Crauel, Hans; Eckmann, Jean-Pierre
1991-01-01
Since the predecessor to this volume (LNM 1186, Eds. L. Arnold, V. Wihstutz)appeared in 1986, significant progress has been made in the theory and applications of Lyapunov exponents - one of the key concepts of dynamical systems - and in particular, pronounced shifts towards nonlinear and infinite-dimensional systems and engineering applications are observable. This volume opens with an introductory survey article (Arnold/Crauel) followed by 26 original (fully refereed) research papers, some of which have in part survey character. From the Contents: L. Arnold, H. Crauel: Random Dynamical Systems.- I.Ya. Goldscheid: Lyapunov exponents and asymptotic behaviour of the product of random matrices.- Y. Peres: Analytic dependence of Lyapunov exponents on transition probabilities.- O. Knill: The upper Lyapunov exponent of Sl (2, R) cocycles:Discontinuity and the problem of positivity.- Yu.D. Latushkin, A.M. Stepin: Linear skew-product flows and semigroups of weighted composition operators.- P. Baxendale: Invariant me...
Experimental Realization of a Multiscroll Chaotic Oscillator with Optimal Maximum Lyapunov Exponent
Esteban Tlelo-Cuautle
2014-01-01
Full Text Available Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. However, those realizations do not consider the value of the maximum Lyapunov exponent, which gives a quantitative measure of the grade of unpredictability of chaotic systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. First, from the mathematical description having four coefficients (a, b, c, d1, an optimization evolutionary algorithm varies them to maximize the value of the positive Lyapunov exponent. Second, a realization of those optimized coefficients using operational amplifiers is given. Herein a, b, c, d1 are implemented with precision potentiometers to tune up to four decimals of the coefficients having the range between 0.0001 and 1.0000. Finally, experimental results of the phase-space portraits for generating from 2 to 10 scrolls are listed to show that their associated value for the optimal maximum Lyapunov exponent increases by increasing the number of scrolls, thus guaranteeing a more complex chaotic behavior.
Experimental realization of a multiscroll chaotic oscillator with optimal maximum Lyapunov exponent.
Tlelo-Cuautle, Esteban; Pano-Azucena, Ana Dalia; Carbajal-Gomez, Victor Hugo; Sanchez-Sanchez, Mauro
2014-01-01
Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. However, those realizations do not consider the value of the maximum Lyapunov exponent, which gives a quantitative measure of the grade of unpredictability of chaotic systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. First, from the mathematical description having four coefficients (a, b, c, d1 ), an optimization evolutionary algorithm varies them to maximize the value of the positive Lyapunov exponent. Second, a realization of those optimized coefficients using operational amplifiers is given. Herein a, b, c, d1 are implemented with precision potentiometers to tune up to four decimals of the coefficients having the range between 0.0001 and 1.0000. Finally, experimental results of the phase-space portraits for generating from 2 to 10 scrolls are listed to show that their associated value for the optimal maximum Lyapunov exponent increases by increasing the number of scrolls, thus guaranteeing a more complex chaotic behavior.
Statistical properties of the maximum Lyapunov exponent calculated via the divergence rate method.
Franchi, Matteo; Ricci, Leonardo
2014-12-01
The embedding of a time series provides a basic tool to analyze dynamical properties of the underlying chaotic system. To this purpose, the choice of the embedding dimension and lag is crucial. Although several methods have been devised to tackle the issue of the optimal setting of these parameters, a conclusive criterion to make the most appropriate choice is still lacking. An accepted procedure to rank different embedding methods relies on the evaluation of the maximum Lyapunov exponent (MLE) out of embedded time series that are generated by chaotic systems with explicit analytic representation. The MLE is evaluated as the local divergence rate of nearby trajectories. Given a system, embedding methods are ranked according to how close such MLE values are to the true MLE. This is provided by the so-called standard method in a way that exploits the mathematical description of the system and does not require embedding. In this paper we study the dependence of the finite-time MLE evaluated via the divergence rate method on the embedding dimension and lag in the case of time series generated by four systems that are widely used as references in the scientific literature. We develop a completely automatic algorithm that provides the divergence rate and its statistical uncertainty. We show that the uncertainty can provide useful information about the optimal choice of the embedding parameters. In addition, our approach allows us to find which systems provide suitable benchmarks for the comparison and ranking of different embedding methods.
Entanglement production and Lyapunov exponents
Hackl, Lucas; Bianchi, Eugenio; Yokomizo, Nelson
2017-01-01
Squeezed vacua play a prominent role in quantum field theory in curved spacetime. Instabilities and resonances that arise from the coupling in the field to the background geometry, result in a large squeezing of the vacuum. In this talk, I discuss the relation between squeezing and Lyapunov exponents of the system. In particular, I derive a new formula for the rate of growth of the entanglement entropy expressed as the sum of the Lyapunov exponents. Examples of such a linear production regime can be found during inflation and in the preheating phase directly after inflation.
Nair, Sandeep P.; Shiau, Deng-Shan; Principe, Jose C.; Iasemidis, Leonidas D.; Pardalos, Panos M.; Norman, Wendy M.; Carney, Paul R.; Sackellares, J. Chris
2009-01-01
Analysis of intracranial electroencephalographic (iEEG) recordings in patients with temporal lobe epilepsy (TLE) has revealed characteristic dynamical features that distinguish the interictal, ictal, and postictal states and inter-state transitions. Experimental investigations into the mechanisms underlying these observations require the use of an animal model. A rat TLE model was used to test for differences in iEEG dynamics between well-defined states and to test specific hypotheses: 1) the short-term maximum Lyapunov exponent (STLmax), a measure of signal order, is lowest and closest in value among cortical sites during the ictal state, and highest and most divergent during the postictal state; 2) STLmax values estimated from the stimulated hippocampus are the lowest among all cortical sites; and 3) the transition from the interictal to ictal state is associated with a convergence in STLmax values among cortical sites. iEEGs were recorded from bilateral frontal cortices and hippocampi. STLmax and T-index (a measure of convergence/divergence of STLmax between recorded brain areas) were compared among the four different periods. Statistical tests (ANOVA and multiple comparisons) revealed that ictal STLmax was lower (p < 0.05) than other periods, STLmax values corresponding to the stimulated hippocampus were lower than those estimated from other cortical regions, and T-index values were highest during the postictal period and lowest during the ictal period. Also, the T-index values corresponding to the preictal period were lower than those during the interictal period (p < 0.05). These results indicate that a rat TLE model demonstrates several important dynamical signal characteristics similar to those found in human TLE and support future use of the model to study epileptic state transitions. PMID:19100262
Diverging Fluctuations of the Lyapunov Exponents.
Pazó, Diego; López, Juan M; Politi, Antonio
2016-07-15
We show that in generic one-dimensional Hamiltonian lattices the diffusion coefficient of the maximum Lyapunov exponent diverges in the thermodynamic limit. We trace this back to the long-range correlations associated with the evolution of the hydrodynamic modes. In the case of normal heat transport, the divergence is even stronger, leading to the breakdown of the usual single-function Family-Vicsek scaling ansatz. A similar scenario is expected to arise in the evolution of rough interfaces in the presence of suitably correlated background noise.
Upper quantum Lyapunov exponent and parametric oscillators
Jauslin, H. R.; Sapin, O.; Guérin, S.; Wreszinski, W. F.
2004-11-01
We introduce a definition of upper Lyapunov exponent for quantum systems in the Heisenberg representation, and apply it to parametric quantum oscillators. We provide a simple proof that the upper quantum Lyapunov exponent ranges from zero to a positive value, as the parameters range from the classical system's region of stability to the instability region. It is also proved that in the instability region the parametric quantum oscillator satisfies the discrete quantum Anosov relations defined by Emch, Narnhofer, Sewell, and Thirring.
Lyapunov exponents for continuous random transformations
无
2010-01-01
In this paper, the concept of Lyapunov exponent is generalized to random transformations that are not necessarily differentiable. For a class of random repellers and of random hyperbolic sets obtained via small perturbations of deterministic ones respectively, the new exponents are shown to coincide with the classical ones.
Short-time Lyapunov exponent analysis
Vastano, J. A.
1990-01-01
A new technique for analyzing complicated fluid flows in numerical simulations has been successfully tested. The analysis uses short time Lyapunov exponent contributions and the associated Lyapunov perturbation fields. A direct simulation of the Taylor-Couette flow just past the onset of chaos demonstrated that this new technique marks important times during the system evolution and identifies the important flow features at those times. This new technique will now be applied to a 'minimal' turbulent channel.
Short-time Lyapunov exponent analysis
Vastano, J. A.
1990-01-01
A new technique for analyzing complicated fluid flows in numerical simulations has been successfully tested. The analysis uses short time Lyapunov exponent contributions and the associated Lyapunov perturbation fields. A direct simulation of the Taylor-Couette flow just past the onset of chaos demonstrated that this new technique marks important times during the system evolution and identifies the important flow features at those times. This new technique will now be applied to a 'minimal' turbulent channel.
Lyapunov exponents computation for hybrid neurons.
Bizzarri, Federico; Brambilla, Angelo; Gajani, Giancarlo Storti
2013-10-01
Lyapunov exponents are a basic and powerful tool to characterise the long-term behaviour of dynamical systems. The computation of Lyapunov exponents for continuous time dynamical systems is straightforward whenever they are ruled by vector fields that are sufficiently smooth to admit a variational model. Hybrid neurons do not belong to this wide class of systems since they are intrinsically non-smooth owing to the impact and sometimes switching model used to describe the integrate-and-fire (I&F) mechanism. In this paper we show how a variational model can be defined also for this class of neurons by resorting to saltation matrices. This extension allows the computation of Lyapunov exponent spectrum of hybrid neurons and of networks made up of them through a standard numerical approach even in the case of neurons firing synchronously.
Lyapunov exponents for synchronous 12-lead ECG signals
无
2002-01-01
The Lyapunov exponents of synchronous 12-lead ECG signals have been investigated for the first time using a multi-sensor (electrode) technique. The results show that the Lyapunov exponents computed from different locations on the body surface are not the same, but have a distribution characteristic for the ECG signals recorded from coronary artery disease (CAD) patients with sinus rhythms and for signals from healthy older people. The maximum Lyapunov exponent L1 of all signals is positive. While all the others are negative, so the ECG signal has chaotic characteristics. With the same leads, L1 of CAD patients is less than that of healthy people, so the CAD patients and healthy people can be classified by L1, L1 therefore has potential values in the diagnosis of heart disease.
Lyapunov exponents for infinite dimensional dynamical systems
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.
Clustering and synchronization with positive Lyapunov exponents
Mendes, R V
1998-01-01
Clustering and correlation effects are frequently observed in chaotic systems in situations where, because of the positivity of the Lyapunov exponents, no dimension reduction is to be expected. In this paper, using a globally coupled network of Bernoulli units, one finds a general mechanism by which strong correlations and slow structures are obtained at the synchronization edge. A structure index is defined, which diverges at the transition points. Some conclusions are drawn concerning the construction of an ergodic theory of self-organization.
Yang, Hong-liu; Radons, Günter; Kantz, Holger
2012-12-14
The estimation of Lyapunov exponents from time series suffers from the appearance of spurious Lyapunov exponents due to the necessary embedding procedure. Separating true from spurious exponents poses a fundamental problem which is not yet solved satisfactorily. We show, in this Letter, analytically and numerically that covariant Lyapunov vectors associated with true exponents lie in the tangent space of the reconstructed attractor. Therefore, we use the angle between the covariant Lyapunov vectors and the tangent space of the reconstructed attractor to identify the true Lyapunov exponents. The usefulness of our method, also for noisy situations, is demonstrated by applications to data from model systems and a NMR laser experiment.
Lyapunov exponents for a Duffing oscillator
Zeni, Andrea R.; Gallas, Jason A. C.
With the help of a parallel computer we perform a systematic computation of Lyapunov exponents for a Duffing oscillator driven externally by a force proportional to cos( t). In contrast to the familiar situation in discrete-time systems where one finds “windows” of regularity embedded in intervals of chaos, we find the continuous-time Duffing oscillator to contain a quite regular epetition of relatively self-similar “islands of chaos” (i.e. regions characterized by positive exponents) embedded in large “seas of regularity” (negative exponents). We also investigate the effect of driving the oscillator with a Jacobian elliptic function cn( t, m). For m = 0 one has cn( t, 0) ≡ cos( t), the usual trigonometric pumping. For m = 1 one has cn( t, 1) ≡ sech( t), a hyperbolic pumping. When 0 displace the islands of chaos in parameter space. Thus, Jacobian pumping provides a possible way of “cleaning chaos” in regions of the parameter space for periodically driven systems.
Lyapunov exponent diagrams of a 4-dimensional Chua system.
Stegemann, Cristiane; Albuquerque, Holokx A; Rubinger, Rero M; Rech, Paulo C
2011-09-01
We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.
Statistics of Lyapunov exponent spectrum in randomly coupled Kuramoto oscillators.
Patra, Soumen K; Ghosh, Anandamohan
2016-03-01
Characterization of spatiotemporal dynamics of coupled oscillatory systems can be done by computing the Lyapunov exponents. We study the spatiotemporal dynamics of randomly coupled network of Kuramoto oscillators and find that the spectral statistics obtained from the Lyapunov exponent spectrum show interesting sensitivity to the coupling matrix. Our results indicate that in the weak coupling limit the gap distribution of the Lyapunov spectrum is Poissonian, while in the limit of strong coupling the gap distribution shows level repulsion. Moreover, the oscillators settle to an inhomogeneous oscillatory state, and it is also possible to infer the random network properties from the Lyapunov exponent spectrum.
The Lyapunov exponents of the Van der Pol oscillator
Grasman, J.; Verhulst, F.; Shih, S.D.
2005-01-01
Lyapunov exponents characterize the dynamics of a system near its attractor. For the Van der Pol oscillator these are quantities for which an approximation should be at hand. Similar to the asymptotic approximation of amplitude and period, expressions are derived for the non-zero Lyapunov exponent
Calculating Lyapunov Exponents: Applying Products and Evaluating Integrals
McCartney, Mark
2010-01-01
Two common examples of one-dimensional maps (the tent map and the logistic map) are generalized to cases where they have more than one control parameter. In the case of the tent map, this still allows the global Lyapunov exponent to be found analytically, and permits various properties of the resulting global Lyapunov exponents to be investigated…
Calculating Lyapunov Exponents: Applying Products and Evaluating Integrals
McCartney, Mark
2010-01-01
Two common examples of one-dimensional maps (the tent map and the logistic map) are generalized to cases where they have more than one control parameter. In the case of the tent map, this still allows the global Lyapunov exponent to be found analytically, and permits various properties of the resulting global Lyapunov exponents to be investigated…
Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations
Kuznetsov, N. V.; Alexeeva, T. A.; Leonov, G. A.
2014-01-01
Nowadays the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. It may lead to a confusion since there are at least two well-known definitions, which are used in computations: the upper bounds of the exponential growth rate of the norms of linearized system solutions (Lyapunov characteristic exponents, LCEs) and the upper bounds of the exponential growth rate of the singula...
Generalized Lyapunov exponent as a unified characterization of dynamical instabilities.
Akimoto, Takuma; Nakagawa, Masaki; Shinkai, Soya; Aizawa, Yoji
2015-01-01
The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent (exponential dynamical instability) is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov exponent, as a unified characterization of nonexponential and exponential dynamical instabilities in one-dimensional maps. Chaos is classified into three different types, i.e., superexponential, exponential, and subexponential chaos. Using one-dimensional maps, we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the Lyapunov pair. In subexponential chaos, we show superweak chaos, which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth. The scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process, which is related to infinite ergodic theory.
Convex Optimization methods for computing the Lyapunov Exponent of matrices
Protasov, Vladimir Yu
2012-01-01
We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative matrices. The method is based on using special positive homogeneous functionals on $R^{d}_+,$ which gives iterative lower and upper bounds for the Lyapunov exponent. They improve previously known bounds and converge to the real value. The rate of convergence is estimated and the efficiency of the algorithm is demonstrated on several problems from applications (in functional analysis, combinatorics, and lan- guage theory) and on numerical examples with randomly generated matrices. The method computes the Lyapunov exponent with a prescribed accuracy in relatively high dimensions (up to 60). We generalize this approach to all matrices, not necessar- ily nonnegative, derive a new universal upper bound for the Lyapunov exponent, and show that such a lower bound, in general, does not exist.
The Lyapunov exponents of C~1 hyperbolic systems
无
2010-01-01
Let f be a C 1 diffeomorphisim of smooth Riemannian manifold and preserve a hyperbolic ergodic measure μ. We prove that if the Osledec splitting is dominated, then the Lyapunov exponents of μ can be approximated by the exponents of atomic measures on hyperbolic periodic orbits.
Short-Term Forecasting of Urban Water Consumption Based on the Largest Lyapunov Exponent
无
2007-01-01
An approach for short-term forecasting of municipal water consumption was presented based on the largest Lyapunov exponent of chaos theory. The chaotic characteristics of time series of urban water consumption were examined by means of the largest Lyapunov exponent and correlation dimension. By using the largest Lyapunov exponent a short-term forecasting model for urban water consumption was developed, which was compared with the artificial neural network (ANN) approach in a case study. The result indicates that the model based on the largest Lyapunov exponent has higher prediction precision and forecasting stability than the ANN method, and its forecasting mean relative error is 9.6% within its maximum predictable time scale while it is 60.6% beyond the scale.
