6.2 Catalogue identifier of previous version: AEAP_v1_0 Journal reference of previous version: Comput. Phys. Comm. 178 (2008) 788 Does the new version supersede the previous version?: Yes Nature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE). Solution method: Numerical solving of

The critical 1-arm exponent for the ferromagnetic Ising model on the Bethe lattice

Science.gov (United States)

Heydenreich, Markus; Kolesnikov, Leonid

2018-04-01

We consider the ferromagnetic nearest-neighbor Ising model on regular trees (Bethe lattice), which is well-known to undergo a phase transition in the absence of an external magnetic field. The behavior of the model at critical temperature can be described in terms of various critical exponents; one of them is the critical 1-arm exponent ρ which characterizes the rate of decay of the (root) magnetization as a function of the distance to the boundary. The crucial quantity we analyze in this work is the thermal expectation of the root spin on a finite subtree, where the expected value is taken with respect to a probability measure related to the corresponding finite-volume Hamiltonian with a fixed boundary condition. The spontaneous magnetization, which is the limit of this thermal expectation in the distance between the root and the boundary (i.e., in the height of the subtree), is known to vanish at criticality. We are interested in a quantitative analysis of the rate of this convergence in terms of the critical 1-arm exponent ρ. Therefore, we rigorously prove that ⟨σ0⟩ n +, the thermal expectation of the root spin at the critical temperature and in the presence of the positive boundary condition, decays as ⟨σ0 ⟩ n +≈n-1/2 (in a rather sharp sense), where n is the height of the tree. This establishes the 1-arm critical exponent for the Ising model on regular trees (ρ =1/2 ).

On nonlinear evolution variational inequalities involving variable exponent

Directory of Open Access Journals (Sweden)

Mingqi Xiang

2013-12-01

Full Text Available In this paper, we discuss a class of quasilinear evolution variational inequalities with variable exponent growth conditions in a generalized Sobolev space. We obtain the existence of weak solutions by means of penalty method. Moreover, we study the extinction properties of weak solutions to parabolic inequalities and provide a sufficient condition that makes the weak solutions vanish in a finite time. The existence of global attractors for weak solutions is also obtained via the theories of multi-valued semiflow.

Analysis and Adaptive Synchronization of Two Novel Chaotic Systems with Hyperbolic Sinusoidal and Cosinusoidal Nonlinearity and Unknown Parameters

Directory of Open Access Journals (Sweden)

S. Vaidyanathan

2013-09-01

Full Text Available This research work describes the modelling of two novel 3-D chaotic systems, the first with a hyperbolic sinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (A and the second with a hyperbolic cosinusoidal nonlinearity and two quadratic nonlinearities (denoted as system (B. In this work, a detailed qualitative analysis of the novel chaotic systems (A and (B has been presented, and the Lyapunov exponents and Kaplan-Yorke dimension of these chaotic systems have been obtained. It is found that the maximal Lyapunov exponent (MLE for the novel chaotic systems (A and (B has a large value, viz. for the system (A and for the system (B. Thus, both the novel chaotic systems (A and (B display strong chaotic behaviour. This research work also discusses the problem of finding adaptive controllers for the global chaos synchronization of identical chaotic systems (A, identical chaotic systems (B and nonidentical chaotic systems (A and (B with unknown system parameters. The adaptive controllers for achieving global chaos synchronization of the novel chaotic systems (A and (B have been derived using adaptive control theory and Lyapunov stability theory. MATLAB simulations have been shown to illustrate the novel chaotic systems (A and (B, and also the adaptive synchronization results derived for the novel chaotic systems (A and (B.

Plethysmogram and EEG: Effects of Music and Voice Sound

Science.gov (United States)

Miao, Tiejun; Oyama-Higa, Mayumi; Sato, Sadaka; Kojima, Junji; Lin, Juan; Reika, Sato

2011-06-01

We studied a relation of chaotic dynamics of finger plethysmogram to complexity of high cerebral center in both theoretical and experimental approaches. We proposed a mathematical model to describe emergence of chaos in finger tip pulse wave, which gave a theoretical prediction indicating increased chaoticity in higher cerebral center leading to an increase of chaos dynamics in plethysmograms. We designed an experiment to observe scalp-EEG and finger plethysmogram using two mental tasks to validate the relationship. We found that scalp-EEG showed an increase of the largest Lyapunov exponents (LLE) during speaking certain voices. Topographical scalp map of LLE showed enhanced arise around occipital and right cerebral area. Whereas there was decreasing tendency during listening music, where LLE scalp map revealed a drop around center cerebral area. The same tendency was found for LLE obtained from finger plethysmograms as ones of EEG under either speaking or listening tasks. The experiment gave results that agreed well with the theoretical relation derived from our proposed model.