Scaling of Lyapunov Exponents in Homogeneous, Isotropic DNS
Fitzsimmons, Nicholas; Malaya, Nicholas; Moser, Robert
2013-11-01
Lyapunov exponents measure the rate of separation of initially infinitesimally close trajectories in a chaotic system. Using the exponents, we are able to probe the chaotic nature of homogeneous isotropic turbulence and study the instabilities of the chaotic field. The exponents are measured by calculating the instantaneous growth rate of a linear disturbance, evolved with the linearized Navier-Stokes equation, at each time step. In this talk, we examine these exponents in the context of homogeneous isotropic turbulence with two goals: 1) to investigate the scaling of the exponents with respect to the parameters of forced homogeneous isotropic turbulence, and 2) to characterize the instabilities that lead to chaos in turbulence. Specifically, we explore the scaling of the Lyapunov exponents with respect to the Reynolds number and with respect to the ratio of the integral length scale and the computational domain size.
Lyapunov exponent in quantum mechanics A phase-space approach
Man'ko, V I
2000-01-01
Using the symplectic tomography map, both for the probability distributionsin classical phase space and for the Wigner functions of its quantumcounterpart, we discuss a notion of Lyapunov exponent for quantum dynamics.Because the marginal distributions, obtained by the tomography map, are alwayswell defined probabilities, the correspondence between classical and quantumnotions is very clear. Then we also obtain the corresponding expressions inHilbert space. Some examples are worked out. Classical and quantum exponentsare seen to coincide for local and non-local time-dependent quadraticpotentials. For non-quadratic potentials classical and quantum exponents aredifferent and some insight is obtained on the taming effect of quantummechanics on classical chaos. A detailed analysis is made for the standard map.Providing an unambiguous extension of the notion of Lyapunov exponent toquantum mechnics, the method that is developed is also computationallyefficient in obtaining analytical results for the Lyapunov expone...
Lyapunov Exponents and Covariant Vectors for Turbulent Flow Simulations
Blonigan, Patrick; Murman, Scott; Fernandez, Pablo; Wang, Qiqi
2016-11-01
As computational power increases, engineers are beginning to use scale-resolving turbulent flow simulations for applications in which jets, wakes, and separation dominate. However, the chaotic dynamics exhibited by scale-resolving simulations poses problems for the conventional sensitivity analysis and stability analysis approaches that are vital for design and control. Lyapunov analysis is used to study the chaotic behavior of dynamical systems, including flow simulations. Lyapunov exponents are the growth or a decay rate of specific flow field perturbations called the Lyapunov covariant vectors. Recently, the authors have used Lyapunov analysis to study the breakdown in conventional sensitivity analysis and the cost of new shadowing-based sensitivity analysis. The current work reviews Lyapunov analysis and presents new results for a DNS of turbulent channel flow, wall-modeled channel flow, and a DNS of a low pressure turbine blade. Additionally, the implications of these Lyapunov analyses for computing sensitivities of these flow simulations will be discussed.
A Hyperchaotic Attractor with Multiple Positive Lyapunov Exponents
HU Guo-Si
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.
Nonlinear local Lyapunov exponent and atmospheric predictability research
CHEN; Baohua; LI; Jianping; DING; Ruiqiang
2006-01-01
Because atmosphere itself is a nonlinear system and there exist some problems using the linearized equations to study the initial error growth, in this paper we try to use the error nonlinear growth theory to discuss its evolution, based on which we first put forward a new concept: nonlinear local Lyapunov exponent. It is quite different from the classic Lyapunov exponent because it may characterize the finite time error local average growth and its value depends on the initial condition,initial error, variables, evolution time, temporal and spatial scales. Based on its definition and the atmospheric features, we provide a reasonable algorithm to the exponent for the experimental data,obtain the atmospheric initial error growth in finite time and gain the maximal prediction time. Lastly,taking 500 hPa height field as example, we discuss the application of the nonlinear local Lyapunov exponent in the study of atmospheric predictability and get some reliable results: atmospheric predictability has a distinct spatial structure. Overall, predictability shows a zonal distribution. Prediction time achieves the maximum over tropics, the second near the regions of Antarctic, it is also longer next to the Arctic and in subtropics and the mid-latitude the predictability is lowest. Particularly speaking, the average prediction time near the equation is 12 days and the maximum is located in the tropical Indian, Indonesia and the neighborhood, tropical eastern Pacific Ocean, on these regions the prediction time is about two weeks. Antarctic has a higher predictability than the neighboring latitudes and the prediction time is about 9 days. This feature is more obvious on Southern Hemispheric summer. In Arctic, the predictability is also higher than the one over mid-high latitudes but it is not pronounced as in Antarctic. Mid-high latitude of both Hemispheres (30°S-60°S, 30°-60°N) have the lowest predictability and the mean prediction time is just 3-4 d. In addition
OBSERVING LYAPUNOV EXPONENTS OF INFINITE-DIMENSIONAL DYNAMICAL SYSTEMS.
Ott, William; Rivas, Mauricio A; West, James
2015-12-01
Can Lyapunov exponents of infinite-dimensional dynamical systems be observed by projecting the dynamics into ℝ (N) using a 'typical' nonlinear projection map? We answer this question affirmatively by developing embedding theorems for compact invariant sets associated with C(1) maps on Hilbert spaces. Examples of such discrete-time dynamical systems include time-T maps and Poincaré return maps generated by the solution semigroups of evolution partial differential equations. We make every effort to place hypotheses on the projected dynamics rather than on the underlying infinite-dimensional dynamical system. In so doing, we adopt an empirical approach and formulate checkable conditions under which a Lyapunov exponent computed from experimental data will be a Lyapunov exponent of the infinite-dimensional dynamical system under study (provided the nonlinear projection map producing the data is typical in the sense of prevalence).
Characterizing weak chaos using time series of Lyapunov exponents.
da Silva, R M; Manchein, C; Beims, M W; Altmann, E G
2015-06-01
We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite-time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered (stickiness), semiordered (or semichaotic), and strongly chaotic motion. The dynamics is then investigated looking at the consecutive time spent in each regime, the transition between different regimes, and the regions in the phase space associated to them. Applying our methodology to a chain of coupled standard maps we obtain (i) that it allows for an improved numerical characterization of stickiness in high-dimensional Hamiltonian systems, when compared to the previous analyses based on the distribution of recurrence times; (ii) that the transition probabilities between different regimes are determined by the phase-space volume associated to the corresponding regions; and (iii) the dependence of the Lyapunov exponents with the coupling strength.
Lyapunov exponents of stochastic systems—from micro to macro
Laffargue, Tanguy; Tailleur, Julien; van Wijland, Frédéric
2016-03-01
Lyapunov exponents of dynamical systems are defined from the rates of divergence of nearby trajectories. For stochastic systems, one typically assumes that these trajectories are generated under the ‘same noise realization’. The purpose of this work is to critically examine what this expression means. For Brownian particles, we consider two natural interpretations of the noise: intrinsic to the particles or stemming from the fluctuations of the environment. We show how they lead to different distributions of the largest Lyapunov exponent as well as different fluctuating hydrodynamics for the collective density field. We discuss, both at microscopic and macroscopic levels, the limits in which these noise prescriptions become equivalent. We close this paper by providing an estimate of the largest Lyapunov exponent and of its fluctuations for interacting particles evolving with Dean-Kawasaki dynamics.
Lyapunov exponents for multi-parameter tent and logistic maps.
McCartney, Mark
2011-12-01
The behaviour of logistic and tent maps is studied in cases where the control parameter is dependent on iteration number. Analytic results for global Lyapunov exponent are presented in the case of the tent map and numerical results are presented in the case of the logistic map. In the case of a tent map with N control parameters, the fraction of parameter space for which the global Lyapunov exponent is positive is calculated. The case of bi-parameter maps of period N are investigated.
Do Finite-Size Lyapunov Exponents detect coherent structures?
Karrasch, Daniel; Haller, George
2013-12-01
Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous mathematical link between the FSLE and LCSs, however, has been missing. Here, we prove that an FSLE ridge satisfying certain conditions does signal a nearby ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn indicates a hyperbolic LCS under further conditions. Other FSLE ridges violating our conditions, however, are seen to be false positives for LCSs. We also find further limitations of the FSLE in Lagrangian coherence detection, including ill-posedness, artificial jump-discontinuities, and sensitivity with respect to the computational time step.
Analysis of human standing balance by largest lyapunov exponent.
Liu, Kun; Wang, Hongrui; Xiao, Jinzhuang; Taha, Zahari
2015-01-01
The purpose of this research is to analyse the relationship between nonlinear dynamic character and individuals' standing balance by the largest Lyapunov exponent, which is regarded as a metric for assessing standing balance. According to previous study, the largest Lyapunov exponent from centre of pressure time series could not well quantify the human balance ability. In this research, two improvements were made. Firstly, an external stimulus was applied to feet in the form of continuous horizontal sinusoidal motion by a moving platform. Secondly, a multiaccelerometer subsystem was adopted. Twenty healthy volunteers participated in this experiment. A new metric, coordinated largest Lyapunov exponent was proposed, which reflected the relationship of body segments by integrating multidimensional largest Lyapunov exponent values. By using this metric in actual standing performance under sinusoidal stimulus, an obvious relationship between the new metric and the actual balance ability was found in the majority of the subjects. These results show that the sinusoidal stimulus can make human balance characteristics more obvious, which is beneficial to assess balance, and balance is determined by the ability of coordinating all body segments.
Lyapunov exponents and particle dispersion in drift wave turbulence
Pedersen, T.S.; Michelsen, Poul; Juul Rasmussen, J.
1996-01-01
The Hasegawa-Wakatani model equations for resistive drift waves are solved numerically for a range of values of the coupling due to the parallel electron motion. The largest Lyapunov exponent, lambda(1), is calculated to quantify the unpredictability of the turbulent flow and compared to other...
Lyapunov exponent for aging process in induction motor
Bayram, Duygu; Ünnü, Sezen Yıdırım; Şeker, Serhat
2012-09-01
Nonlinear systems like electrical circuits and systems, mechanics, optics and even incidents in nature may pass through various bifurcations and steady states like equilibrium point, periodic, quasi-periodic, chaotic states. Although chaotic phenomena are widely observed in physical systems, it can not be predicted because of the nature of the system. On the other hand, it is known that, chaos is strictly dependent on initial conditions of the system [1-3]. There are several methods in order to define the chaos. Phase portraits, Poincaré maps, Lyapunov Exponents are the most common techniques. Lyapunov Exponents are the theoretical indicator of the chaos, named after the Russian mathematician Aleksandr Lyapunov (1857-1918). Lyapunov Exponents stand for the average exponential divergence or convergence of nearby system states, meaning estimating the quantitive measure of the chaotic attractor. Negative numbers of the exponents stand for a stable system whereas zero stands for quasi-periodic systems. On the other hand, at least if one of the exponents is positive, this situation is an indicator of the chaos. For estimating the exponents, the system should be modeled by differential equation but even in that case mathematical calculation of Lyapunov Exponents are not very practical and evaluation of these values requires a long signal duration [4-7]. For experimental data sets, it is not always possible to acquire the differential equations. There are several different methods in literature for determining the Lyapunov Exponents of the system [4, 5]. Induction motors are the most important tools for many industrial processes because they are cheap, robust, efficient and reliable. In order to have healthy processes in industrial applications, the conditions of the machines should be monitored and the different working conditions should be addressed correctly. To the best of our knowledge, researches related to Lyapunov exponents and electrical motors are mostly
From Lyapunov modes to their exponents for hard disk systems.
Chung, Tony; Truant, Daniel; Morriss, Gary P
2010-06-01
We demonstrate the preservation of the Lyapunov modes in a system of hard disks by the underlying tangent space dynamics. This result is exact for the Zero modes and correct to order ϵ for the Transverse and Longitudinal-Momentum modes, where ϵ is linear in the mode number. For sufficiently large mode numbers, the ϵ terms become significant and the dynamics no longer preserves the mode structure. We propose a modified Gram-Schmidt procedure based on orthogonality with respect to the center zero space that produces the exact numerical mode. This Gram-Schmidt procedure can also exploit the orthogonality between conjugate modes and their symplectic structure in order to find a simple relation that determines the Lyapunov exponent from the Lyapunov mode. This involves a reclassification of the modes into either direction preserving or form preserving. These analytic methods assume a knowledge of the ordering of the modes within the Lyapunov spectrum, but gives both predictive power for the values of the exponents from the modes and describes the modes in greater detail than was previously achievable. Thus the modes and the exponents contain the same information.
Integral expressions of Lyapunov exponents for autonomous ordinary differential systems
DAI XiongPing
2009-01-01
In the paper,the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space Rd,not necessarily compact,by Liaowise spectral theorems that give integral expressions of Lyapunov exponents.In the context of smooth linear skew-product flows with Polish driving systems,the results are still valid.This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.
Integral expressions of Lyapunov exponents for autonomous ordinary differential systems
无
2009-01-01
In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space Rd, not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.
Lyapunov exponents for one-dimensional aperiodic photonic bandgap structures
Kissel, Glen J.
2011-10-01
Existing in the "gray area" between perfectly periodic and purely randomized photonic bandgap structures are the socalled aperoidic structures whose layers are chosen according to some deterministic rule. We consider here a onedimensional photonic bandgap structure, a quarter-wave stack, with the layer thickness of one of the bilayers subject to being either thin or thick according to five deterministic sequence rules and binary random selection. To produce these aperiodic structures we examine the following sequences: Fibonacci, Thue-Morse, Period doubling, Rudin-Shapiro, as well as the triadic Cantor sequence. We model these structures numerically with a long chain (approximately 5,000,000) of transfer matrices, and then use the reliable algorithm of Wolf to calculate the (upper) Lyapunov exponent for the long product of matrices. The Lyapunov exponent is the statistically well-behaved variable used to characterize the Anderson localization effect (exponential confinement) when the layers are randomized, so its calculation allows us to more precisely compare the purely randomized structure with its aperiodic counterparts. It is found that the aperiodic photonic systems show much fine structure in their Lyapunov exponents as a function of frequency, and, in a number of cases, the exponents are quite obviously fractal.
On finite-size Lyapunov exponents in multiscale systems
Mitchell, Lewis
2012-01-01
We study the effect of regime switches on finite size Lyapunov exponents (FSLEs) in determining the error growth rates and predictability of multiscale systems. We consider a dynamical system involving slow and fast regimes and switches between them. The surprising result is that due to the presence of regimes the error growth rate can be a non-monotonic function of initial error amplitude. In particular, troughs in the large scales of FSLE spectra is shown to be a signature of slow regimes, whereas fast regimes are shown to cause large peaks in the spectra where error growth rates far exceed those estimated from the maximal Lyapunov exponent. We present analytical results explaining these signatures and corroborate them with numerical simulations. We show further that these peaks disappear in stochastic parametrizations of the fast chaotic processes, and the associated FSLE spectra reveal that large scale predictability properties of the full deterministic model are well approximated whereas small scale feat...
Behavior of the Lyapunov Exponent and Phase Transition in Nuclei
WANG Nan; WU Xi-Zhen; LI Zhu-Xia; WANG Ning; ZHUO Yi-Zhong; SUN Xiu-Quan
2000-01-01
Based on the quantum molecular dynamics model, we investigate the dynamical behaviors of the excited nuclear system to simulate the latter stage of heavy ion reactions, which associate with a liquid-gas phase transition. We try to search a microscopic way to describe the phase transition in realnuclei. The Lyapunov exponent is employed and examined for our purpose. We find out that the Lyapunov exponent is one of good microscopic quantities to describe the phase transition in hot nuclei. Coulomb potential and the finite size effect may give a strong influence on the critical temperature. However, the collision term plays a minor role in the process of the liquid-gas phase transition in finite systems.
Lyapunov exponents a tool to explore complex dynamics
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...
Geometry of dynamics, Lyapunov exponents and phase transitions
Caiani, L; Clementi, C; Pettini, M; Caiani, Lando; Casetti, Lapo; Clementi, Cecilia; Pettini, Marco
1997-01-01
The Hamiltonian dynamics of classical planar Heisenberg model is numerically investigated in two and three dimensions. By considering the dynamics as a geodesic flow on a suitable Riemannian manifold, it is possible to analytically estimate the largest Lyapunov exponent in terms of some curvature fluctuations. The agreement between numerical and analytical values for Lyapunov exponents is very good in a wide range of temperatures. Moreover, in the three dimensional case, in correspondence with the second order phase transition, the curvature fluctuations exibit a singular behaviour which is reproduced in an abstract geometric model suggesting that the phase transition might correspond to a change in the topology of the manifold whose geodesics are the motions of the system.
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
Casetti, L; Pettini, M; Casetti, Lapo; Clementi, Cecilia; Pettini, Marco
1996-01-01
This paper deals with the problem of analytically computing the largest Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is succesfully reached within a theoretical framework that makes use of a geometrization of newtonian dynamics in the language of Riemannian geometry. A new point of view about the origin of chaos in these systems is obtained independently of homoclinic intersections. Chaos is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of Jacobi equation for geodesic spread. Under general conditions ane effective stability equation is derived; an analytic formula for the growth-rate of its solutions is worked out and applied to the Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent agreement is found the theoretical prediction and the values of the Lyapunov exponent obtained by numerical simulations for both models.
A local Echo State Property through the largest Lyapunov exponent.
Wainrib, Gilles; Galtier, Mathieu N
2016-04-01
Echo State Networks are efficient time-series predictors, which highly depend on the value of the spectral radius of the reservoir connectivity matrix. Based on recent results on the mean field theory of driven random recurrent neural networks, enabling the computation of the largest Lyapunov exponent of an ESN, we develop a cheap algorithm to establish a local and operational version of the Echo State Property.
Bohmian quantum mechanical and classical Lyapunov exponents for kicked rotor
Zheng Yindong [Department of Physics, University of North Texas, Denton, TX 76203-1427 (United States); Kobe, Donald H. [Department of Physics, University of North Texas, Denton, TX 76203-1427 (United States)], E-mail: kobe@unt.edu
2008-04-15
Using de Broglie-Bohm approach to quantum theory, we show that the kicked rotor at quantum resonance exhibits quantum chaos for the control parameter K above a threshold. Lyapunov exponents are calculated from the method of Benettin et al. for bounded systems for both the quantum and classical kicked rotor. In the chaotic regime we find stability regions for control parameters equal to even and odd multiples of {pi}, but the quantum regions are only remnants of the classical ones.
Local Lyapunov exponents sublimiting growth rates of linear random differential equations
Siegert, Wolfgang
2009-01-01
Establishing a new concept of local Lyapunov exponents the author brings together two separate theories, namely Lyapunov exponents and the theory of large deviations. Specifically, a linear differential system is considered which is controlled by a stochastic process that during a suitable noise-intensity-dependent time is trapped near one of its so-called metastable states. The local Lyapunov exponent is then introduced as the exponential growth rate of the linear system on this time scale. Unlike classical Lyapunov exponents, which involve a limit as time increases to infinity in a fixed system, here the system itself changes as the noise intensity converges, too.
A Simple Method for the Computation of the COnditional Lyapunov Exponents
无
1999-01-01
An handy method of calculating the conditional Lyapunov exponents is put forward.Lyapunov exponents of differential dynamical systems and the conditional Lyapunov exponents can be acquired easily with the method.The method has been successfully used in kinds of synchronization ,such as continuous driving synchronization,impulsive(sporadic)driving synchronization,intermittently driving synchronization.The conditional Lyapunov exponents obtained with our method can give the largest and the best time interval for impulsive synchronization that can hardly be settled in other ways.
GPU and APU computations of Finite Time Lyapunov Exponent fields
Conti, Christian; Rossinelli, Diego; Koumoutsakos, Petros
2012-03-01
We present GPU and APU accelerated computations of Finite-Time Lyapunov Exponent (FTLE) fields. The calculation of FTLEs is a computationally intensive process, as in order to obtain the sharp ridges associated with the Lagrangian Coherent Structures an extensive resampling of the flow field is required. The computational performance of this resampling is limited by the memory bandwidth of the underlying computer architecture. The present technique harnesses data-parallel execution of many-core architectures and relies on fast and accurate evaluations of moment conserving functions for the mesh to particle interpolations. We demonstrate how the computation of FTLEs can be efficiently performed on a GPU and on an APU through OpenCL and we report over one order of magnitude improvements over multi-threaded executions in FTLE computations of bluff body flows.
Characterizing heart rate variability by scale-dependent Lyapunov exponent
Hu, Jing; Gao, Jianbo; Tung, Wen-wen
2009-06-01
Previous studies on heart rate variability (HRV) using chaos theory, fractal scaling analysis, and many other methods, while fruitful in many aspects, have produced much confusion in the literature. Especially the issue of whether normal HRV is chaotic or stochastic remains highly controversial. Here, we employ a new multiscale complexity measure, the scale-dependent Lyapunov exponent (SDLE), to characterize HRV. SDLE has been shown to readily characterize major models of complex time series including deterministic chaos, noisy chaos, stochastic oscillations, random 1/f processes, random Levy processes, and complex time series with multiple scaling behaviors. Here we use SDLE to characterize the relative importance of nonlinear, chaotic, and stochastic dynamics in HRV of healthy, congestive heart failure, and atrial fibrillation subjects. We show that while HRV data of all these three types are mostly stochastic, the stochasticity is different among the three groups.
Lyapunov exponents of linear cocycles continuity via large deviations
Duarte, Pedro
2016-01-01
The aim of this monograph is to present a general method of proving continuity of Lyapunov exponents of linear cocycles. The method uses an inductive procedure based on a general, geometric version of the Avalanche Principle. The main assumption required by this method is the availability of appropriate large deviation type estimates for quantities related to the iterates of the base and fiber dynamics associated with the linear cocycle. We establish such estimates for various models of random and quasi-periodic cocycles. Our method has its origins in a paper of M. Goldstein and W. Schlag. Our present work expands upon their approach in both depth and breadth. We conclude this monograph with a list of related open problems, some of which may be treated using a similar approach.
Refining and classifying finite-time Lyapunov exponent ridges
Allshouse, Michael R
2015-01-01
While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of an FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by an FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model.
Computation of entropy and Lyapunov exponent by a shift transform.
Matsuoka, Chihiro; Hiraide, Koichi
2015-10-01
We present a novel computational method to estimate the topological entropy and Lyapunov exponent of nonlinear maps using a shift transform. Unlike the computation of periodic orbits or the symbolic dynamical approach by the Markov partition, the method presented here does not require any special techniques in computational and mathematical fields to calculate these quantities. In spite of its simplicity, our method can accurately capture not only the chaotic region but also the non-chaotic region (window region) such that it is important physically but the (Lebesgue) measure zero and usually hard to calculate or observe. Furthermore, it is shown that the Kolmogorov-Sinai entropy of the Sinai-Ruelle-Bowen measure (the physical measure) coincides with the topological entropy.
Computation of entropy and Lyapunov exponent by a shift transform
Matsuoka, Chihiro, E-mail: matsuoka.chihiro.mm@ehime-u.ac.jp [Department of Physics, Graduate School of Science and Technology, Ehime University, Matsuyama, Ehime 790-8577 (Japan); Hiraide, Koichi [Department of Mathematics, Graduate School of Science and Technology, Ehime University, Matsuyama, Ehime 790-8577 (Japan)
2015-10-15
We present a novel computational method to estimate the topological entropy and Lyapunov exponent of nonlinear maps using a shift transform. Unlike the computation of periodic orbits or the symbolic dynamical approach by the Markov partition, the method presented here does not require any special techniques in computational and mathematical fields to calculate these quantities. In spite of its simplicity, our method can accurately capture not only the chaotic region but also the non-chaotic region (window region) such that it is important physically but the (Lebesgue) measure zero and usually hard to calculate or observe. Furthermore, it is shown that the Kolmogorov-Sinai entropy of the Sinai-Ruelle-Bowen measure (the physical measure) coincides with the topological entropy.
Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics.
Vannitsem, Stéphane
2017-03-01
The deterministic equations describing the dynamics of the atmosphere (and of the climate system) are known to display the property of sensitivity to initial conditions. In the ergodic theory of chaos, this property is usually quantified by computing the Lyapunov exponents. In this review, these quantifiers computed in a hierarchy of atmospheric models (coupled or not to an ocean) are analyzed, together with their local counterparts known as the local or finite-time Lyapunov exponents. It is shown in particular that the variability of the local Lyapunov exponents (corresponding to the dominant Lyapunov exponent) decreases when the model resolution increases. The dynamics of (finite-amplitude) initial condition errors in these models is also reviewed, and in general found to display a complicated growth far from the asymptotic estimates provided by the Lyapunov exponents. The implications of these results for operational (high resolution) atmospheric and climate modelling are also discussed.
Dünki, Rudolf M.
2000-11-01
Limited predictability is one of the remarkable features of deterministic chaos and this feature may be quantized in terms of Lyapunov exponents. Accordingly, Lyapunov-exponent estimates may be expected to follow in a natural way from forecast algorithms. Exploring this idea, we propose a method estimating the largest Lyapunov exponent from a time series which uses the behavior of so-called simplex forecasts. The method considers the estimation of properties of the distribution of local simplex expansion coefficients. These are also used for the definition of error bars for the Lyapunov-exponent estimates and allows for selective forecasts with improved prediction accuracy. We demonstrate these concepts on standard test examples and three realistic applications to time series concerning largest Lyapunov-exponent estimation of an experimentally obtained hyperchaotic NMR signal, brain state differentiation, and stock-market prediction.
Entropy, Lyapunov Exponents and Escape Rates in Open Systems
Demers, Mark; Young, Lai-Sang
2011-01-01
We study the relation between escape rates and pressure in general dynamical systems with holes, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents. Central to the discussion is the formulation of a class of invariant measures supported on the survivor set over which we take the supremum to measure the pressure. Upper bounds for escape rates are proved for general diffeomorphisms of manifolds, possibly with singularities, for arbitrary holes and natural initial distributions including Lebesgue and SRB measures. Lower bounds do not hold in such generality, but for systems admitting Markov tower extensions with spectral gaps, we prove the equality of the escape rate with the absolute value of the pressure and the existence of an invariant measure realizing the escape rate, i.e. we prove a full variational principle. As an application of our results, we prove a variational principle for the billiard map associated with a planar Lorentz gas of finite horizon ...
Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents
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.
[A Standing Balance Evaluation Method Based on Largest Lyapunov Exponent].
Liu, Kun; Wang, Hongrui; Xiao, Jinzhuang; Zhao, Qing
2015-12-01
In order to evaluate the ability of human standing balance scientifically, we in this study proposed a new evaluation method based on the chaos nonlinear analysis theory. In this method, a sinusoidal acceleration stimulus in forward/backward direction was forced under the subjects' feet, which was supplied by a motion platform. In addition, three acceleration sensors, which were fixed to the shoulder, hip and knee of each subject, were applied to capture the balance adjustment dynamic data. Through reconstructing the system phase space, we calculated the largest Lyapunov exponent (LLE) of the dynamic data of subjects' different segments, then used the sum of the squares of the difference between each LLE (SSDLLE) as the balance capabilities evaluation index. Finally, 20 subjects' indexes were calculated, and compared with evaluation results of existing methods. The results showed that the SSDLLE were more in line with the subjects' performance during the experiment, and it could measure the body's balance ability to some extent. Moreover, the results also illustrated that balance level was determined by the coordinate ability of various joints, and there might be more balance control strategy in the process of maintaining balance.
He, Jianbin; Yu, Simin; Cai, Jianping
2016-12-01
Lyapunov exponent is an important index for describing chaotic systems behavior, and the largest Lyapunov exponent can be used to determine whether a system is chaotic or not. For discrete-time dynamical systems, the Lyapunov exponents are calculated by an eigenvalue method. In theory, according to eigenvalue method, the more accurate calculations of Lyapunov exponent can be obtained with the increment of iterations, and the limits also exist. However, due to the finite precision of computer and other reasons, the results will be numeric overflow, unrecognized, or inaccurate, which can be stated as follows: (1) The iterations cannot be too large, otherwise, the simulation result will appear as an error message of NaN or Inf; (2) If the error message of NaN or Inf does not appear, then with the increment of iterations, all Lyapunov exponents will get close to the largest Lyapunov exponent, which leads to inaccurate calculation results; (3) From the viewpoint of numerical calculation, obviously, if the iterations are too small, then the results are also inaccurate. Based on the analysis of Lyapunov-exponent calculation in discrete-time systems, this paper investigates two improved algorithms via QR orthogonal decomposition and SVD orthogonal decomposition approaches so as to solve the above-mentioned problems. Finally, some examples are given to illustrate the feasibility and effectiveness of the improved algorithms.
Pseudo-Lyapunov exponents and predictability of Hodgkin-Huxley neuronal network dynamics.
Sun, Yi; Zhou, Douglas; Rangan, Aaditya V; Cai, David
2010-04-01
We present a numerical analysis of the dynamics of all-to-all coupled Hodgkin-Huxley (HH) neuronal networks with Poisson spike inputs. It is important to point out that, since the dynamical vector of the system contains discontinuous variables, we propose a so-called pseudo-Lyapunov exponent adapted from the classical definition using only continuous dynamical variables, and apply it in our numerical investigation. The numerical results of the largest Lyapunov exponent using this new definition are consistent with the dynamical regimes of the network. Three typical dynamical regimes-asynchronous, chaotic and synchronous, are found as the synaptic coupling strength increases from weak to strong. We use the pseudo-Lyapunov exponent and the power spectrum analysis of voltage traces to characterize the types of the network behavior. In the nonchaotic (asynchronous or synchronous) dynamical regimes, i.e., the weak or strong coupling limits, the pseudo-Lyapunov exponent is negative and there is a good numerical convergence of the solution in the trajectory-wise sense by using our numerical methods. Consequently, in these regimes the evolution of neuronal networks is reliable. For the chaotic dynamical regime with an intermediate strong coupling, the pseudo-Lyapunov exponent is positive, and there is no numerical convergence of the solution and only statistical quantifications of the numerical results are reliable. Finally, we present numerical evidence that the value of pseudo-Lyapunov exponent coincides with that of the standard Lyapunov exponent for systems we have been able to examine.
Backward Finite-Time Lyapunov Exponents in Inertial Flows.
Gunther, Tobias; Theisel, Holger
2017-01-01
Inertial particles are finite-sized objects that are carried by fluid flows and in contrast to massless tracer particles they are subject to inertia effects. In unsteady flows, the dynamics of tracer particles have been extensively studied by the extraction of Lagrangian coherent structures (LCS), such as hyperbolic LCS as ridges of the Finite-Time Lyapunov Exponent (FTLE). The extension of the rich LCS framework to inertial particles is currently a hot topic in the CFD literature and is actively under research. Recently, backward FTLE on tracer particles has been shown to correlate with the preferential particle settling of small inertial particles. For larger particles, inertial trajectories may deviate strongly from (massless) tracer trajectories, and thus for a better agreement, backward FTLE should be computed on inertial trajectories directly. Inertial backward integration, however, has not been possible until the recent introduction of the influence curve concept, which - given an observation and an initial velocity - allows to recover all sources of inertial particles as tangent curves of a derived vector field. In this paper, we show that FTLE on the influence curve vector field is in agreement with preferential particle settling and more importantly it is not only valid for small (near-tracer) particles. We further generalize the influence curve concept to general equations of motion in unsteady spatio-velocity phase spaces, which enables backward integration with more general equations of motion. Applying the influence curve concept to tracer particles in the spatio-velocity domain emits streaklines in massless flows as tangent curves of the influence curve vector field. We demonstrate the correlation between inertial backward FTLE and the preferential particle settling in a number of unsteady vector fields.
Joint Statistics of Finite Time Lyapunov Exponents in Isotropic Turbulence
Johnson, Perry; Meneveau, Charles
2014-11-01
Recently, the notion of Lagrangian Coherent Structures (LCS) has gained attention as a tool for qualitative visualization of flow features. LCS visualize repelling and attracting manifolds marked by local ridges in the field of maximal and minimal finite-time Lyapunov exponents (FTLE), respectively. To provide a quantitative characterization of FTLEs, the statistical theory of large deviations can be used based on the so-called Cramér function. To obtain the Cramér function from data, we use both the method based on measuring moments and measuring histograms (with finite-size correction). We generalize the formalism to characterize the joint distributions of the two independent FTLEs in 3D. The ``joint Cramér function of turbulence'' is measured from the Johns Hopkins Turbulence Databases (JHTDB) isotropic simulation at Reλ = 433 and results are compared with those computed using only the symmetric part of the velocity gradient tensor, as well as with those of instantaneous strain-rate eigenvalues. We also extend the large-deviation theory to study the statistics of the ratio of FTLEs. When using only the strain contribution of the velocity gradient, the maximal FTLE nearly doubles in magnitude and the most likely ratio of FTLEs changes from 4:1:-5 to 8:3:-11, highlighting the role of rotation in de-correlating the fluid deformations along particle paths. Supported by NSF Graduate Fellowship (DGE-1232825), a JHU graduate Fellowship, and NSF Grant CMMI-0941530. CM thanks Prof. Luca Biferale for useful discussions on the subject.
Lyapunov exponent of chaos generated by acousto-optic modulators with feedback
Ghosh, Anjan K.; Verma, Pramode
2011-01-01
Generation of chaos from acousto-optic modulators with an electronic feedback has been studied for several years. Such chaotic signals have an important application in providing secure encryption in free-space optical communication systems. Lyapunov exponent is an important parameter for analysis of chaos generated by a nonlinear system. The Lyapunov exponent of an acousto-optic system is determined and calculated in this paper to understand the dependence of the chaotic response on the system parameters such as bias, feedback gain, input intensity and initial condition exciting the cell. Analysis of chaos using Lyapunov exponent is consistent with bifurcation analysis and is useful in encrypting data signals.
Morawetz, K
1999-07-01
Within the frame of kinetic theory a response function is derived for finite Fermi systems which includes dissipation in relaxation time approximation and a contribution from additional chaotic processes characterized by the largest Lyapunov exponent. A generalized local density approximation is presented including the effect of many particle relaxation and the additional chaotic scattering. For small Lyapunov exponents relative to the product of wave vector and Fermi time. Therefore the transport coefficients can be connected with the largest positive Lyapunov exponent in the same way as known the transport theory in relaxation time approximation. (author)
Finite-time Lyapunov exponents in time-delayed nonlinear dynamical systems.
Kanno, Kazutaka; Uchida, Atsushi
2014-03-01
We introduce a method for the calculation of finite-time Lyapunov exponents in time-delayed nonlinear dynamical systems. We apply the method to the Mackey-Glass model with time-delayed feedback. We investigate the standard deviation of the probability distribution of the finite-time Lyapunov exponents when the finite time or the delay time is changed. It is found that the standard deviation decreases in a power-law scaling with the exponent ∼0.5 as the finite time or the delay time is increased. Similar results are obtained for the finite-time Lyapunov spectrum.
Lyapunov exponents for small aspect ratio Rayleigh-Bénard convection.
Scheel, J D; Cross, M C
2006-12-01
Leading order Lyapunov exponents and their corresponding eigenvectors have been computed numerically for small aspect ratio, three-dimensional Rayleigh-Benard convection cells with no-slip boundary conditions. The parameters are the same as those used by Ahlers and Behringer [Phys. Rev. Lett. 40, 712 (1978)] and Gollub and Benson [J. Fluid Mech. 100, 449 (1980)] in their work on a periodic time dependence in Rayleigh-Benard convection cells. Our work confirms that the dynamics in these cells truly are chaotic as defined by a positive Lyapunov exponent. The time evolution of the leading order Lyapunov eigenvector in the chaotic regime will also be discussed. In addition we study the contributions to the leading order Lyapunov exponent for both time periodic and aperiodic states and find that while repeated dynamical events such as dislocation creation/annihilation and roll compression do contribute to the short time Lyapunov exponent dynamics, they do not contribute to the long time Lyapunov exponent. We find instead that nonrepeated events provide the most significant contribution to the long time leading order Lyapunov exponent.