Time evolution of coarse-grained entropy in classical and quantum motions of strongly chaotic systems

Science.gov (United States)

Gu, Yan; Wang, Jiao

1997-02-01

We study relaxation of an ensemble of cat maps with initially localized phase-space distributions. Calculations of the coarse-grained entropy Sɛ ( t) for both classical and quantum motions are presented. It is shown that, within the relaxation period, both classical and quantum entropies increase with a nearly constant rate which can be identified as the largest Lyapunov exponent of the classical cat. After an empirical relaxation time, the time behavior for two entropies becomes different. While the classical entropy increases to the equilibrium entropy Seqm and stays there, its quantum analogue fluctuates incessantly around a mean overlineSɛ which is less than Seqm. We regard the entropy difference ΔS = S eqm - overlineSɛ as a measure of nonergodicity of the quantum motion of strongly chaotic systems and investigate its dependence on the Planck constant h. For fixed initial phase-space distributions, numerical results suggest that there is a scaling law ΔSαhβ with β ≈ 0.72 in the semiclassical regime.

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

International Nuclear Information System (INIS)

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

Nonlinear Dynamic in an Ecological System with Impulsive Effect and Optimal Foraging

Directory of Open Access Journals (Sweden)

Min Zhao

2014-01-01

Full Text Available The population dynamics of a three-species ecological system with impulsive effect are investigated. Using the theories of impulsive equations and small-amplitude perturbation scales, the conditions for the system to be permanent when the number of predators released is less than some critical value can be obtained. Furthermore, because the predator in the system follows the predictions of optimal foraging theory, it follows that optimal foraging promotes species coexistence. In particular, the less beneficial prey can support the predator alone when the more beneficial prey goes extinct. Moreover, the influences of the impulsive effect and optimal foraging on inherent oscillations are studied using simulation, which reveals rich dynamic behaviors such as period-halving bifurcations, a chaotic band, a periodic window, and chaotic crises. In addition, the largest Lyapunov exponent and the power spectra of the strange attractor, which can help analyze the chaotic dynamic behavior of the model, are investigated. This information will be useful for studying the dynamic complexity of ecosystems.

Oscillations, complex spatiotemporal behavior, and information transport in networks of excitatory and inhibitory neurons

International Nuclear Information System (INIS)

Destexhe, A.

1994-01-01

Various types of spatiotemporal behavior are described for two-dimensional networks of excitatory and inhibitory neurons with time delayed interactions. It is described how the network behaves as several structural parameters are varied, such as the number of neurons, the connectivity, and the values of synaptic weights. A transition from spatially uniform oscillations to spatiotemporal chaos via intermittentlike behavior is observed. The properties of spatiotemporally chaotic solutions are investigated by evaluating the largest positive Lyapunov exponent and the loss of correlation with distance. Finally, properties of information transport are evaluated during uniform oscillations and spatiotemporal chaos. It is shown that the diffusion coefficient increases significantly in the spatiotemporal phase similar to the increase of transport coefficients at the onset of fluid turbulence. It is proposed that such a property should be seen in other media, such as chemical turbulence or networks of oscillators. The possibility of measuring information transport from appropriate experiments is also discussed

Synchronization in slowly switching networks of coupled oscillators

Science.gov (United States)

Zhou, Jie; Zou, Yong; Guan, Shuguang; Liu, Zonghua; Boccaletti, S.

2016-01-01

Networks whose structure of connections evolves in time constitute a big challenge in the study of synchronization, in particular when the time scales for the evolution of the graph topology are comparable with (or even longer than) those pertinent to the units’ dynamics. We here focus on networks with a slow-switching structure, and show that the necessary conditions for synchronization, i.e. the conditions for which synchronization is locally stable, are determined by the time average of the largest Lyapunov exponents of transverse modes of the switching topologies. Comparison between fast- and slow-switching networks allows elucidating that slow-switching processes prompt synchronization in the cases where the Master Stability Function is concave, whereas fast-switching schemes facilitate synchronization for convex curves. Moreover, the condition of slow-switching enables the introduction of a control strategy for inducing synchronization in networks with arbitrary structure and coupling strength, which is of evident relevance for broad applications in real world systems. PMID:27779253

Soft Measurement Modeling Based on Chaos Theory for Biochemical Oxygen Demand (BOD

Directory of Open Access Journals (Sweden)

Junfei Qiao

2016-12-01

Full Text Available The precision of soft measurement for biochemical oxygen demand (BOD is always restricted due to various factors in the wastewater treatment plant (WWTP. To solve this problem, a new soft measurement modeling method based on chaos theory is proposed and is applied to BOD measurement in this paper. Phase space reconstruction (PSR based on Takens embedding theorem is used to extract more information from the limited datasets of the chaotic system. The WWTP is first testified as a chaotic system by the correlation dimension (D, the largest Lyapunov exponents (λ1, the Kolmogorov entropy (K of the BOD and other water quality parameters time series. Multivariate chaotic time series modeling method with principal component analysis (PCA and artificial neural network (ANN is then adopted to estimate the value of the effluent BOD. Simulation results show that the proposed approach has higher accuracy and better prediction ability than the corresponding modeling approaches not based on chaos theory.