Gustavsson, K
2013-01-01
We calculate the Lyapunov exponents describing spatial clustering of particles advected in one- and two-dimensional random velocity fields at finite Kubo number Ku (a dimensionless parameter characterising the correlation time of the velocity field). In one dimension we obtain accurate results up to Ku ~ 1 by resummation of a perturbation expansion in Ku. At large Kubo numbers we compute the Lyapunov exponent by taking into account the fact that the particles follow the minima of the potential function corresponding to the velocity field. In two dimensions we compute the first four non-vanishing terms in the small-Ku expansion of the Lyapunov exponents. For large Kubo numbers we estimate the Lyapunov exponents by assuming that the particles sample stagnation points of the velocity field with det A > 0 and Tr A < 0 where A is the matrix of flow-velocity gradients.
Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem
Anderson, Rodney L.; Lo, Martin W.; Born, George H.
2003-01-01
Dynamical systems theory has recently been employed to design trajectories within the three-body problem for several missions. This research has applied one stability technique, the calculation of local Lyapunov exponents, to such trajectories. Local Lyapunov exponents give an indication of the effects that perturbations or maneuvers will have on trajectories over a specified time. A numerical comparison of local Lyapunov exponents was first made with the distance random perturbations traveled from a nominal trajectory, and the local Lyapunov exponents were found to correspond well with the perturbations that caused the greatest deviation from the nominal. This would allow them to be used as an indicator of the points where it would be important to reduce navigation uncertainties.
On the bound of the Lyapunov exponents for the fractional differential systems.
Li, Changpin; Gong, Ziqing; Qian, Deliang; Chen, YangQuan
2010-03-01
In recent years, fractional(-order) differential equations have attracted increasing interests due to their applications in modeling anomalous diffusion, time dependent materials and processes with long range dependence, allometric scaling laws, and complex networks. Although an autonomous system cannot define a dynamical system in the sense of semigroup because of the memory property determined by the fractional derivative, we can still use the Lyapunov exponents to discuss its dynamical evolution. In this paper, we first define the Lyapunov exponents for fractional differential systems then estimate the bound of the corresponding Lyapunov exponents. For linear fractional differential system, the bounds of its Lyapunov exponents are conveniently derived which can be regarded as an example for the theoretical results established in this paper. Numerical example is also included which supports the theoretical analysis.
Application of local Lyapunov exponents to maneuver design and navigation in the three-body problem
Anderson, Rodney L.; Lo, Martin W.; Born, George H.
2003-01-01
Dynamical systems theory has recently been employed to design trajectories within the three-body problem for several missions. This research has applied one stability technique, the calculation of local Lyapunov exponents, to such trajectories. Local Lyapunov exponents give an indication of the effects that perturbations or maneuvers will have on trajectories over a specified time. A numerical comparison of local Lyapunov exponents was first made with the distance random perturbations traveled from a nominal trajectory, and the local Lyapunov exponents were found to correspond well with the perturbations that caused the greatest deviation from the nominal. This would allow them to be used as an indicator of the points where it would be important to reduce navigation uncertainties.
Wang, Zhikang; Lou, Haifang; Sun, Jianzhong
2011-07-01
Attempting to use nonlinear spatiotemporal Lyapunov exponent to characterize fMRI brain functional connectivity of anxiety disease patients, we adopted the methods of nonlinear spatiotemporal Lyapunov exponent and linear correlation coefficients to analyses fMRI datum of 11 anxiety disease patients and 11 healthy volunteers, respectively. The results show that there are significant normalized variance exponent (NVE) differences in Inferior Frontal Gyrus (rIFG) and Medial Frontal Gyrus (MFG) between the two groups (PLyapunov exponent method had higher sensitivity than the correlation coefficient method in the characterization of functional connectivity; Anxiety disease patients have abnormal functional connectivity in rIFG and MFG during our experiment.
Zeta function for the Lyapunov exponent of a product of random matrices
Mainieri, R. (Neils Bohr Institute, Blegdamsvej 17, Copenhagen O, 2100 (Denmark) Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States))
1992-03-30
A cycle expansion for the Lyapunov exponent of a product of random matrices is derived. The formula is nonperturbative and numerically effective, which allows the Lyapunov exponent to be computed to high accuracy. In particular, the free energy and heat capacity are computed for the one-dimensional Ising model with quenched disorder. The formula is derived by using a Bernoulli dynamical system to mimic the randomness.
Lyapunov Exponents and Kolmogorov-Sinai Entropy for the Lorentz Gas at Low Densities
van Beijeren, Henk; Dorfman, J. R.
1995-05-01
The Lyapunov exponents and the Kolmogorov-Sinai (KS) entropy for a two-dimensional Lorentz gas at low densities are defined for general nonequilibrium states and calculated with the use of a Lorentz-Boltzmann type equation. In equilibrium the density dependence of these quantities, predicted by Krylov, is recovered and explicit expressions are obtained. The relationship between KS entropy, Lyapunov exponents, and diffusion coefficients, developed by Gaspard and Nicolis, is generalized to a wide class of nonequilibrium states.
Lyapunov exponent evaluation of a digital watermarking scheme proven to be secure
Bahi, Jacques M; Guyeux, Christophe
2012-01-01
In our previous researches, a new digital watermarking scheme based on chaotic iterations has been introduced. This scheme was both stego-secure and topologically secure. The stego-security is to face an attacker in the "watermark only attack" category, whereas the topological security concerns other categories of attacks. Its Lyapunov exponent is evaluated here, to quantify the chaos generated by this scheme. Keywords : Lyapunov exponent; Information hiding; Security; Chaotic iterations; Digital Watermarking.
Symmetry of Lyapunov exponents in bifurcation structures of one-dimensional maps.
Shimada, Yutaka; Takagi, Emiko; Ikeguchi, Tohru
2016-12-01
We observe a symmetry of Lyapunov exponents in bifurcation structures of one-dimensional maps in which there exists a pair of parameter values in a dynamical system such that two dynamical systems with these paired parameter values have the same Lyapunov exponent. We show that this is a consequence of the presence of an invariant transformation from a dynamical system with one of the two paired parameter values to that with another parameter value, which does not change natures of dynamical systems.
Critical behavior of the Lyapunov exponent in type-III intermittency
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.
Mizuta, Keisuke; Tokita, Takashi; Ito, Yatsuji; Aoki, Mitsuhiro; Kuze, Bunya
2009-12-01
In the present study, we investigated the body sway in patients with unilateral vestibular dysfunction by the largest Lyapunov exponents using a chaotic time series analysis. The largest Lyapunov exponent is regarded as a parameter indexing an orbital instability. Subjects consisted of 55 normal healthy subjects, 11 patients diagnosed as having vestibular neuritis (VN), 6 patients diagnosed as having sudden deafness (SD) with vertigo, 23 patients diagnosed as having Meniere disease (MD), 11 patients diagnosed as having benign paroxysmal positional vertigo (BPPV) and 14 patients diagnosed as having other vestibular disorders. Using a stabilometer, the sway of the body center of gravity in an upright standing position was recorded with eyes open and closed for 60 seconds under each condition. From the time series data obtained, the largest Lyapunov exponents were calculated using a chaos analysis program. In normal healthy subjects and patients with unilateral vestibular dysfunction, the largest Lyapunov exponents on right-left sway were larger than those on forward-backward sway with eyes open and closed. The largest Lyapunov exponents in patients with unilateral vestibular dysfunction on forward-backward sway with eyes closed were significantly larger than those in normal healthy subjects. A few patients with the instability of standing posture judged from conventional analysis (area of sway, locus length per time) showed higher values of the LLE. We investigated the variation of the values of the largest Lyapunov exponents in patients with unilateral vestibular dysfunction at each stage during recovery from their vestibular damage. The largest Lyapunov exponents at the early stage with stable standing posture were significantly higher than those at the late stable stage with stable standing posture. Some patients at the very early stage had lower values of the largest Lyapunov exponents. We speculate that the orbital instability indicated by the values of the
Study on the expression of systematic Lyapunov exponent based on UPOs
岳毅宏; 韩文秀; 程国平
2004-01-01
The natural measure of a certain area in phase space is defined firstly. On the basis of natural measure, the expression of Lyapunov exponent based on unstable periodic orbits (UPOs) of chaotic systems is deduced from theoretical aspect. Then, by means of the inherent relation between UPOs and systematic Lyapunov exponent, the transitional mechanism and route of chaotic systems from low-dimensional chaos to high-dimensional chaos are explained. In the end,a novel method for computing systematic Lyapunov exponents based on UPOs is proposed. Its computing procedure is also summarized. The chaotic system described by Henon map is taken as example. Through calculating the Lypunov exponents of this system, validity of the suggested method is verified.
Geometrical constraints on finite-time Lyapunov exponents in two and three dimensions.
Thiffeault, Jean-Luc; Boozer, Allen H.
2001-03-01
Constraints are found on the spatial variation of finite-time Lyapunov exponents of two- and three-dimensional systems of ordinary differential equations. In a chaotic system, finite-time Lyapunov exponents describe the average rate of separation, along characteristic directions, of neighboring trajectories. The solution of the equations is a coordinate transformation that takes initial conditions (the Lagrangian coordinates) to the state of the system at a later time (the Eulerian coordinates). This coordinate transformation naturally defines a metric tensor, from which the Lyapunov exponents and characteristic directions are obtained. By requiring that the Riemann curvature tensor vanish for the metric tensor (a basic result of differential geometry in a flat space), differential constraints relating the finite-time Lyapunov exponents to the characteristic directions are derived. These constraints are realized with exponential accuracy in time. A consequence of the relations is that the finite-time Lyapunov exponents are locally small in regions where the curvature of the stable manifold is large, which has implications for the efficiency of chaotic mixing in the advection-diffusion equation. The constraints also modify previous estimates of the asymptotic growth rates of quantities in the dynamo problem, such as the magnitude of the induced current. (c) 2001 American Institute of Physics.
The Multivariate Largest Lyapunov Exponent as an Age-Related Metric of Quiet Standing Balance
Kun Liu
2015-01-01
Full Text Available The largest Lyapunov exponent has been researched as a metric of the balance ability during human quiet standing. However, the sensitivity and accuracy of this measurement method are not good enough for clinical use. The present research proposes a metric of the human body’s standing balance ability based on the multivariate largest Lyapunov exponent which can quantify the human standing balance. The dynamic multivariate time series of ankle, knee, and hip were measured by multiple electrical goniometers. Thirty-six normal people of different ages participated in the test. With acquired data, the multivariate largest Lyapunov exponent was calculated. Finally, the results of the proposed approach were analysed and compared with the traditional method, for which the largest Lyapunov exponent and power spectral density from the centre of pressure were also calculated. The following conclusions can be obtained. The multivariate largest Lyapunov exponent has a higher degree of differentiation in differentiating balance in eyes-closed conditions. The MLLE value reflects the overall coordination between multisegment movements. Individuals of different ages can be distinguished by their MLLE values. The standing stability of human is reduced with the increment of age.
The Multivariate Largest Lyapunov Exponent as an Age-Related Metric of Quiet Standing Balance.
Liu, Kun; Wang, Hongrui; Xiao, Jinzhuang
2015-01-01
The largest Lyapunov exponent has been researched as a metric of the balance ability during human quiet standing. However, the sensitivity and accuracy of this measurement method are not good enough for clinical use. The present research proposes a metric of the human body's standing balance ability based on the multivariate largest Lyapunov exponent which can quantify the human standing balance. The dynamic multivariate time series of ankle, knee, and hip were measured by multiple electrical goniometers. Thirty-six normal people of different ages participated in the test. With acquired data, the multivariate largest Lyapunov exponent was calculated. Finally, the results of the proposed approach were analysed and compared with the traditional method, for which the largest Lyapunov exponent and power spectral density from the centre of pressure were also calculated. The following conclusions can be obtained. The multivariate largest Lyapunov exponent has a higher degree of differentiation in differentiating balance in eyes-closed conditions. The MLLE value reflects the overall coordination between multisegment movements. Individuals of different ages can be distinguished by their MLLE values. The standing stability of human is reduced with the increment of age.
A method to calculate finite-time Lyapunov exponents for inertial particles in incompressible flows
Garaboa-Paz, D.; Pérez-Muñuzuri, V.
2015-10-01
The present study aims to improve the calculus of finite-time Lyapunov exponents (FTLEs) applied to describe the transport of inertial particles in a fluid flow. To this aim, the deformation tensor is modified to take into account that the stretching rate between particles separated by a certain distance is influenced by the initial velocity of the particles. Thus, the inertial FTLEs (iFTLEs) are defined in terms of the maximum stretching between infinitesimally close trajectories that have different initial velocities. The advantages of this improvement, if compared to the standard method (Shadden et al., 2005), are discussed for the double-gyre flow and the meandering jet flow. The new method allows one to identify the initial velocity that inertial particles must have in order to maximize their dispersion.
Ding, Ruiqiang; Li, Jianping; Li, Baosheng
2017-09-01
For an n-dimensional chaotic system, we extend the definition of the nonlinear local Lyapunov exponent (NLLE) from one- to n-dimensional spectra, and present a method for computing the NLLE spectrum. The method is tested on three chaotic systems with different complexity. The results indicate that the NLLE spectrum realistically characterizes the growth rates of initial error vectors along different directions from the linear to nonlinear phases of error growth. This represents an improvement over the traditional Lyapunov exponent spectrum, which only characterizes the error growth rates during the linear phase of error growth. In addition, because the NLLE spectrum can effectively separate the slowly and rapidly growing perturbations, it is shown to be more suitable for estimating the predictability of chaotic systems, as compared to the traditional Lyapunov exponent spectrum.
ALI M.; SAHA L.M.
2005-01-01
A chaotic dynamical system is characterized by a positive averaged exponential separation of two neighboring trajectories over a chaotic attractor. Knowledge of the Largest Lyapunov Exponent λ1 of a dynamical system over a bounded attractor is necessary and sufficient for determining whether it is chaotic (λ1＞0) or not (λ1≤0). We intended in this work to elaborate the connection between Local Lyapunov Exponents and the Largest Lyapunov Exponent where an altemative method to calculate λ1has emerged. Finally, we investigated some characteristics of the fixed points and periodic orbits embedded within a chaotic attractor which led to the conclusion of the existence of chaotic attractors that may not embed in any fixed point or periodic orbit within it.
New prediction of chaotic time series based on local Lyapunov exponent
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.
Lyapunov exponent corresponding to enslaved phase dynamics: Estimation from time series.
Moskalenko, Olga I; Koronovskii, Alexey A; Hramov, Alexander E
2015-07-01
A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the supercritical region of the control parameters. The main results are illustrated with the help of the examples of the noised circle map, the nonautonomous Van der Pole oscillator in the presence of noise, and coupled chaotic Rössler systems.
Shihui Fu; Qi Wang
2006-01-01
Using the properties of chaos synchronization. the method for estimating the largest Lyapunov exponent in a multibody system with dry friction is presented in this paper. The Lagrange equations with multipliers of the systems are given in matrix form. which is adequate for numerical calculation. The approach for calculating the generalized velocity and acceleration of the slider is given to determine slipping or sticking of the slider in the systems. For slip-slip and stick-slip multibody systems, their largest Lyapunov exponents are calculated to characterize their dynamics.
Structured scale dependence in the Lyapunov exponent of a Boolean chaotic map.
Cohen, Seth D
2015-04-01
We report on structures in a scale-dependent Lyapunov exponent of an experimental chaotic map that arise due to discontinuities in the map. The chaos is realized in an autonomous Boolean network, which is constructed using asynchronous logic gates to form a map operator that outputs an unclocked pulse-train of varying widths. The map operator executes pulse-width stretching and folding and the operator's output is fed back to its input to continuously iterate the map. Using a simple model, we show that the structured scale-dependence in the system's Lyapunov exponent is the result of the discrete logic elements in the map operator's stretching function.
An Isomorphism between Lyapunov Exponents and Shannon's Channel Capacity
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.
Universal scaling of Lyapunov-exponent fluctuations in space-time chaos.
Pazó, Diego; López, Juan M; Politi, Antonio
2013-06-01
Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase space. A recent numerical study of spatially extended systems has revealed that the diffusion coefficient D of the Lyapunov exponents (LEs) exhibits a nontrivial scaling behavior, D(L)~L(-γ), with the system size L. Here, we show that the wandering exponent γ can be expressed in terms of the roughening exponents associated with the corresponding "Lyapunov surface." Our theoretical predictions are supported by the numerical analysis of several spatially extended systems. In particular, we find that the wandering exponent of the first LE is universal: in view of the known relationship with the Kardar-Parisi-Zhang equation, γ can be expressed in terms of known critical exponents. Furthermore, our simulations reveal that the bulk of the spectrum exhibits a clearly different behavior and suggest that it belongs to a possibly unique universality class, which has, however, yet to be identified.
No ISCOs in Charged Myers Perry Spacetimes by Measuring Lyapunov Exponent
Pradhan, Parthapratim
2015-01-01
By computing coordinate time Lyapunov exponent, we prove that for more than four spacetime dimensions (N ≥ 3), there are no Innermost Stable Circular Orbit (ISCO) in charged Myers Perry blackhole spacetime.Using it, we show that the instability of equatorial circular geodesics, both massive and massless particles for such types of blackhole space-times.
Predicting Traffic Flow in Local Area Networks by the Largest Lyapunov Exponent
Yan Liu
2016-01-01
Full Text Available The dynamics of network traffic are complex and nonlinear, and chaotic behaviors and their prediction, which play an important role in local area networks (LANs, are studied in detail, using the largest Lyapunov exponent. With the introduction of phase space reconstruction based on the time sequence, the high-dimensional traffic is projected onto the low dimension reconstructed phase space, and a reduced dynamic system is obtained from the dynamic system viewpoint. Then, a numerical method for computing the largest Lyapunov exponent of the low-dimensional dynamic system is presented. Further, the longest predictable time, which is related to chaotic behaviors in the system, is studied using the largest Lyapunov exponent, and the Wolf method is used to predict the evolution of the traffic in a local area network by both Dot and Interval predictions, and a reliable result is obtained by the presented method. As the conclusion, the results show that the largest Lyapunov exponent can be used to describe the sensitivity of the trajectory in the reconstructed phase space to the initial values. Moreover, Dot Prediction can effectively predict the flow burst. The numerical simulation also shows that the presented method is feasible and efficient for predicting the complex dynamic behaviors in LAN traffic, especially for congestion and attack in networks, which are the main two complex phenomena behaving as chaos in networks.
Phase space reconstruction and estimation of the largest Lyapunov exponent for gait kinematic data
Josiński, Henryk [Silesian University of Technology, Akademicka 16, 44-100 Gliwice (Poland); Świtoński, Adam [Polish-Japanese Institute of Information Technology, Aleja Legionów 2, 41-902 Bytom (Poland); Silesian University of Technology, Akademicka 16, 44-100 Gliwice (Poland); Michalczuk, Agnieszka; Wojciechowski, Konrad [Polish-Japanese Institute of Information Technology, Aleja Legionów 2, 41-902 Bytom (Poland)
2015-03-10
The authors describe an example of application of nonlinear time series analysis directed at identifying the presence of deterministic chaos in human motion data by means of the largest Lyapunov exponent. The method was previously verified on the basis of a time series constructed from the numerical solutions of both the Lorenz and the Rössler nonlinear dynamical systems.
Lyapunov exponent of magnetospheric activity from AL time series
Vassiliadis, D.; Sharma, A. S.; Papadopoulos, K.
1991-01-01
A correlation dimension analysis of the AE index indicates that the magnetosphere behaves as a low-dimensional chaotic system with a dimension close to 4. Similar techniques are used to determine if the system's behavior is due to an intrinsic sensitivity to initial conditions and thus is truly chaotic. The quantity used to measure the sensitivity to initial conditions is the Liapunov exponent. Its calculation for AL shows that it is nonzero (0.11 + or - 0.05/min). This gives the exponential rate at which initially similar configurations of the magnetosphere evolve into completely different states. Also, predictions of deterministic nonlinear models are expected to deviate from the observed behavior at the same rate.
Lyapunov exponent of magnetospheric activity from AL time series
Vassiliadis, D.; Sharma, A. S.; Papadopoulos, K.
1991-01-01
A correlation dimension analysis of the AE index indicates that the magnetosphere behaves as a low-dimensional chaotic system with a dimension close to 4. Similar techniques are used to determine if the system's behavior is due to an intrinsic sensitivity to initial conditions and thus is truly chaotic. The quantity used to measure the sensitivity to initial conditions is the Liapunov exponent. Its calculation for AL shows that it is nonzero (0.11 + or - 0.05/min). This gives the exponential rate at which initially similar configurations of the magnetosphere evolve into completely different states. Also, predictions of deterministic nonlinear models are expected to deviate from the observed behavior at the same rate.
Extracting Lyapunov exponents from the echo dynamics of Bose-Einstein condensates on a lattice
Tarkhov, Andrei E.; Wimberger, Sandro; Fine, Boris V.
2017-08-01
We propose theoretically an experimentally realizable method to demonstrate the Lyapunov instability and to extract the value of the largest Lyapunov exponent for a chaotic many-particle interacting system. The proposal focuses specifically on a lattice of coupled Bose-Einstein condensates in the classical regime describable by the discrete Gross-Pitaevskii equation. We suggest to use imperfect time reversal of the system's dynamics known as the Loschmidt echo, which can be realized experimentally by reversing the sign of the Hamiltonian of the system. The routine involves tracking and then subtracting the noise of virtually any observable quantity before and after the time reversal. We support the theoretical analysis by direct numerical simulations demonstrating that the largest Lyapunov exponent can indeed be extracted from the Loschmidt echo routine. We also discuss possible values of experimental parameters required for implementing this proposal.
Cheng Changjun; Fan Xiaojun
2000-01-01
The relation between the Lyapunov exponent spectrun of a periodically excited non-autono mous dynamical system and the Lyapunov exponent spectrum of the corresponding autonomous system is given and the validity of the relation is verified theoretically and computationally. A direct method for calculating the Lyapunov exponent spectrum of non-autonomous dynamical systems is suggested in this paper, which makes it more convenient to calculate the Lyapunov exponent spectrum of the dynamical system periodically excited. Following the defi nition of the Lyapunov dimension D(LA) of the autonomous system, the definition of the Lyapunov dimension Dl of the non-autonomous dynamical system is also given, and the difference be- tween them is the integer 1, namely, D(A)L - DL = 1. For a quasi-poriodically excited dynamical system, similar conclusions are formed.
Symmetry properties of orthogonal and covariant Lyapunov vectors and their exponents
Posch, Harald A.
2013-06-01
Lyapunov exponents are indicators for the chaotic properties of a classical dynamical system. They are most naturally defined in terms of the time evolution of a set of so-called covariant vectors, co-moving with the linearized flow in tangent space. Taking a simple spring pendulum and the Hénon-Heiles system as examples, we demonstrate the consequences of symplectic symmetry and of time-reversal invariance for such vectors, and study the transformation between different parameterizations of the flow. 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’.
Rui Wang
2014-01-01
Full Text Available A modified multiple structural changes model is built to test structural breaks of the financial system based on calculating the largest Lyapunov exponents of the financial time series. Afterwards, the Lorenz system is used as a simulation example to inspect the new model. As the Lorenz system has strong nonlinearity, the verification results show that the new model has good capability in both finding the breakpoint and revealing the changes in nonlinear characteristics of the time series. The empirical study based on the model used daily data from the S&P 500 stock index during the global financial crisis from 2005 to 2012. The results provide four breakpoints of the period, which divide the contagion into four stages: stationary, local outbreak, global outbreak, and recovery period. An additional significant result is the obvious chaos characteristic difference in the largest Lyapunov exponents and the standard deviation at various stages, particularly at the local outbreak stage.
Quantification of the degree of mixing in chaotic micromixers using finite time Lyapunov exponents
Sarkar, Aniruddha; Harting, Jens
2010-01-01
Chaotic micromixers such as the staggered herringbone mixer developed by Stroock et al. allow efficient mixing of fluids even at low Reynolds number by repeated stretching and folding of the fluid interfaces. The ability of the fluid to mix well depends on the rate at which "chaotic advection" occurs in the mixer. An optimization of mixer geometries based on the quantification of chaotic advection is a non trivial task which is often performed by time consuming and expensive trial and error experiments. In this paper it is shown that the concept of finite-time Lyapunov exponents is a suitable tool to provide a quantitative measure of the chaotic advection of the flow. By performing lattice Boltzmann simulations of the flow inside a mixer geometry, introducing massless and non-interacting tracer particles and following their trajectories the finite time Lyapunov exponents can be calculated. The applicability of the method is demonstrated by optimizing the geometrical structure of the staggered herringbone mixe...
Using largest Lyapunov exponent to confirm the intrinsic stability of boiling water reactors
Gavilian-Moreno, Carlos [Iberdrola Generacion, S.A., Cofrentes Nuclear Power Plant, Project Engineering Department, Paraje le Plano S/N, Valencia (Spain); Espinosa-Paredes, Gilberto [Area de ingeniera en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa, Mexico city (Mexico)
2016-04-15
The aim of this paper is the study of instability state of boiling water reactors with a method based in largest Lyapunov exponents (LLEs). Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the LLE. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. This method was applied to a set of signals from several nuclear power plant (NPP) reactors under commercial operating conditions that experienced instabilities events, apparently each of a different nature. Laguna Verde and Forsmark NPPs with in-phase instabilities, and Cofrentes NPP with out-of-phases instability. This study presents the results of intrinsic instability in the boiling water reactors of three NPPs. In the analyzed cases the limit cycle was not reached, which implies that the point of equilibrium exerts influence and attraction on system evolution.
Romero, K M F; Parreira, J E; Souza, L A M; Wreszinski, W F
2007-01-01
We study and compare the information loss of a large class of gaussian bipartite systems. It includes the usual Caldeira-Leggett type model as well as Anosov models (parametric oscillators, the inverted oscillator environment, etc), which exhibit instability, one of the most important characteristics of chaotic systems. We establish a rigorous connection between the quantum Lyapunov exponents and coherence loss. We show that in the case of unstable environments, coherence loss is completely determined by the upper quantum Lyapunov exponent, a behavior dramatically different to that of the Caldeira-Leggett type model. For this class of systems we have been able to prove a long standing conjecture that for information loss the complexity of even a few (one) degrees of freedom is far more effective in destroying quantum coherence than stable many-body environments.
Xavier, J C; Strunz, W T; Beims, M W
2015-08-01
We consider the energy flow between a classical one-dimensional harmonic oscillator and a set of N two-dimensional chaotic oscillators, which represents the finite environment. Using linear response theory we obtain an analytical effective equation for the system harmonic oscillator, which includes a frequency dependent dissipation, a shift, and memory effects. The damping rate is expressed in terms of the environment mean Lyapunov exponent. A good agreement is shown by comparing theoretical and numerical results, even for environments with mixed (regular and chaotic) motion. Resonance between system and environment frequencies is shown to be more efficient to generate dissipation than larger mean Lyapunov exponents or a larger number of bath chaotic oscillators.
THE RELATION OF DIMENSION,ENTROPY AND LYAPUNOV EXPONENT IN RANDOM CASE
Yun Zhao
2008-01-01
We consider random systems generated by two-sided compositions of random surface diffeomorphisms,together with an ergodic Borel probability measure μ.Let D(μω)be its dimension of the sample measure,then we prove a formula relating D(μω)to the entropy and Lyapunov exponents of the random system,where D(μω)is dimHμω,-/dinBμω,or-/dimBμω.
ZHANGYing-xun; WUXi-zhen; LIZhu-xia
2003-01-01
The largest Lyapunov exponent (LLE) has been widely used to measure the levelof chaos of a system and was used to study the “solid-like” to “liquid-like” phase transition. Nuclear multifragmentation has been considered to be associated with a liquid-gas phase transition. Thus, in this paper we want to extend the study to the energy regime that encompasses fragmentation phenomena.
Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System.
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 tt_{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.
A Lower Bound on the Lyapunov Exponent for the Generalized Harper's Model
Jitomirskaya, Svetlana; Liu, Wencai
2017-02-01
We obtain a lower bound for the Lyapunov exponent of a family of discrete Schrödinger operators (Hu)_n=u_{n+1}+u_{n-1}+2a_1 cos 2π (θ +nα )u_n+2a_2 cos 4π (θ +nα )u_n, that incorporates both a_1 and a_2, thus going beyond the Herman's bound.
Perturbation theory for Lyapunov exponents of an Anderson model on a strip
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.
A Lower Bound on the Lyapunov Exponent for the Generalized Harper's Model
Jitomirskaya, Svetlana; Liu, Wencai
2016-05-01
We obtain a lower bound for the Lyapunov exponent of a family of discrete Schrödinger operators (Hu)_n=u_{n+1}+u_{n-1}+2a_1 cos 2π (θ +nα )u_n+2a_2 cos 4π (θ +nα )u_n , that incorporates both a_1 and a_2, thus going beyond the Herman's bound.
Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System
Rozenbaum, Efim B.; Ganeshan, Sriram; Galitski, Victor
2017-02-01
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 tE: transitioning from a time-independent value of t-1ln C (t ) at t tE. 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), 10.1103/PhysRevB.54.14423; Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004), 10.1103/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.
Effective power-law dependence of Lyapunov exponents on the central mass in galaxies
Delis, N; Kalapotharakos, C
2015-01-01
Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation $L\\propto m^p$ between the mean Lyapunov exponent $L$ of stellar orbits chaotically scattered by a supermassive black hole in the center of a galaxy and the mass parameter $m$, i.e. ratio of the mass of the black hole over the mass of the galaxy. The exponent $p$ is found numerically to obtain values in the range $p \\approx 0.3$--$0.5$. 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 black hole's sphere of influence. We thus predict $p=2/3-q$ with $q\\approx 0.1$--$0.2$. 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...
Using Lyapunov exponents to predict the onset of chaos in nonlinear oscillators.
Ryabov, Vladimir B
2002-07-01
An analytic technique for predicting the emergence of chaotic instability in nonlinear nonautonomous dissipative oscillators is proposed. The method is based on the Lyapunov-type stability analysis of an arbitrary phase trajectory and the standard procedure of calculating the Lyapunov characteristic exponents. The concept of temporally local Lyapunov exponents is then utilized for specifying the area in the phase space where any trajectory is asymptotically stable, and, therefore, the existence of chaotic attractors is impossible. The procedure of linear coordinate transform optimizing the linear part of the vector field is developed for the purpose of maximizing the stability area in the vicinity of a stable fixed point. By considering the inverse conditions of asymptotic stability, this approach allows formulating a necessary condition for chaotic motion in a broad class of nonlinear oscillatory systems, including many cases of practical interest. The examples of externally excited one- and two-well Duffing oscillators and a planar pendulum demonstrate efficiency of the proposed method, as well as a good agreement of the theoretical predictions with the results of numerical experiments. The comparison of the proposed method with Melnikov's criterion shows a potential advantage of using the former one at high values of dissipation parameter and/or multifrequency type of excitation in dynamical systems.
Valenza, Gaetano; Citi, Luca; Barbieri, Riccardo
2014-01-01
Measures of nonlinearity and complexity, and in particular the study of Lyapunov exponents, have been increasingly used to characterize dynamical properties of a wide range of biological nonlinear systems, including cardiovascular control. In this work, we present a novel methodology able to effectively estimate the Lyapunov spectrum of a series of stochastic events in an instantaneous fashion. The paradigm relies on a novel point-process high-order nonlinear model of the event series dynamics. The long-term information is taken into account by expanding the linear, quadratic, and cubic Wiener-Volterra kernels with the orthonormal Laguerre basis functions. Applications to synthetic data such as the Hénon map and Rössler attractor, as well as two experimental heartbeat interval datasets (i.e., healthy subjects undergoing postural changes and patients with severe cardiac heart failure), focus on estimation and tracking of the Instantaneous Dominant Lyapunov Exponent (IDLE). The novel cardiovascular assessment demonstrates that our method is able to effectively and instantaneously track the nonlinear autonomic control dynamics, allowing for complexity variability estimations.
Fazanaro, Filipe I; Soriano, Diogo C; Suyama, Ricardo; Attux, Romis; Madrid, Marconi K; de Oliveira, José Raimundo
2013-06-01
The present work aims to apply a recently proposed method for estimating Lyapunov exponents to characterize-with the aid of the metric entropy and the fractal dimension-the degree of information and the topological structure associated with multiscroll attractors. In particular, the employed methodology offers the possibility of obtaining the whole Lyapunov spectrum directly from the state equations without employing any linearization procedure or time series-based analysis. As a main result, the predictability and the complexity associated with the phase trajectory were quantified as the number of scrolls are progressively increased for a particular piecewise linear model. In general, it is shown here that the trajectory tends to increase its complexity and unpredictability following an exponential behaviour with the addition of scrolls towards to an upper bound limit, except for some degenerated situations where a non-uniform grid of scrolls is attained. Moreover, the approach employed here also provides an easy way for estimating the finite time Lyapunov exponents of the dynamics and, consequently, the Lagrangian coherent structures for the vector field. These structures are particularly important to understand the stretching/folding behaviour underlying the chaotic multiscroll structure and can provide a better insight of phase space partition and exploration as new scrolls are progressively added to the attractor.
Control of chaos in permanent magnet synchronous motor by using optimal Lyapunov exponents placement
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.
A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents
Doan, T. S.; Karrasch, D.; Nguyen, T. Y.; Siegmund, S.
A hyperbolicity notion for linear differential equations x˙=A(t)x, t∈[t-,t+], is defined which unifies different existing notions like finite-time Lyapunov exponents (Haller, 2001, [13], Shadden et al., 2005, [24]), uniform or M-hyperbolicity (Haller, 2001, [13], Berger et al., 2009, [6]) and (t-,(t+-t-))-dichotomy (Rasmussen, 2010, [21]). Its relation to the dichotomy spectrum (Sacker and Sell, 1978, [23], Siegmund, 2002, [26]), D-hyperbolicity (Berger et al., 2009, [6]) and real parts of the eigenvalues (in case A is constant) is described. We prove a spectral theorem and provide an approximation result for the spectral intervals.
Principal component cluster analysis of ECG time series based on Lyapunov exponent spectrum
WANG Nai; RUAN Jiong
2004-01-01
In this paper we propose an approach of principal component cluster analysis based on Lyapunov exponent spectrum (LES) to analyze the ECG time series. Analysis results of 22 sample-files of ECG from the MIT-BIH database confirmed the validity of our approach. Another technique named improved teacher selecting student (TSS) algorithm is presented to analyze unknown samples by means of some known ones, which is of better accuracy. This technique combines the advantages of both statistical and nonlinear dynamical methods and is shown to be significant to the analysis of nonlinear ECG time series.
Effective Power-Law Dependence of Lyapunov Exponents on the Central Mass in Galaxies
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.
Effective Power-Law Dependence of Lyapunov Exponents on the Central Mass in Galaxies
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.
Lyapunov exponents from CHUA's circuit time series using artificial neural networks
Gonzalez, J. Jesus; Espinosa, Ismael E.; Fuentes, Alberto M.