Spontaneous fluctuations in a zero-noise model of flocking

Science.gov (United States)

Chakraborty, Abhijit; Bhattacharya, Kunal

2016-11-01

Investigations into the complex structure and dynamics of collectively moving groups of living organisms have provided valuable insights. Understanding the emergent features, especially, the origin of fluctuations, appears to be challenging in the current scheme of models. It has been argued that flocks are poised at criticality. We present a two-dimensional self-propelled particle model where neighbourhoods and forces are defined through topology-based rules. The attractive forces are modeled in order to maintain cohesion in the flock in open-boundary conditions. We find that fluctuations occur spontaneously in the absence of any external noise. For certain values of the parameters the flock shows a high degree of order as well as scale-free decay of spatial correlations in velocity and speed. We characterize the dynamical behaviour of the system using the Lyapunov spectrum. Largest exponents being positive but small in magnitude suggest that the apparent high susceptibility may result from the system operating near the borderline of order and chaos.

Bearing Health Assessment Based on Chaotic Characteristics

Directory of Open Access Journals (Sweden)

Chen Lu

2013-01-01

Full Text Available Vibration signals extracted from rotating parts of machinery carry a lot of useful information about the condition of operating machine. Due to the strong non-linear, complex and non-stationary characteristics of vibration signals from working bearings, an accurate and reliable health assessment method for bearing is necessary. This paper proposes to utilize the selected chaotic characteristics of vibration signal for health assessment of a bearing by using self-organizing map (SOM. Both Grassberger-Procaccia algorithm and Takens' theory are employed to calculate the characteristic vector which includes three chaotic characteristics, such as correlation dimension, largest Lyapunov exponent and Kolmogorov entropy. After that, SOM is used to map the three corresponding characteristics into a confidence value (CV which represents the health state of the bearing. Finally, a case study based on vibration datasets of a group of testing bearings was conducted to demonstrate that the proposed method can reliably assess the health state of bearing.

A Lyapunov theory based UPFC controller for power flow control

Energy Technology Data Exchange (ETDEWEB)

Zangeneh, Ali; Kazemi, Ahad; Hajatipour, Majid; Jadid, Shahram [Center of Excellence for Power Systems Automation and Operation, Iran University of Science and Technology, Tehran (Iran)

2009-09-15

Unified power flow controller (UPFC) is the most comprehensive multivariable device among the FACTS controllers. Capability of power flow control is the most important responsibility of UPFC. According to high importance of power flow control in transmission lines, the proper controller should be robust against uncertainty and disturbance and also have suitable settling time. For this purpose, a new controller is designed based on the Lyapunov theory and its stability is also evaluated. The Main goal of this paper is to design a controller which enables a power system to track reference signals precisely and to be robust in the presence of uncertainty of system parameters and disturbances. The performance of the proposed controller is simulated on a two bus test system and compared with a conventional PI controller. The simulation results show the power and accuracy of the proposed controller. (author)

Synchronization of hyperchaotic oscillators

DEFF Research Database (Denmark)

Tamasevicius, A.; Cenys, A.; Mykolaitis, G.

1997-01-01

Synchronization of chaotic oscillators is believed to have promising applications in secure communications. Hyperchaotic systems with multiple positive Lyapunov exponents (LEs) have an advantage over common chaotic systems with only one positive LE. Three different types of hyperchaotic electronic...... oscillators are investigated demonstrating synchronization by means of only one properly selected variable....

Effect of speed on local dynamic stability of locomotion under different task constraints in running.

Science.gov (United States)

Mehdizadeh, Sina; Arshi, Ahmed Reza; Davids, Keith

2014-01-01

A number of studies have investigated effects of speed on local dynamic stability of walking, although this relationship has been rarely investigated under changing task constraints, such as during forward and backward running. To rectify this gap in the literature, the aim of this study was to investigate the effect of running speed on local dynamic stability of forward and backward running on a treadmill. Fifteen healthy male participants took part in this study. Participants ran in forward and backward directions at speeds of 80%, 100% and 120% of their preferred running speed. The three-dimensional motion of a C7 marker was recorded using a motion capture system. Local dynamic stability of the marker was quantified using short- and long-term largest finite-time Lyapunov exponents (LyE). Results showed that short-term largest finite-time LyE values increased with participant speed meaning that local dynamic stability decreased with increasing speed. Long-term largest finite-time LyEs, however, remained unaffected as speed increased. Results of this study indicated that, as in walking, slow running is more stable than fast running. These findings improve understanding of how stability is regulated when constraints on the speed of movements is altered. Implications for the design of rehabilitation or sport practice programmes suggest how task constraints could be manipulated to facilitate adaptations in locomotion stability during athletic training.

Homogeneous Stabilizer by State Feedback for Switched Nonlinear Systems Using Multiple Lyapunov Functions’ Approach

Directory of Open Access Journals (Sweden)

Hui Ye

2017-01-01

Full Text Available This paper investigates the problem of global stabilization for a class of switched nonlinear systems using multiple Lyapunov functions (MLFs. The restrictions on nonlinearities are neither linear growth condition nor Lipschitz condition with respect to system states. Based on adding a power integrator technique, we design homogeneous state feedback controllers of all subsystems and a switching law to guarantee that the closed-loop system is globally asymptotically stable. Finally, an example is given to illustrate the validity of the proposed control scheme.

Exponent and scrambling index of double alternate circular snake graphs

Science.gov (United States)

Rahmayanti, Sri; Pasaribu, Valdo E.; Nasution, Sawaluddin; Liani Salnaz, Sishi

2018-01-01

A graph is primitive if it contains a cycle of odd length. The exponent of a primitive graph G, denoted by exp(G), is the smallest positive integer k such that for each pair of vertices u and v in G there is a uv-walk length k. The scrambling index of a primitive graph G, denoted by k(G), is the smallest positive integer k such that for each pair of vertices u and v in G there is a uv-walk of length 2k. For an even positive integer n and an odd positive integer r, a (n,r)-double alternate circular snake graph, denoted by DA(C r,n ), is a graph obtained from a path u 1 u 2 ... u n by replacing each edge of the form u 2i u 2i+1 by two different r-cycles. We study the exponent and scrambling index of DA(C r,n ) and show that exp(DA(C r,n )) = n + r - 4 and k(DA(C r,n )) = (n + r - 3)/2.

Magnetic entropy change and critical exponents in double perovskite Y{sub 2}NiMnO{sub 6}

Energy Technology Data Exchange (ETDEWEB)

Sharma, G. [School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067 (India); Tripathi, T.S. [Inter-University Accelerator Centre, New Delhi-110067 (India); Saha, J. [School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067 (India); Patnaik, S., E-mail: spatnaik@mail.jnu.ac.in [School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067 (India)

2014-11-15

We report the magnetic entropy change (ΔS{sub M}) and the critical exponents in the double perovskite manganite Y{sub 2}NiMnO{sub 6} with a ferromagnetic to paramagnetic transition T{sub C}∼85K. For a magnetic field change ΔH=80kOe, a maximum magnetic entropy change ΔS{sub M}=−6.57J/kgK is recorded around T{sub C}. The critical exponents β=0.363±0.05 and γ=1.331±0.09 obtained from power law fitting to spontaneous magnetization M{sub S}(T) and the inverse initial susceptibility χ{sub 0}{sup −1}(T) satisfy well to values derived for a 3D-Heisenberg ferromagnet. The critical exponent δ=4.761±0.129 is determined from the isothermal magnetization at T{sub C}. The scaling exponents corresponding to second order phase transition are consistent with the exponents from Kouvel–Fisher analysis and satisfy Widom's scaling relation δ=1+(γ/β). Additionally, they also satisfy the single scaling equation M(H,ϵ)=ϵ{sup β}f±(H/ϵ{sup β+γ}) according to which the magnetization-field-temperature data around T{sub C} should collapse into two curves for temperatures below and above T{sub C}. - Highlights: • The magneto-caloric (MC) effect and the critical exponent analysis in Y{sub 2}NiMnO{sub 6} are studied. • Methods such as Kouvel–Fisher, Widom's and Mean-Field scaling are used. • The magnetic ground state in Y{sub 2}NiMnO{sub 6} is based on isotropic 3D Heisenberg model. • The large MC effect can be utilized towards magnetic refrigeration around 77 K. • The nearest neighbor interaction in Y{sub 2}NiMnO{sub 6} rules out ferroelectricity.

Gaussian fluctuation of the diffusion exponent of virus capsid in a living cell nucleus

Science.gov (United States)

Itto, Yuichi

2018-05-01

In their work [4], Bosse et al. experimentally showed that virus capsid exhibits not only normal diffusion but also anomalous diffusion in nucleus of a living cell. There, it was found that the distribution of fluctuations of the diffusion exponent characterizing them takes the Gaussian form, which is, quite remarkably, the same form for two different types of the virus. This suggests high robustness of such fluctuations. Here, the statistical property of local fluctuations of the diffusion exponent of the virus capsid in the nucleus is studied. A maximum-entropy-principle approach (originally proposed for a different virus in a different cell) is applied for obtaining the fluctuation distribution of the exponent. Largeness of the number of blocks identified with local areas of interchromatin corrals is also examined based on the experimental data. It is shown that the Gaussian distribution of the local fluctuations can be derived, in accordance with the above form. In addition, it is quantified how the fluctuation distribution on a long time scale is different from the Gaussian distribution.