1995-01-01
In this paper we present the general problem of identifying if a nonlinear dynamic system has a chaotic behavior. If the answer is positive the system will be sensitive to small perturbations in the initial conditions which will imply that there is a chaotic attractor in its state space. A particular problem would be that of identifying a chaotic oscillator. We present an example of three well known different chaotic oscillators where we have knowledge of the equations that govern the dynamical systems and from there we can obtain the corresponding time series. In a similar example we assume that we only know the time series and, finally, in another example we have to take measurements in the Chua's circuit to obtain sample points of the time series. With the knowledge about the time series the phase plane portraits are plotted and from them, by visual inspection, it is concluded whether or not the system is chaotic. This method has the problem of uncertainty and subjectivity and for that reason a different approach is needed. A quantitative approach is the computation of the Lyapunov exponents. We describe several methods for obtaining them and apply a little known method of artificial neural networks to the different examples mentioned above. We end the paper discussing the importance of the Lyapunov exponents in the interpretation of the dynamic behavior of biological neurons and biological neural networks.
Babaee, Hessam; Farazmand, Mohamad; Haller, George; Sapsis, Themistoklis P.
2017-06-01
High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g., long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here, we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy-Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.
Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series
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.
Lyapunov exponents and phase diagrams reveal multi-factorial control over TRAIL-induced apoptosis
Aldridge, Bree B; Gaudet, Suzanne; Lauffenburger, Douglas A; Sorger, Peter K
2011-01-01
Receptor-mediated apoptosis proceeds via two pathways: one requiring only a cascade of initiator and effector caspases (type I behavior) and the second requiring an initiator–effector caspase cascade and mitochondrial outer membrane permeabilization (type II behavior). Here, we investigate factors controlling type I versus II phenotypes by performing Lyapunov exponent analysis of an ODE-based model of cell death. The resulting phase diagrams predict that the ratio of XIAP to pro-caspase-3 concentrations plays a key regulatory role: type I behavior predominates when the ratio is low and type II behavior when the ratio is high. Cell-to-cell variability in phenotype is observed when the ratio is close to the type I versus II boundary. By positioning multiple tumor cell lines on the phase diagram we confirm these predictions. We also extend phase space analysis to mutations affecting the rate of caspase-3 ubiquitylation by XIAP, predicting and showing that such mutations abolish all-or-none control over activation of effector caspases. Thus, phase diagrams derived from Lyapunov exponent analysis represent a means to study multi-factorial control over a complex biochemical pathway. PMID:22108795
Babaee, Hessam; Farazmand, Mohamad; Haller, George; Sapsis, Themistoklis P
2017-06-01
High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g., long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here, we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy-Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.
Classification of Heart Rate Signals during Meditation using Lyapunov Exponents and Entropy
Ateke Goshvarpour
2012-03-01
Full Text Available Meditation is commonly perceived as an alternative medicine method of psychological diseases management tool that assist in alleviating depression and anxiety disorders. The purpose of this study is to evaluate the accuracy of different classifiers on the heart rate signals in a specific psychological state. Two types of heart rate time series (before, and during meditation of 25 healthy women are collected in the meditation clinic in Mashhad. Nonlinear features such as Lyapunov Exponents and Entropy were extracted. To evaluate performance of the classifiers, the classification accuracies and mean square error (MSE of the classifiers were examined. Different classifiers were tested and the studies confirmed that for the heart rate signals, Quadratic classifier trained on Lyapunov Exponents and Entropy results in higher classification accuracy. The classification accuracy of the Quadratic classifier is 92.31%. However, the accuracies of Fisher and k-Nearest Neighbor (k-NN classifiers are encouraging. The classification results demonstrate that the dynamical measures are useful parameters which contain comprehensive information about signals and the Quadratic classifier using nonlinear features can be useful in analyzing the heart rate signals in a specific psychological state.
Determining the sub-Lyapunov exponent of delay systems from time series.
Jüngling, Thomas; Soriano, Miguel C; Fischer, Ingo
2015-06-01
For delay systems the sign of the sub-Lyapunov exponent (sub-LE) determines key dynamical properties. This includes the properties of strong and weak chaos and of consistency. Here we present a robust algorithm based on reconstruction of the local linearized equations of motion, which allows for calculating the sub-LE from time series. The algorithm is inspired by a method introduced by Pyragas for a nondelayed drive-response scheme [K. Pyragas, Phys. Rev. E 56, 5183 (1997)]. In the presented extension to delay systems, the delayed feedback takes over the role of the drive, whereas the response of the low-dimensional node leads to the sub-Lyapunov exponent. Our method is based on a low-dimensional representation of the delay system. We introduce the basic algorithm for a discrete scalar map, extend the concept to scalar continuous delay systems, and give an outlook to the case of a full vector-state system, from which only a scalar observable is recorded.
Liu, Xiuling; Du, Haiman; Wang, Guanglei; Zhou, Suiping; Zhang, Hong
2015-10-01
Premature ventricular contraction (PVC) is a common type of abnormal heartbeat. Without early diagnosis and proper treatment, PVC may result in serious harms. Diagnosis of PVC is of great importance in goal-directed treatment and preoperation prognosis. This paper proposes a novel diagnostic method for PVC based on Lyapunov exponents of electrocardiogram (ECG) beats. The methodology consists of preprocessing, feature extraction and classification integrated into the system. PVC beats can be classified and differentiated from other types of abnormal heartbeats by analyzing Lyapunov exponents and training a learning vector quantization (LVQ) neural network. Our algorithm can obtain a good diagnostic result with little features by using single lead ECG data. The sensitivity, positive predictability, and the overall accuracy of the automatic diagnosis of PVC is 90.26%, 92.31%, and 98.90%, respectively. The effectiveness of the new method is validated through extensive tests using data from MIT-BIH database. The experimental results show that the proposed method is efficient and robust.
Lyapunov exponents and phase diagrams reveal multi-factorial control over TRAIL-induced apoptosis.
Aldridge, Bree B; Gaudet, Suzanne; Lauffenburger, Douglas A; Sorger, Peter K
2011-11-22
Receptor-mediated apoptosis proceeds via two pathways: one requiring only a cascade of initiator and effector caspases (type I behavior) and the second requiring an initiator-effector caspase cascade and mitochondrial outer membrane permeabilization (type II behavior). Here, we investigate factors controlling type I versus II phenotypes by performing Lyapunov exponent analysis of an ODE-based model of cell death. The resulting phase diagrams predict that the ratio of XIAP to pro-caspase-3 concentrations plays a key regulatory role: type I behavior predominates when the ratio is low and type II behavior when the ratio is high. Cell-to-cell variability in phenotype is observed when the ratio is close to the type I versus II boundary. By positioning multiple tumor cell lines on the phase diagram we confirm these predictions. We also extend phase space analysis to mutations affecting the rate of caspase-3 ubiquitylation by XIAP, predicting and showing that such mutations abolish all-or-none control over activation of effector caspases. Thus, phase diagrams derived from Lyapunov exponent analysis represent a means to study multi-factorial control over a complex biochemical pathway.
The Calculation of Lyapunov Exponent of Water Molecules Vibration System%基于水分子振动体系的Lyapunov指数的计算
刘松红; 庞成群
2012-01-01
采用wolf重构法改进了水分子振动体系最大Lyapunov指数的计算，通过对水分子振动体系的最大Lyapunov指数的计算，得到了计算水分子振动体系的最大Lyapunov指数合适的初始长度、延迟时间以及总的演化时间。%By adopting wolf reconstruction method and improving the method, we gained the expression of the maximum Lyapunov exponent of water molecules vibration system o From the results of calculating the maximum Lyapunov exponent of water molecules vibration system, we received the appropriate initial length, duration and the total evolution time.
Li Qun-Hong; Tan Jie-Yan
2011-01-01
A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process,the Poincar6 map of the system is constructed. Using the Poincare map and the Gram-Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.
Computing Finite-Time Lyapunov Exponents with Optimally Time Dependent Reduction
Babaee, Hessam; Farazmand, Mohammad; Sapsis, Themis; Haller, George
2016-11-01
We present a method to compute Finite-Time Lyapunov Exponents (FTLE) of a dynamical system using Optimally Time-Dependent (OTD) reduction recently introduced by H. Babaee and T. P. Sapsis. The OTD modes are a set of finite-dimensional, time-dependent, orthonormal basis {ui (x , t) } |i=1N that capture the directions associated with transient instabilities. The evolution equation of the OTD modes is derived from a minimization principle that optimally approximates the most unstable directions over finite times. To compute the FTLE, we evolve a single OTD mode along with the nonlinear dynamics. We approximate the FTLE from the reduced system obtained from projecting the instantaneous linearized dynamics onto the OTD mode. This results in a significant reduction in the computational cost compared to conventional methods for computing FTLE. We demonstrate the efficiency of our method for double Gyre and ABC flows. ARO project 66710-EG-YIP.
How reliable are Finite-Size Lyapunov Exponents for the assessment of ocean dynamics?
Hernández-Carrasco, Ismael; López, Cristóbal; Turiel, Antonio
2010-01-01
Much of atmospheric and oceanic transport is associated with coherent structures. Lagrangian methods are emerging as optimal tools for their identification and analysis. An important Lagrangian technique which is starting to be widely used in oceanography is that of Finite-Size Lyapunov Exponents (FSLEs). Despite this growing relevance there are still many open questions concerning the reliability of the FSLEs in order to analyse the ocean dynamics. In particular, it is still unclear how robust they are when confronted with real data. In this paper we analyze the effect on this Lagrangian technique of the two most important effects when facing real data, namely noise and dynamics of unsolved scales. Our results, using as a benchmarch data from a primitive numerical model of the Mediterranean Sea, show that even when some dynamics is missed the FSLEs results still give an accurate picture of the oceanic transport properties.
Ota, Y; Ota, Yukihiro; Ohba, Ichiro
2003-01-01
We discuss the quantum--classical correspondence in a specific dissipative chaotic system, Duffing oscillator. We quantize it on the basis of quantum state diffusion (QSD) which is a certain formulation for open quantum systems and an effective tool for analyzing complex problems numerically. We consider a sensitivity to initial conditions, `` pseudo-Lyapunov exponent '', and investigate it in detail, varying Planck constant effectively. We show that in a dissipative system there exists a certain critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we show that an effect of dissipation suppresses the occurrence of chaos in the quantum region, while it, combined with the periodic external force, plays a crucial role in the chaotic behaviors of classical system.
Mean field theory for Lyapunov exponents and KS entropy in Lorentz lattice gases
Ernst, M H; Nix, R; Jacobs, D; Ernst, M H; Dorfman, J R; Nix, R; Jacobs, D
1994-01-01
automata lattice gases are useful systems for systematically exploring the connections between non-equilibrium statistical mechanics and dynamical systems theory. Here the chaotic properties of a Lorentz lattice gas are studied analytically and by means of computer simulations. The escape-rates, Lyapunov exponents, and KS entropies are estimated for a one- dimensional example using a mean field theory. The results are compared with simulations for a range of densities and scattering parameters of the lattice gas. The computer results show a distribution of values for the dynamical quantities with average values that are in good agreement with the mean field theory and consistent with the escape-rate formalism for the coefficient of diffusion.
On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents
Bonetto, F; Pugh, C
1998-01-01
For Hamiltonian systems subject to an external potential, which in the presence of a thermostat will reach a nonequilibrium stationary state, Dettmann and Morriss proved a strong conjugate pairing rule (SCPR) for pairs of Lyapunov exponents in the case of isokinetic (IK) stationary states which have a given kinetic energy. This SCPR holds for all initial phases of the system, all times t and all numbers of particles N. This proof was generalized by Wojtkovski and Liverani to include hard interparticle potentials. A geometrical reformulation of those results is presented. The present paper proves numerically, using periodic orbits for the Lorentz gas, that SCPR cannot hold for isoenergetic (IE) stationary states, which have a given total internal energy. In that case strong evidence is obtained for CPR to hold for large N and t, where it can be conjectured that the larger N, the smaller t will be. This suffices for statistical mechanics.
Analysis of the Emergence in Swarm Model Based on Largest Lyapunov Exponent
Yu Wu
2011-01-01
Full Text Available Emergent behaviors of collective intelligence systems, exemplified by swarm model, have attracted broad interests in recent years. However, current research mostly stops at observational interpretations and qualitative descriptions of emergent phenomena and is essentially short of quantitative analysis and evaluation. In this paper, we conduct a quantitative study on the emergence of swarm model by using chaos analysis of complex dynamic systems. This helps to achieve a more exact understanding of emergent phenomena. In particular, we evaluate the emergent behaviors of swarm model quantitatively by using the chaos and stability analysis of swarm model based on largest Lyapunov exponent. It is concluded that swarm model is at the edge of chaos when emergence occurs, and whether chaotic or stable at the beginning, swarm model will converge to stability with the elapse of time along with interactions among agents.
Look, Nicole [Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309 (United States); Arellano, Christopher J.; Grabowski, Alena M.; Kram, Rodger [Department of Integrative Physiology, University of Colorado Boulder, Boulder, Colorado 80309 (United States); McDermott, William J. [The Orthopedic Specialty Hospital, Murray, Utah 84107 (United States); Bradley, Elizabeth [Department of Computer Science, University of Colorado Boulder, Boulder, Colorado 80309, USA and Santa Fe Institute, Santa Fe, New Mexico 87501 (United States)
2013-12-15
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.
Ghayoumi Zadeh, Hossein; Haddadnia, Javad; Montazeri, Alimohammad
2016-05-01
The segmentation of cancerous areas in breast images is important for the early detection of disease. Thermal imaging has advantages, such as being non-invasive, non-radiation, passive, quick, painless, inexpensive, and non-contact. Imaging technique is the focus of this research. The proposed model in this paper is a combination of surf and corners that are very resistant. Obtained features are resistant to changes in rotation and revolution then with the help of active contours, this feature has been used for segmenting cancerous areas. Comparing the obtained results from the proposed method and mammogram show that proposed method is Accurate and appropriate. Benign and malignance of segmented areas are detected by Lyapunov exponent. Values obtained include TP=91.31%, FN=8.69%, FP=7.26%. The proposed method can classify those abnormally segmented areas of the breast, to the Benign and malignant cancer.
Detection of the onset of numerical chaotic instabilities by lyapunov exponents
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.
A perturbation method to the tent map based on Lyapunov exponent and its application
曹绿晨; 罗玉玲; 丘森辉; 刘俊秀
2015-01-01
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.
Refining finite-time Lyapunov exponent ridges and the challenges of classifying them.
Allshouse, Michael R; Peacock, Thomas
2015-08-01
While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model.
Positivity of Lyapunov exponents for Anderson-type models on two coupled strings
Hakim Boumaza
2007-03-01
Full Text Available We study two models of Anderson-type random operators on two deterministically coupled continuous strings. Each model is associated with independent, identically distributed four-by-four symplectic transfer matrices, which describe the asymptotics of solutions. In each case we use a criterion by Gol'dsheid and Margulis (i.e. Zariski denseness of the group generated by the transfer matrices in the group of symplectic matrices to prove positivity of both leading Lyapunov exponents for most energies. In each case this implies almost sure absence of absolutely continuous spectrum (at all energies in the first model and for sufficiently large energies in the second model. The methods used allow for singularly distributed random parameters, including Bernoulli distributions.
Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment
Gärtner, Jürgen; Maillard, Grégory
2010-01-01
We continue our study of the parabolic Anderson equation $\\partial u/\\partial t = \\kappa\\Delta u + \\gamma\\xi u$ for the space-time field $u\\colon\\,\\Z^d\\times [0,\\infty)\\to\\R$, where $\\kappa \\in [0,\\infty)$ is the diffusion constant, $\\Delta$ is the discrete Laplacian, $\\gamma\\in (0,\\infty)$ is the coupling constant, and $\\xi\\colon\\,\\Z^d\\times [0,\\infty)\\to\\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $\\xi$, both living on $\\Z^d$. In earlier work we considered three choices for $\\xi$: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the \\emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\\ $\\xi$, and showed that these exponents display an interesting dependence on the diffusion constant $\\kappa$, with qualitatively different beha...
The Comparison for Lyapunov Exponents Calculation Methods%关于Lyapunov指数计算方法的比较
张海龙; 闵富红; 王恩荣
2012-01-01
针对常用的几种Lyapunov指数数值计算方法,即定义法、正交法、wolf法和小数据量法,以典型的Lorenz系统为例,分别计算Lorenz混沌吸引子的Lyapunov指数谱或者最大Lyapunov指数,比较各种方法的计算精度、计算复杂度,并且对含噪声的混沌时间序列给出Lyapunov指数计算结果,比较各种抗干扰能力.给出了不同计算方法的性能差异、适用场合和选择依据.%In this paper,the several computational methods of Lyapunov exponents are compared,i.e.,the definition method,the orthogonal method,the wolf method and the small data sets.The Lyapunov exponent power and the max-Lyapunov exponent are computed through the above methods for Lorenz system.From the results,the accuracies and the complexity of the above methods are investigated.Furthermore,the max-Lyapunov exponents are also calculated for the chaotic time series including the noise.Finally,numerical results demonstrate that the performances of different computational methods have differences,and some summaries will be presented.
Truant, Daniel P; Morriss, Gary P
2014-11-01
The covariant Lyapunov analysis is generalized to systems attached to deterministic thermal reservoirs that create a heat current across the system and perturb it away from equilibrium. The change in the Lyapunov exponents as a function of heat current is described and explained. Both the nonequilibrium backward and covariant hydrodynamic Lyapunov modes are analyzed and compared. The movement of the converged angle between the hydrodynamic stable and unstable conjugate manifolds with the free flight time of the dynamics is accurately predicted for any nonequilibrium system simply as a function of their exponent. The nonequilibrium positive and negative LP mode frequencies are found to be asymmetrical, causing the negative mode to oscillate between the two functional forms of each mode in the positive conjugate mode pair. This in turn leads to the angular distributions between the conjugate modes to oscillate symmetrically about π/2 at a rate given by the difference between the positive and negative mode frequencies.
Wang, Qiqi; Rigas, Georgios; Esclapez, Lucas; Magri, Luca; Blonigan, Patrick
2016-11-01
Bluff body flows are of fundamental importance to many engineering applications involving massive flow separation and in particular the transport industry. Coherent flow structures emanating in the wake of three-dimensional bluff bodies, such as cars, trucks and lorries, are directly linked to increased aerodynamic drag, noise and structural fatigue. For low Reynolds laminar and transitional regimes, hydrodynamic stability theory has aided the understanding and prediction of the unstable dynamics. In the same framework, sensitivity analysis provides the means for efficient and optimal control, provided the unstable modes can be accurately predicted. However, these methodologies are limited to laminar regimes where only a few unstable modes manifest. Here we extend the stability analysis to low-dimensional chaotic regimes by computing the Lyapunov covariant vectors and their associated Lyapunov exponents. We compare them to eigenvectors and eigenvalues computed in traditional hydrodynamic stability analysis. Computing Lyapunov covariant vectors and Lyapunov exponents also enables the extension of sensitivity analysis to chaotic flows via the shadowing method. We compare the computed shadowing sensitivities to traditional sensitivity analysis. These Lyapunov based methodologies do not rely on mean flow assumptions, and are mathematically rigorous for calculating sensitivities of fully unsteady flow simulations.
Presence of nonlinearity in intracranial EEG recordings: detected by Lyapunov exponents
Liu, Chang-Chia; Shiau, Deng-Shan; Chaovalitwongse, W. Art; Pardalos, Panos M.; Sackellares, J. C.
2007-11-01
In this communication, we performed nonlinearity analysis in the EEG signals recorded from patients with temporal lobe epilepsy (TLE). The largest Lyapunov exponent (Lmax) and phase randomization surrogate data technique were employed to form the statistical test. EEG recordings were acquired invasively from three patients in six brain regions (left and right temporal depth, sub-temporal and orbitofrontal) with 28-32 depth electrodes placed in depth and subdural of the brain. All three patients in this study have unilateral epileptic focus region on the right hippocampus(RH). Nonlinearity was detected by comparing the Lmax profiles of the EEG recordings to its surrogates. The nonlinearity was seen in all different states of the patient with the highest found in post-ictal state. Further our results for all patients exhibited higher degree of differences, quantified by paired t-test, in Lmax values between original and its surrogate from EEG signals recorded from epileptic focus regions. The results of this study demonstrated the Lmax is capable to capture spatio-temporal dynamics that may not be able to detect by linear measurements in the intracranial EEG recordings.
A perturbation method to the tent map based on Lyapunov exponent and its application
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).
Moura, R. C.; Silva, A. F. C.; Bigarella, E. D. V.; Fazenda, A. L.; Ortega, M. A.
2016-08-01
This paper proposes two important improvements to shock-capturing strategies using a discontinuous Galerkin scheme, namely, accurate shock identification via finite-time Lyapunov exponent (FTLE) operators and efficient shock treatment through a point-implicit discretization of a PDE-based artificial viscosity technique. The advocated approach is based on the FTLE operator, originally developed in the context of dynamical systems theory to identify certain types of coherent structures in a flow. We propose the application of FTLEs in the detection of shock waves and demonstrate the operator's ability to identify strong and weak shocks equally well. The detection algorithm is coupled with a mesh refinement procedure and applied to transonic and supersonic flows. While the proposed strategy can be used potentially with any numerical method, a high-order discontinuous Galerkin solver is used in this study. In this context, two artificial viscosity approaches are employed to regularize the solution near shocks: an element-wise constant viscosity technique and a PDE-based smooth viscosity model. As the latter approach is more sophisticated and preferable for complex problems, a point-implicit discretization in time is proposed to reduce the extra stiffness introduced by the PDE-based technique, making it more competitive in terms of computational cost.
Sun, Yuming; Wu, Christine Qiong
2012-12-01
Balancing control is important for biped standing. In spite of large efforts, it is very difficult to design balancing control strategies satisfying three requirements simultaneously: maintaining postural stability, improving energy efficiency and satisfying the constraints between the biped feet and the ground. In this article, a proportional-derivative (PD) controller is proposed for a standing biped, which is simplified as a two-link inverted pendulum with one additional rigid foot-link. The genetic algorithm (GA) is used to search for the control gain meeting all three requirements. The stability analysis of such a deterministic biped control system is carried out using the concept of Lyapunov exponents (LEs), based on which, the system stability, where the disturbance comes from the initial states, and the structural stability, where the disturbance comes from the PD gains, are examined quantitively in terms of stability region. This article contributes to the biped balancing control, more significantly, the method shown in the studied case of biped provides a general framework of systematic stability analysis for certain deterministic nonlinear dynamical systems.
Gauss map and Lyapunov exponents of interacting particles in a billiard
Manchein, C. [Departamento de Fisica, Universidade Federal do Parana, 81531-990 Curitiba, PR (Brazil); Beims, M.W. [Departamento de Fisica, Universidade Federal do Parana, 81531-990 Curitiba, PR (Brazil); Max Planck Institute for the Physics of Complex Systems, Noethnitzer Strasse 38, D-01187 Dresden (Germany)], E-mail: mbeims@fisica.ufpr.br
2009-03-15
We show that the Lyapunov exponent (LE) of periodic orbits with Lebesgue measure zero from the Gauss map can be used to determine the main qualitative behavior of the LE of a Hamiltonian system. The Hamiltonian system is a one-dimensional box with two particles interacting via a Yukawa potential and does not possess Kolmogorov-Arnold-Moser (KAM) curves. In our case the Gauss map is applied to the mass ratio ({gamma} = m{sub 2}/m{sub 1}) between particles. Besides the main qualitative behavior, some unexpected peaks in the {gamma} dependence of the mean LE and the appearance of 'stickness' in phase space can also be understand via LE from the Gauss map. This shows a nice example of the relation between the 'instability' of the continued fraction representation of a number with the stability of non-periodic curves (no KAM curves) from the physical model. Our results also confirm the intuition that pseudo-integrable systems with more complicated invariant surfaces of the flow (higher genus) should be more unstable under perturbation.
Influence of finite-time Lyapunov exponents on winter precipitation over the Iberian Peninsula
D. Garaboa-Paz
2017-05-01
Full Text Available 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.
Designing Hyperchaotic Cat Maps With Any Desired Number of Positive Lyapunov Exponents.
Hua, Zhongyun; Yi, Shuang; Zhou, Yicong; Li, Chengqing; Wu, Yue
2017-01-04
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 n-dimensional (n-D) hyperchaotic Cat map (HCM) with any desired number of positive LEs. The method first generates two basic n-D Cat matrices iteratively and then constructs the final n-D Cat matrix by performing similarity transformation on one basic n-D Cat matrix by the other. Given any number of positive LEs, it can generate an n-D HCM with desired hyperchaotic complexity. Two illustrative examples of n-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 n-D HCMs with lower computation complexity and their outputs demonstrate strong randomness and complex ergodicity.
Finite-Time Lyapunov Exponents and Lagrangian Coherent Structures in Uncertain Unsteady Flows.
Guo, Hanqi; He, Wenbin; Peterka, Tom; Shen, Han-Wei; Collis, Scott; Helmus, Jonathan
2016-02-29
The objective of this paper is to understand transport behavior in uncertain time-varying flow fields by redefining the finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structure (LCS) as stochastic counterparts of their traditional deterministic definitions. Three new concepts are introduced: the distribution of the FTLE (D-FTLE), the FTLE of distributions (FTLE-D), and uncertain LCS (U-LCS). The D-FTLE is the probability density function of FTLE values for every spatiotemporal location, which can be visualized with different statistical measurements. The FTLE-D extends the deterministic FTLE by measuring the divergence of particle distributions. It gives a statistical overview of how transport behaviors vary in neighborhood locations. The U-LCS, the probabilities of finding LCSs over the domain, can be extracted with stochastic ridge finding and density estimation algorithms. We show that our approach produces better results than existing variance-based methods do. Our experiments also show that the combination of D-FTLE, FTLE-D, and U-LCS can help users understand transport behaviors and find separatrices in ensemble simulations of atmospheric processes.
Valenza, Gaetano; Allegrini, Paolo; Lanatà, Antonio; Scilingo, Enzo Pasquale
2012-01-01
In this work we characterized the non-linear complexity of Heart Rate Variability (HRV) in short time series. The complexity of HRV signal was evaluated during emotional visual elicitation by using Dominant Lyapunov Exponents (DLEs) and Approximate Entropy (ApEn). We adopted a simplified model of emotion derived from the Circumplex Model of Affects (CMAs), in which emotional mechanisms are conceptualized in two dimensions by the terms of valence and arousal. Following CMA model, a set of standardized visual stimuli in terms of arousal and valence gathered from the International Affective Picture System (IAPS) was administered to a group of 35 healthy volunteers. Experimental session consisted of eight sessions alternating neutral images with high arousal content images. Several works can be found in the literature showing a chaotic dynamics of HRV during rest or relax conditions. The outcomes of this work showed a clear switching mechanism between regular and chaotic dynamics when switching from neutral to arousal elicitation. Accordingly, the mean ApEn decreased with statistical significance during arousal elicitation and the DLE became negative. Results showed a clear distinction between the neutral and the arousal elicitation and could be profitably exploited to improve the accuracy of emotion recognition systems based on HRV time series analysis. PMID:22393320
The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields
Vladimirov, Igor G
2012-01-01
We consider an "elastic" version of the statistical mechanical monomer-dimer problem on the n-dimensional integer lattice. Our setting includes the classical "rigid" formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan" EDM systems where the dimer potential is a weighted l1-distance and the auxiliary GRF is a Markov random fie...
Multiscale analysis of biological data by scale-dependent lyapunov exponent.
Gao, Jianbo; Hu, Jing; Tung, Wen-Wen; Blasch, Erik
2011-01-01
Physiological signals often are highly non-stationary (i.e., mean and variance change with time) and multiscaled (i.e., dependent on the spatial or temporal interval lengths). They may exhibit different behaviors, such as non-linearity, sensitive dependence on small disturbances, long memory, and extreme variations. Such data have been accumulating in all areas of health sciences and rapid analysis can serve quality testing, physician assessment, and patient diagnosis. To support patient care, it is very desirable to characterize the different signal behaviors on a wide range of scales simultaneously. The Scale-Dependent Lyapunov Exponent (SDLE) is capable of such a fundamental task. In particular, SDLE can readily characterize all known types of signal data, including deterministic chaos, noisy chaos, random 1/f(α) processes, stochastic limit cycles, among others. SDLE also has some unique capabilities that are not shared by other methods, such as detecting fractal structures from non-stationary data and detecting intermittent chaos. In this article, we describe SDLE in such a way that it can be readily understood and implemented by non-mathematically oriented researchers, develop a SDLE-based consistent, unifying theory for the multiscale analysis, and demonstrate the power of SDLE on analysis of heart-rate variability (HRV) data to detect congestive heart failure and analysis of electroencephalography (EEG) data to detect seizures.
Characteristic distribution of finite-time Lyapunov exponents for chimera states.
Botha, André E
2016-07-04
Our fascination with chimera states stems partially from the somewhat paradoxical, yet fundamental trait of identical, and identically coupled, oscillators to split into spatially separated, coherently and incoherently oscillating groups. While the list of systems for which various types of chimeras have already been detected continues to grow, there is a corresponding increase in the number of mathematical analyses aimed at elucidating the fundamental reasons for this surprising behaviour. Based on the model systems, there are strong indications that chimera states may generally be ubiquitous in naturally occurring systems containing large numbers of coupled oscillators - certain biological systems and high-Tc superconducting materials, for example. In this work we suggest a new way of detecting and characterising chimera states. Specifically, it is shown that the probability densities of finite-time Lyapunov exponents, corresponding to chimera states, have a definite characteristic shape. Such distributions could be used as signatures of chimera states, particularly in systems for which the phases of all the oscillators cannot be measured directly. For such cases, we suggest that chimera states could perhaps be detected by reconstructing the characteristic distribution via standard embedding techniques, thus making it possible to detect chimera states in systems where they could otherwise exist unnoticed.
Valenza, Gaetano; Allegrini, Paolo; Lanatà, Antonio; Scilingo, Enzo Pasquale
2012-01-01
In this work we characterized the non-linear complexity of Heart Rate Variability (HRV) in short time series. The complexity of HRV signal was evaluated during emotional visual elicitation by using Dominant Lyapunov Exponents (DLEs) and Approximate Entropy (ApEn). We adopted a simplified model of emotion derived from the Circumplex Model of Affects (CMAs), in which emotional mechanisms are conceptualized in two dimensions by the terms of valence and arousal. Following CMA model, a set of standardized visual stimuli in terms of arousal and valence gathered from the International Affective Picture System (IAPS) was administered to a group of 35 healthy volunteers. Experimental session consisted of eight sessions alternating neutral images with high arousal content images. Several works can be found in the literature showing a chaotic dynamics of HRV during rest or relax conditions. The outcomes of this work showed a clear switching mechanism between regular and chaotic dynamics when switching from neutral to arousal elicitation. Accordingly, the mean ApEn decreased with statistical significance during arousal elicitation and the DLE became negative. Results showed a clear distinction between the neutral and the arousal elicitation and could be profitably exploited to improve the accuracy of emotion recognition systems based on HRV time series analysis.
Stability analysis and quasinormal modes of Reissner–Nordstrøm space-time via Lyapunov exponent
PRADHAN PARTHAPRATIM
2016-07-01
We explicitly derive the proper-time (τ ) principal Lyapunov exponent (λp) and coordinate-time (t ) principal Lyapunov exponent $(\\lambda_c)$ for Reissner–Nordstrøm (RN) black hole (BH). We also compute their ratio. For RN space-time, it is shown that the ratio is $(\\lambda_{p}/\\lambda_{c}) = r_{0}/\\sqrt{r^{2}0 − 3Mr_{0} + 2Q^{2}}$ for time-like circulargeodesics and for Schwarzschild BH, it is $(\\lambda_{p}/\\lambda_{c}) = \\sqrt{r_{0}}/\\sqrt{r_{0} − 3M}. We further show that their ratio $\\lambda_{p}/\\lambda_{c}$ may vary from orbit to orbit. For instance, for Schwarzschild BH at the innermost stable circular orbit (ISCO), the ratio is $(\\lambda_{p}/\\lambda_{c})_{|rISCO}=6M = \\sqrt{2}$ and at marginally bound circular orbit (MBCO) the ratio is calculated to be $(\\lambda_{p}/\\lambda_{c})|_{rmb}=4M = 2$. Similarly, for extremal RN BH, the ratio at ISCO is $(\\lambda_{p}/\\lambda_{c})|_{rISCO}=4M = 2\\sqrt{2}/\\sqrt{3}$. We also further analyse the geodesic stability via this exponent. By evaluating the Lyapunov exponent, it is shown that in the eikonal limit, the real and imaginary parts of the quasinormal modes of RN BH is given by the frequency and instability time-scale of the unstable null circular geodesics.
Smith, Beth A.; Stergiou, Nicholas; Ulrich, Beverly D.
2010-01-01
In previous studies we found that while preadolescents with Down syndrome (DS) produce higher amounts of variability (Smith et al., 2007) and larger Lyapunov exponent (LyE) values (indicating more instability) during walking than peers with typical development (TD) (Buzzi & Ulrich, 2004), they also partition more of this into goal-equivalent variability (UCM//), that can be exploited to increase options for success when perturbed (Black et al., 2007). Here we use nonlinear methods to examine the patterns that characterize gait variability as it emerges, in toddlers with TD and with DS, rather than after years of practice. We calculated Lyapunov exponent (LyE) values to assess stability of leg trajectories. We also tested the use of 3 algorithms for surrogation analysis to investigate mathematical periodicity of toddlers’ strides. Results show that toddlers’ LyE values were not different between groups or with practice and strides of both groups become more periodic with practice. PMID:20237407
M Seidi
2016-12-01
Full Text Available Lyapunov exponent method is one of the best tools for investigating the range of stability and the transient behavior of the dynamical systems. In beryllium-moderated and heavy water-moderated reactors, photo-neutron plays an important role in dynamic behavior of the reactor. Therefore, stability analysis for changes in the control parameters of the reactor in order to guarantee safety and control nuclear reactor is important. In this work, the range of stability has been investigated using Lyapunov exponent method in response to step, ramp and sinusoidal external reactivities regarding six groups of delayed neutrons plus nine groups of photo-neutrons. The qualitative results are in good agreement with quantitative results of other works
Finite-time Lyapunov exponent-based analysis for compressible flows
González, D. R.; Speth, R. L.; Gaitonde, D. V.; Lewis, M. J.
2016-08-01
The finite-time Lyapunov exponent (FTLE) technique has shown substantial success in analyzing incompressible flows by capturing the dynamics of coherent structures. Recent applications include river and ocean flow patterns, respiratory tract dynamics, and bio-inspired propulsors. In the present work, we extend FTLE to the compressible flow regime so that coherent structures, which travel at convective speeds, can be associated with waves traveling at acoustic speeds. This is particularly helpful in the study of jet acoustics. We first show that with a suitable choice of integration time interval, FTLE can extract wave dynamics from the velocity field. The integration time thus acts as a pseudo-filter separating coherent structures from waves. Results are confirmed by examining forward and backward FTLE coefficients for several simple, well-known acoustic fields. Next, we use this analysis to identify events associated with intermittency in jet noise pressure probe data. Although intermittent events are known to be dominant causes of jet noise, their direct source in the turbulent jet flow has remained unexplained. To this end, a Large-Eddy Simulation of a Mach 0.9 jet is subjected to FTLE to simultaneously examine, and thus expose, the causal relationship between coherent structures and the corresponding acoustic waves. Results show that intermittent events are associated with entrainment in the initial roll up region and emissive events downstream of the potential-core collapse. Instantaneous acoustic disturbances are observed to be primarily induced near the collapse of the potential core and continue propagating towards the far-field at the experimentally observed, approximately 30° angle relative to the jet axis.