A Lyapunov Stability Theory-Based Control Strategy for Three-Level Shunt Active Power Filter

Directory of Open Access Journals (Sweden)

Yijia Cao

2017-01-01

Full Text Available The three-phase three-wire neutral-point-clamped shunt active power filter (NPC-SAPF, which most adopts classical closed-loop feedback control methods such as proportional-integral (PI, proportional-resonant (PR and repetitive control, can only output 1st–25th harmonic currents with 10–20 kHz switching frequency. The reason for this is that the controller design must make a compromise between system stability and harmonic current compensation ability under the condition of less than 20 kHz switching frequency. To broaden the bandwidth of the compensation current, a Lyapunov stability theory-based control strategy is presented in this paper for NPC-SAPF. The proposed control law is obtained by constructing the switching function on the basis of the mathematical model and the Lyapunov candidate function, which can avoid introducing closed-loop feedback control and keep the system globally asymptotically stable. By means of the proposed method, the NPC-SAPF has compensation ability for the 1st–50th harmonic currents, the total harmonic distortion (THD and each harmonic content of grid currents satisfy the requirements of IEEE Standard 519-2014. In order to verify the superiority of the proposed control strategy, stability conditions of the proposed strategy and the representative PR controllers are compared. The simulation results in MATLAB/Simulink (MathWorks, Natick, MA, USA and the experimental results obtained on a 6.6 kVA NPC-SAPF laboratory prototype validate the proposed control strategy.

Nonlinear anisotropic elliptic equations with variable exponents and degenerate coercivity

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Hocine Ayadi

2018-02-01

Full Text Available In this article, we prove the existence and the regularity of distributional solutions for a class of nonlinear anisotropic elliptic equations with $p_i(x$ growth conditions, degenerate coercivity and $L^{m(\\cdot}$ data, with $m(\\cdot$ being small, in appropriate Lebesgue-Sobolev spaces with variable exponents. The obtained results extend some existing ones [8,10].

High-speed image analysis reveals chaotic vibratory behaviors of pathological vocal folds

Energy Technology Data Exchange (ETDEWEB)

Zhang Yu, E-mail: yuzhang@xmu.edu.c [Key Laboratory of Underwater Acoustic Communication and Marine Information Technology of the Ministry of Education, Xiamen University, Xiamen Fujian 361005 (China); Shao Jun [Shanghai EENT Hospital of Fudan University, Shanghai (China); Krausert, Christopher R. [Department of Surgery, Division of Otolaryngology - Head and Neck Surgery, University of Wisconsin School of Medicine and Public Health, Madison, WI 53792-7375 (United States); Zhang Sai [Key Laboratory of Underwater Acoustic Communication and Marine Information Technology of the Ministry of Education, Xiamen University, Xiamen Fujian 361005 (China); Jiang, Jack J. [Shanghai EENT Hospital of Fudan University, Shanghai (China); Department of Surgery, Division of Otolaryngology - Head and Neck Surgery, University of Wisconsin School of Medicine and Public Health, Madison, WI 53792-7375 (United States)

2011-01-15

Research highlights: Low-dimensional human glottal area data. Evidence of chaos in human laryngeal activity from high-speed digital imaging. Traditional perturbation analysis should be cautiously applied to aperiodic high speed image signals. Nonlinear dynamic analysis may be helpful for understanding disordered behaviors in pathological laryngeal systems. - Abstract: Laryngeal pathology is usually associated with irregular dynamics of laryngeal activity. High-speed imaging facilitates direct observation and measurement of vocal fold vibrations. However, chaotic dynamic characteristics of aperiodic high-speed image data have not yet been investigated in previous studies. In this paper, we will apply nonlinear dynamic analysis and traditional perturbation methods to quantify high-speed image data from normal subjects and patients with various laryngeal pathologies including vocal fold nodules, polyps, bleeding, and polypoid degeneration. The results reveal the low-dimensional dynamic characteristics of human glottal area data. In comparison to periodic glottal area series from a normal subject, aperiodic glottal area series from pathological subjects show complex reconstructed phase space, fractal dimension, and positive Lyapunov exponents. The estimated positive Lyapunov exponents provide the direct evidence of chaos in pathological human vocal folds from high-speed digital imaging. Furthermore, significant differences between the normal and pathological groups are investigated for nonlinear dynamic and perturbation analyses. Jitter in the pathological group is significantly higher than in the normal group, but shimmer does not show such a difference. This finding suggests that the traditional perturbation analysis should be cautiously applied to high speed image signals. However, the correlation dimension and the maximal Lyapunov exponent reveal a statistically significant difference between normal and pathological groups. Nonlinear dynamic analysis is capable of

High-speed image analysis reveals chaotic vibratory behaviors of pathological vocal folds

International Nuclear Information System (INIS)

Zhang Yu; Shao Jun; Krausert, Christopher R.; Zhang Sai; Jiang, Jack J.

2011-01-01

Research highlights: → Low-dimensional human glottal area data. → Evidence of chaos in human laryngeal activity from high-speed digital imaging. → Traditional perturbation analysis should be cautiously applied to aperiodic high speed image signals. → Nonlinear dynamic analysis may be helpful for understanding disordered behaviors in pathological laryngeal systems. - Abstract: Laryngeal pathology is usually associated with irregular dynamics of laryngeal activity. High-speed imaging facilitates direct observation and measurement of vocal fold vibrations. However, chaotic dynamic characteristics of aperiodic high-speed image data have not yet been investigated in previous studies. In this paper, we will apply nonlinear dynamic analysis and traditional perturbation methods to quantify high-speed image data from normal subjects and patients with various laryngeal pathologies including vocal fold nodules, polyps, bleeding, and polypoid degeneration. The results reveal the low-dimensional dynamic characteristics of human glottal area data. In comparison to periodic glottal area series from a normal subject, aperiodic glottal area series from pathological subjects show complex reconstructed phase space, fractal dimension, and positive Lyapunov exponents. The estimated positive Lyapunov exponents provide the direct evidence of chaos in pathological human vocal folds from high-speed digital imaging. Furthermore, significant differences between the normal and pathological groups are investigated for nonlinear dynamic and perturbation analyses. Jitter in the pathological group is significantly higher than in the normal group, but shimmer does not show such a difference. This finding suggests that the traditional perturbation analysis should be cautiously applied to high speed image signals. However, the correlation dimension and the maximal Lyapunov exponent reveal a statistically significant difference between normal and pathological groups. Nonlinear dynamic

Some chaotic behaviors in a MCA learning algorithm with a constant learning rate

International Nuclear Information System (INIS)

Lv Jiancheng; Yi Zhang

2007-01-01

Douglas's minor component analysis algorithm with a constant learning rate has both stability and chaotic dynamical behavior under some conditions. The paper explores such dynamical behavior of this algorithm. Certain stability and chaos of this algorithm are derived. Waveform plots, Lyapunov exponents and bifurcation diagrams are presented to illustrate the existence of chaotic behavior

Chaotic dynamic characteristics of pressure fluctuation signals in hydro-turbine

Energy Technology Data Exchange (ETDEWEB)

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.

On the entropy of a hidden Markov process.

Science.gov (United States)

Jacquet, Philippe; Seroussi, Gadiel; Szpankowski, Wojciech

2008-05-01

We study the entropy rate of a hidden Markov process (HMP) defined by observing the output of a binary symmetric channel whose input is a first-order binary Markov process. Despite the simplicity of the models involved, the characterization of this entropy is a long standing open problem. By presenting the probability of a sequence under the model as a product of random matrices, one can see that the entropy rate sought is equal to a top Lyapunov exponent of the product. This offers an explanation for the elusiveness of explicit expressions for the HMP entropy rate, as Lyapunov exponents are notoriously difficult to compute. Consequently, we focus on asymptotic estimates, and apply the same product of random matrices to derive an explicit expression for a Taylor approximation of the entropy rate with respect to the parameter of the binary symmetric channel. The accuracy of the approximation is validated against empirical simulation results. We also extend our results to higher-order Markov processes and to Rényi entropies of any order.

Parameter space of experimental chaotic circuits with high-precision control parameters

Energy Technology Data Exchange (ETDEWEB)

Sousa, Francisco F. G. de; Rubinger, Rero M. [Instituto de Física e Química, Universidade Federal de Itajubá, Itajubá, MG (Brazil); Sartorelli, José C., E-mail: sartorelli@if.usp.br [Universidade de São Paulo, São Paulo, SP (Brazil); Albuquerque, Holokx A. [Departamento de Física, Universidade do Estado de Santa Catarina, Joinville, SC (Brazil); Baptista, Murilo S. [Institute of Complex Systems and Mathematical Biology, SUPA, University of Aberdeen, Aberdeen (United Kingdom)

2016-08-15

We report high-resolution measurements that experimentally confirm a spiral cascade structure and a scaling relationship of shrimps in the Chua's circuit. Circuits constructed using this component allow for a comprehensive characterization of the circuit behaviors through high resolution parameter spaces. To illustrate the power of our technological development for the creation and the study of chaotic circuits, we constructed a Chua circuit and study its high resolution parameter space. The reliability and stability of the designed component allowed us to obtain data for long periods of time (∼21 weeks), a data set from which an accurate estimation of Lyapunov exponents for the circuit characterization was possible. Moreover, this data, rigorously characterized by the Lyapunov exponents, allows us to reassure experimentally that the shrimps, stable islands embedded in a domain of chaos in the parameter spaces, can be observed in the laboratory. Finally, we confirm that their sizes decay exponentially with the period of the attractor, a result expected to be found in maps of the quadratic family.

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

Directory of Open Access Journals (Sweden)

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.

On Chaotic Nature of the Emerging European Forex Markets

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Anoop S Kumar

2014-06-01

Full Text Available This study attempts to analyze the presence deterministic chaos in the forex markets of select European countries namely Bulgaria, Croatia, Czech Republic, Hungary Poland, Romania, Russia, Slovakia and Slovenia. Monthly NEER data ranging from jan-1994 to Dec-2013 is used for the purpose of analysis. A two step methodology is employed where in the first step, non-linear dependence structure in the underlying time series is verified using BDS test. The results show that all the markets under study exhibit non-linear dependence. In the next stage, it is enquired whether this non-linear behavior is due to the presence of chaotic dynamics in the markets. This is achieved by estimating Lyapunov exponents for the time series under analysis. An EGARCH (1, 1 filter is applied to see if the non-linearity could be explained by a GARCH process. From the Lyapunov exponent values, it is found that the GARCH process is unable to explain the forex markets behavior in a satisfying manner. It is concluded that the forex markets under study exhibit deterministic chaotic behavior.

Linear growth of the entanglement entropy and the Kolmogorov-Sinai rate

Science.gov (United States)

Bianchi, Eugenio; Hackl, Lucas; Yokomizo, Nelson

2018-03-01

The rate of entropy production in a classical dynamical system is characterized by the Kolmogorov-Sinai entropy rate h KS given by the sum of all positive Lyapunov exponents of the system. We prove a quantum version of this result valid for bosonic systems with unstable quadratic Hamiltonian. The derivation takes into account the case of time-dependent Hamiltonians with Floquet instabilities. We show that the entanglement entropy S A of a Gaussian state grows linearly for large times in unstable systems, with a rate Λ A ≤ h KS determined by the Lyapunov exponents and the choice of the subsystem A. We apply our results to the analysis of entanglement production in unstable quadratic potentials and due to periodic quantum quenches in many-body quantum systems. Our results are relevant for quantum field theory, for which we present three applications: a scalar field in a symmetry-breaking potential, parametric resonance during post-inflationary reheating and cosmological perturbations during inflation. Finally, we conjecture that the same rate Λ A appears in the entanglement growth of chaotic quantum systems prepared in a semiclassical state.

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

Science.gov (United States)

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.

Dynamic analysis, controlling chaos and chaotification of a SMIB power system

International Nuclear Information System (INIS)

Chen, H.-K.; Lin, T.-N.; Chen, J.-H.

2005-01-01

The dynamic behaviors of a SMIB power system are studied in this paper. A single modal equation is used to analyze the qualitative behaviors of the system. The famous equation of motion is called 'swing equation'. The Lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system. The bifurcation of the parameter dependent system is studied numerically. Besides, the phase portraits, the Poincare maps, and the Lyapunov exponents are presented to observe periodic and chaotic motions. Further, the addition of periodic force and the feedback control are used to control chaos effectively. Finally, the chaotification problem of the SMIB power system is also issued

Logarithmic bred vectors in spatiotemporal chaos: structure and growth.

Science.gov (United States)

Hallerberg, Sarah; Pazó, Diego; López, Juan M; Rodríguez, Miguel A

2010-06-01

Bred vectors are a type of finite perturbation used in prediction studies of atmospheric models that exhibit spatially extended chaos. We study the structure, spatial correlations, and the growth rates of logarithmic bred vectors (which are constructed by using a given norm). We find that, after a suitable transformation, logarithmic bred vectors are roughly piecewise copies of the leading Lyapunov vector. This fact allows us to deduce a scaling law for the bred vector growth rate as a function of its amplitude. In addition, we relate growth rates with the spectrum of Lyapunov exponents corresponding to the most expanding directions. We illustrate our results with simulations of the Lorenz 1996 model.

Brief communication: Possible explanation of the values of Hack's drainage basin, river length scaling exponent

Science.gov (United States)

Hunt, Allen G.

2016-04-01

Percolation theory can be used to find water flow paths of least resistance. Application of percolation theory to drainage networks allows identification of the range of exponent values that describe the tortuosity of rivers in real river networks, which is then used to generate the observed scaling between drainage basin area and channel length, a relationship known as Hack's law. Such a theoretical basis for Hack's law may allow interpretation of the range of exponent values based on an assessment of the heterogeneity of the substrate.

Sliding Mode Controller and Lyapunov Redesign Controller to Improve Microgrid Stability

DEFF Research Database (Denmark)

Hossain, Eklas; Perez, Ron; Padmanaban, Sanjeevikumar

2017-01-01

technique is used to enhance stability of microgrids. Besides adopting this technique here, Sliding Mode Controller (SMC) and Lyapunov Redesign Controller (LRC), two of the most prominent nonlinear control techniques, are individually implemented to control microgrid system stability with desired robustness....... CPL power is then varied to compare robustness of these two control techniques. This investigation revealed the better performance of the LRC system compared to SMC to retain stability in microgrid with dense CPL load. All the necessary results are simulated in Matlab/Simulink platform for authentic......To mitigate the microgrid instability despite the presence of dense Constant Power Load (CPL) loads in the system, a number of compensation techniques have already been gone through extensive research, proposed, and implemented around the world. In this paper, a storage based load side compensation...

Relation between the Hurst Exponent and the Efficiency of Self-organization of a Deformable System

Science.gov (United States)

Alfyorova, E. A.; Lychagin, D. V.

2018-04-01

We have established the degree of self-organization of a system under plastic deformation at different scale levels. Using fractal analysis, we have determined the Hurst exponent and correlation lengths in the region of formation of a corrugated (wrinkled) structure in [111] nickel single crystals under compression. This has made it possible to single out two (micro-and meso-) levels of self-organization in the deformable system. A qualitative relation between the values of the Hurst exponent and the stages of the stress-strain curve has been established.

Characterization of the local instability in the Henon-Heiles Hamiltonian

International Nuclear Information System (INIS)

Vallejo, Juan C.; Aguirre, Jacobo; Sanjuan, Miguel A.F.

2003-01-01

Several prototypical distributions of finite-time Lyapunov exponents have been computed for the two-dimensional Henon-Heiles Hamiltonian system. Different shapes are obtained for each dynamical state. Even when an evolution is observed in the morphology of the distributions for the smallest integration intervals, they can still serve for characterizing the dynamical state of the system

Anti-control of chaos of single time-scale brushless DC motor.

Science.gov (United States)

Ge, Zheng-Ming; Chang, Ching-Ming; Chen, Yen-Sheng

2006-09-15

Anti-control of chaos of single time-scale brushless DC motors is studied in this paper. In order to analyse a variety of periodic and chaotic phenomena, we employ several numerical techniques such as phase portraits, bifurcation diagrams and Lyapunov exponents. Anti-control of chaos can be achieved by adding an external constant term or an external periodic term.

Critical exponents for diluted resistor networks.

Science.gov (United States)

Stenull, O; Janssen, H K; Oerding, K

1999-05-01

An approach by Stephen [Phys. Rev. B 17, 4444 (1978)] is used to investigate the critical properties of randomly diluted resistor networks near the percolation threshold by means of renormalized field theory. We reformulate an existing field theory by Harris and Lubensky [Phys. Rev. B 35, 6964 (1987)]. By a decomposition of the principal Feynman diagrams, we obtain diagrams which again can be interpreted as resistor networks. This interpretation provides for an alternative way of evaluating the Feynman diagrams for random resistor networks. We calculate the resistance crossover exponent phi up to second order in epsilon=6-d, where d is the spatial dimension. Our result phi=1+epsilon/42+4epsilon(2)/3087 verifies a previous calculation by Lubensky and Wang, which itself was based on the Potts-model formulation of the random resistor network.

Loy Yang A - Australia's largest privatisation

International Nuclear Information System (INIS)

Yenckin, C.

1997-01-01

The recent A$4,746 million privatisation of the 2000MW Loy Yang A power station and the Loy Yang coal mine by the Victorian Government is Australia's largest privatisation and one of 1997's largest project financing deals. (author)

A Lyapunov Function Based Remedial Action Screening Tool Using Real-Time Data

Energy Technology Data Exchange (ETDEWEB)

Mitra, Joydeep [Michigan State Univ., East Lansing, MI (United States); Ben-Idris, Mohammed [Univ. of Nevada, Reno, NV (United States); Faruque, Omar [Florida State Univ., Tallahassee, FL (United States); Backhaus, Scott [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Deb, Sidart [LCG Consulting, Los Altos, CA (United States)

2016-03-30

This report summarizes the outcome of a research project that comprised the development of a Lyapunov function based remedial action screening tool using real-time data (L-RAS). The L-RAS is an advanced computational tool that is intended to assist system operators in making real-time redispatch decisions to preserve power grid stability. The tool relies on screening contingencies using a homotopy method based on Lyapunov functions to avoid, to the extent possible, the use of time domain simulations. This enables transient stability evaluation at real-time speed without the use of massively parallel computational resources. The project combined the following components. 1. Development of a methodology for contingency screening using a homotopy method based on Lyapunov functions and real-time data. 2. Development of a methodology for recommending remedial actions based on the screening results. 3. Development of a visualization and operator interaction interface. 4. Testing of screening tool, validation of control actions, and demonstration of project outcomes on a representative real system simulated on a Real-Time Digital Simulator (RTDS) cluster. The project was led by Michigan State University (MSU), where the theoretical models including homotopy-based screening, trajectory correction using real-time data, and remedial action were developed and implemented in the form of research-grade software. Los Alamos National Laboratory (LANL) contributed to the development of energy margin sensitivity dynamics, which constituted a part of the remedial action portfolio. Florida State University (FSU) and Southern California Edison (SCE) developed a model of the SCE system that was implemented on FSU's RTDS cluster to simulate real-time data that was streamed over the internet to MSU where the L-RAS tool was executed and remedial actions were communicated back to FSU to execute stabilizing controls on the simulated system. LCG Consulting developed the visualization

Scaling exponents of the velocity structure functions in the interplanetary medium

Directory of Open Access Journals (Sweden)

V. Carbone

Full Text Available We analyze the scaling exponents of the velocity structure functions, obtained from the velocity fluctuations measured in the interplanetary space plasma. Using the expression for the energy transfer rate which seems the most relevant in describing the evolution of the pseudo-energy densities in the interplanetary medium, we introduce an energy cascade model derived from a simple fragmentation process, which takes into account the intermittency effect. In the absence and in the presence of the large-scale magnetic field decorrelation effect the model reduces to the fluid and the hydromagnetic p-model, respectively. We show that the scaling exponents of the q-th power of the velocity structure functions, as obtained by the model in the absence of the decorrelation effect, furnishes the best-fit to the data analyzed from the Voyager 2 velocity field measurements at 8.5 AU. Our results allow us to hypothesize a new kind of scale-similarity for magnetohydrodynamic turbulence when the decorrelation effect is at work, related to the fourth-order velocity structure function.