Finite-time Lyapunov exponent-based analysis for compressible flows.
González, D R; Speth, R L; Gaitonde, D V; Lewis, M J
2016-08-01
The finite-time Lyapunov exponent (FTLE) technique has shown substantial success in analyzing incompressible flows by capturing the dynamics of coherent structures. Recent applications include river and ocean flow patterns, respiratory tract dynamics, and bio-inspired propulsors. In the present work, we extend FTLE to the compressible flow regime so that coherent structures, which travel at convective speeds, can be associated with waves traveling at acoustic speeds. This is particularly helpful in the study of jet acoustics. We first show that with a suitable choice of integration time interval, FTLE can extract wave dynamics from the velocity field. The integration time thus acts as a pseudo-filter separating coherent structures from waves. Results are confirmed by examining forward and backward FTLE coefficients for several simple, well-known acoustic fields. Next, we use this analysis to identify events associated with intermittency in jet noise pressure probe data. Although intermittent events are known to be dominant causes of jet noise, their direct source in the turbulent jet flow has remained unexplained. To this end, a Large-Eddy Simulation of a Mach 0.9 jet is subjected to FTLE to simultaneously examine, and thus expose, the causal relationship between coherent structures and the corresponding acoustic waves. Results show that intermittent events are associated with entrainment in the initial roll up region and emissive events downstream of the potential-core collapse. Instantaneous acoustic disturbances are observed to be primarily induced near the collapse of the potential core and continue propagating towards the far-field at the experimentally observed, approximately 30° angle relative to the jet axis.
Zeren, Tamer; Özbek, Mustafa; Kutlu, Necip; Akilli, Mahmut
2016-01-05
Pneumocardiography (PNCG) is the recording method of cardiac-induced tracheal air flow and pressure pulsations in the respiratory airways. PNCG signals reflect both the lung and heart actions and could be accurately recorded in spontaneously breathing anesthetized rats. Nonlinear analysis methods, including the Lyapunov exponent, can be used to explain the biological dynamics of systems such as the cardiorespiratory system. In this study, we recorded tracheal air flow signals, including PNCG signals, from 3 representative anesthetized rats and analyzed the nonlinear behavior of these complex signals using Lyapunov exponents. Lyapunov exponents may also be used to determine the normal and pathological structure of biological systems. If the signals have at least one positive Lyapunov exponent, the signals reflect chaotic activity, as seen in PNCG signals in rats; the largest Lyapunov exponents of the signals of the healthy rats were greater than zero in this study. A method was proposed to determine the diagnostic and prognostic values of the cardiorespiratory system of rats using the arrangement of the PNCG and Lyapunov exponents, which may be monitored as vitality indicators.
Duc, Luu Hoang; Chávez, Joseph Páez; Son, Doan Thai; Siegmund, Stefan
2016-01-01
In biochemical networks transient dynamics plays a fundamental role, since the activation of signalling pathways is determined by thresholds encountered during the transition from an initial state (e.g. an initial concentration of a certain protein) to a steady-state. These thresholds can be defined in terms of the inflection points of the stimulus-response curves associated to the activation processes in the biochemical network. In the present work, we present a rigorous discussion as to the suitability of finite-time Lyapunov exponents and metabolic control coefficients for the detection of inflection points of stimulus-response curves with sigmoidal shape.
Pratap R. Patnaik
2008-04-01
Full Text Available Large-scale fed-batch fermentations are often subject to noise carried by the feed streams. This noise corrupts the process data and may destabilize the fermentation. So it is important to retrieve clear signals from noisy data. This is done by noise filters. The performances of some commonly used filters have been studied for poly-β-hydroxybutyrate production by Ralstonia eutropha. In simulated experiments, Gaussian noise was added to the flow rates of the carbon and nitrogen substrates. The filters were compared by means of the Lyapunov exponents of the outputs and their closeness to the noise-free performance. Negative exponents indicate a stable fermentation. An auto-associative neural filter performed the best, followed by a combination of a cusum filter and an extended Kalman filter. Butterworth filters were inferior and inadequate.
Short-time Lyapunov exponent analysis and the transition to chaos in Taylor-Couette flow
Vastano, John A.; Moser, Robert D.
1991-01-01
The physical mechanism driving the weakly chaotic Taylor-Couette flow is investigated using the short-time Liapunov exponent analysis. In this procedure, the transition from quasi-periodicity to chaos is studied using direct numerical 3D simulations of axially periodic Taylor-Couette flow, and a partial Liapunov exponent spectrum for the flow is computed by simultaneously advancing the full solution and a set of perturbations. It is shown that the short-time Liapunov exponent analysis yields more information on the exponents and dimension than that obtained from the common Liapunov exponent calculations. Results show that the chaotic state studied here is caused by a Kelvin-Helmholtz-type instability of the outflow boundary jet of Taylor vortices.
Short-time Lyapunov exponent analysis and the transition to chaos in Taylor-Couette flow
Vastano, John A.; Moser, Robert D.
1991-01-01
The physical mechanism driving the weakly chaotic Taylor-Couette flow is investigated using the short-time Liapunov exponent analysis. In this procedure, the transition from quasi-periodicity to chaos is studied using direct numerical 3D simulations of axially periodic Taylor-Couette flow, and a partial Liapunov exponent spectrum for the flow is computed by simultaneously advancing the full solution and a set of perturbations. It is shown that the short-time Liapunov exponent analysis yields more information on the exponents and dimension than that obtained from the common Liapunov exponent calculations. Results show that the chaotic state studied here is caused by a Kelvin-Helmholtz-type instability of the outflow boundary jet of Taylor vortices.
Reducible linear quasi-periodic systems with positive Lyapunov exponent and varying rotation number
Broer, HW; Simo, C
2000-01-01
A linear system in two dimensions is studied. The coefficients are 2 pi -periodic in three angles, 0(j), = 1, 2, 3, and these angles are linear with respect to time, with incommensurable frequencies. The system has positive Lyapunov coefficients and the rotation number changes in a continuous way wh
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
The maximal Lyapunov exponent of a co-dimension two-bifurcation system excited by a bounded noise
Sheng-Hong Li; Xian-Bin Liu
2012-01-01
In the present paper,the maximal Lyapunov exponent is investigated for a co-dimension two bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by a bounded noise.By using a perturbation method,the expressions of the invariant measure of a one-dimensional phase diffusion process are obtained for three cases,in which different forms of the matrix B,that is included in the noise excitation term,are assumed and then,as a result,all the three kinds of singular boundaries for one-dimensional phase diffusion process are analyzed.Via Monte-Carlo simulation,we find that the analytical expressions of the invariant measures meet well the numerical ones.And furthermore,the P-bifurcation behaviors are investigated for the one-dimensional phase diffusion process.Finally,for the three cases of singular boundaries for one-dimensional phase diffusion process,analytical expressions of the maximal Lyapunov exponent are presented for the stochastic bifurcation system.
Hramov, Alexander E; Maximenko, Vladimir A; Moskalenko, Olga I; 10.1063/1.4740063
2013-01-01
The spectrum of Lyapunov exponents is powerful tool for the analysis of the complex system dynamics. In the general framework of nonlinear dynamical systems a number of the numerical technics have been developed to obtain the spectrum of Lyapunov exponents for the complex temporal behavior of the systems with a few degree of freedom. Unfortunately, these methods can not apply directly to analysis of complex spatio-temporal dynamics in plasma devices which are characterized by the infinite phase space, since they are the spatially extended active media. In the present paper we propose the method for the calculation of the spectrum of the spatial Lyapunov exponents (SLEs) for the spatially extended beam-plasma systems. The calculation technique is applied to the analysis of chaotic spatio-temporal oscillations in three different beam-plasma model: (1) simple plasma Pierce diode, (2) coupled Pierce diodes, and (3) electron-wave system with backward electromagnetic wave. We find an excellent agreement between the...
Beijeren, H. van; Zon, R. van; Dorfman, J.R.
2000-01-01
The kinetic theory of gases provides methods for calculating Lyapunov exponents and other quantities, such as Kolmogorov-Sinai entropies, that characterize the chaotic behavior of hard-ball gases. Here we illustrate the use of these methods for calculating the Kolmogorov-Sinai entropy, and the
Ankışhan, Haydar; Yılmaz, Derya
2013-01-01
Snoring, which may be decisive for many diseases, is an important indicator especially for sleep disorders. In recent years, many studies have been performed on the snore related sounds (SRSs) due to producing useful results for detection of sleep apnea/hypopnea syndrome (SAHS). The first important step of these studies is the detection of snore from SRSs by using different time and frequency domain features. The SRSs have a complex nature that is originated from several physiological and physical conditions. The nonlinear characteristics of SRSs can be examined with chaos theory methods which are widely used to evaluate the biomedical signals and systems, recently. The aim of this study is to classify the SRSs as snore/breathing/silence by using the largest Lyapunov exponent (LLE) and entropy with multiclass support vector machines (SVMs) and adaptive network fuzzy inference system (ANFIS). Two different experiments were performed for different training and test data sets. Experimental results show that the multiclass SVMs can produce the better classification results than ANFIS with used nonlinear quantities. Additionally, these nonlinear features are carrying meaningful information for classifying SRSs and are able to be used for diagnosis of sleep disorders such as SAHS. PMID:24194786
Branicki, Michal
2009-01-01
We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of "transport barriers" in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagra...
Lueptow, Richard M.; Schlick, Conor P.; Umbanhowar, Paul B.; Ottino, Julio M.
2013-11-01
We investigate chaotic advection and diffusion in competitive autocatalytic reactions. To study this subject, we use a computationally efficient method for solving advection-reaction-diffusion equations for periodic flows using a mapping method with operator splitting. In competitive autocatalytic reactions, there are two species, B and C, which both react autocatalytically with species A (A +B -->2B and A +C -->2C). If there is initially a small amount of spatially localized B and C and a large amount of A, all three species will be advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that the small scale interactions associated with the chaotic velocity field, specifically the local finite-time Lyapunov exponents (FTLEs), can accurately predict the final average concentrations of B and C after the reaction is complete. The species, B or C, that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If species B and C start in regions having similar FTLEs, their average concentrations at the end of the reaction will also be similar. Funded by NSF Grant CMMI-1000469.
Shayegh, F; Sadri, S; Amirfattahi, R; Ansari-Asl, K
2014-01-01
In order to predict epileptic seizures many precursory features, extracted from the EEG signals, have been introduced. Before checking out the performance of features in detection of pre-seizure state, it is required to see whether these features are accurately extracted. Evaluation of feature estimation methods has been less considered, mainly due to the lack of a ground truth for the real EEG signals' features. In this paper, some simulated long-term depth-EEG signals, with known state spaces, are generated via a realistic neural mass model with physiological parameters. Thanks to the known ground truth of these synthetic signals, they are suitable for evaluating different algorithms used to extract the features. It is shown that conventional methods of estimating correlation dimension, the largest Lyapunov exponent, and phase coherence have non-negligible errors. Then, a parameter identification-based method is introduced for estimating the features, which leads to better estimation results for synthetic signals. It is shown that the neural mass model is able to reproduce real depth-EEG signals accurately; thus, assuming this model underlying real depth-EEG signals, can improve the accuracy of features' estimation.
Vaidyanathan Sundarapandian
2014-12-01
Full Text Available In this research work, a twelve-term novel 5-D hyperchaotic Lorenz system with three quadratic nonlinearities has been derived by adding a feedback control to a ten-term 4-D hyperchaotic Lorenz system (Jia, 2007 with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system (Jia, 2007 has the Lyapunov exponents L1 = 0.3684,L2 = 0.2174,L3 = 0 and L4 =−12.9513, and the Kaplan-Yorke dimension of this 4-D system is found as DKY =3.0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L1 = 0.4195,L2 = 0.2430,L3 = 0.0145,L4 = 0 and L5 = −13.0405, and the Kaplan-Yorke dimension of this 5-D system is found as DKY =4.0159. Thus, the novel 5-D hyperchaotic Lorenz system has a maximal Lyapunov exponent (MLE, which is greater than the maximal Lyapunov exponent (MLE of the 4-D hyperchaotic Lorenz system. The 5-D novel hyperchaotic Lorenz system has a unique equilibrium point at the origin, which is a saddle-point and hence unstable. Next, an adaptive controller is designed to stabilize the novel 5-D hyperchaotic Lorenz system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 5-D hyperchaotic Lorenz systems with unknown system parameters. Finally, an electronic circuit realization of the novel 5-D hyperchaotic Lorenz system using SPICE is described in detail to confirm the feasibility of the theoretical model.
Liu Wei-Dong(刘卫东); K.F.Ren; S.Meunier-Guttin-Cluzel; G.Gouesbet
2003-01-01
A method for the global vector-field reconstruction of nonlinear dynamical systems from a time series is studied in this paper. It employs a complete set of polynomials and singular value decomposition (SVD) to estimate a standard function which is central to the algorithm. Lyapunov exponents and dimension, calculated from the differential equations of a standard system, are used for the validation of the reconstruction. The algorithm is proven to be practical by applying it to a Rossler system.
M. Branicki
2010-01-01
Full Text Available We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.
Wang, Aixing; Fang, Chao; Liu, Yibao
2017-01-07
In this article the dynamic features of the highly excited vibrational states of the hypochlorous acid (HOCl) non-integrable system are studied using the dynamic potential and Lyapunov exponent approaches. On the condition that the 3:1 resonance between the H-O stretching and H-O-Cl bending modes accompany the 2:1 Fermi resonance between the O-Cl stretching and H-O-Cl bending modes, it is found that the dynamic potentials of the highly excited vibrational states vary regularly with different Polyad numbers (P numbers). As the P number increases, the dynamic potentials of the H-O stretching mode remain the same, but those of the H-O-Cl bending mode gradually become complex. In order to investigate the chaotic and stable features of the highly excited vibrational states of the HOCl non-integrable system, the Lyapunov exponents of different energy levels lying in the dynamic potentials of the H-O-Cl bending mode (P = 4 and 5) are calculated. It is shown that the Lyapunov exponents of the energy levels staying in the junction of Morse potential and inverse Morse potential are relative large, which indicates the degrees of chaos for these energy levels is relatively high, but the stabilities of the corresponding states are good. These results could be interpreted as the intramolecular vibrational relaxation (IVR) acting strongly via the HOCl bending motion and causing energy transfers among different modes. Based on the previous studies, these conclusions seem to be generally valid to some extent for non-integrable triatomic molecules.
Angelin Jeba, K.; Latha, M. M., E-mail: lathaisaac@yahoo.com [Department of Physics, Women' s Christian College, Nagercoil 629 001 (India); Jain, Sudhir R. [Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085 (India)
2015-11-15
The nonlinear dynamics of intra- and inter-spine interaction models of alpha-helical proteins is investigated by proposing a Hamiltonian using the first quantized operators. Hamilton's equations of motion are derived, and the dynamics is studied by constructing the trajectories and phase space plots in both cases. The phase space plots display a chaotic behaviour in the dynamics, which opens questions about the relationship between the chaos and exciton-exciton and exciton-phonon interactions. This is verified by plotting the Lyapunov characteristic exponent curves.
Angelin Jeba, K; Latha, M M; Jain, Sudhir R
2015-11-01
The nonlinear dynamics of intra- and inter-spine interaction models of alpha-helical proteins is investigated by proposing a Hamiltonian using the first quantized operators. Hamilton's equations of motion are derived, and the dynamics is studied by constructing the trajectories and phase space plots in both cases. The phase space plots display a chaotic behaviour in the dynamics, which opens questions about the relationship between the chaos and exciton-exciton and exciton-phonon interactions. This is verified by plotting the Lyapunov characteristic exponent curves.
金俐; 王琪; 陆启韶
2001-01-01
A numerical algorithm for calculating Lyapunov exponents of Hamiltonian multibody systems with topological tree configuration is studied. The algorithms for Lyapunov exponents of Hamiltonian multibody systems using the canonical equations of the system and symplectic algorithm for ordinary differential equations are presented, which are used to study the stability of the Hamiltonian multibody systems. An example is given to analyze the stability of a typical Hamiltonian multibody system, including periodic solution and chaos.%研究了树形多体Hamilton系统Lyapunov指数的数值方法.利用多体Hamilton系统的正则方程和辛算法, 给出了多体Hamilton系统Lyapunov指数的计算方法,该算法具有较好的计算精度和通用性.利用该算法可对系统的运动稳定性进行分析.最后用算例说明了该算法的有效性.
Lyapunov modes in extended systems.
Yang, Hong-Liu; Radons, Günter
2009-08-28
Hydrodynamic Lyapunov modes, which have recently been observed in many extended systems with translational symmetry, such as hard sphere systems, dynamic XY models or Lennard-Jones fluids, are nowadays regarded as fundamental objects connecting nonlinear dynamics and statistical physics. We review here our recent results on Lyapunov modes in extended system. The solution to one of the puzzles, the appearance of good and 'vague' modes, is presented for the model system of coupled map lattices. The structural properties of these modes are related to the phase space geometry, especially the angles between Oseledec subspaces, and to fluctuations of local Lyapunov exponents. In this context, we report also on the possible appearance of branches splitting in the Lyapunov spectra of diatomic systems, similar to acoustic and optical branches for phonons. The final part is devoted to the hyperbolicity of partial differential equations and the effective degrees of freedom of such infinite-dimensional systems.
Lyapunov decay in quantum irreversibility.
García-Mata, Ignacio; Roncaglia, Augusto J; Wisniacki, Diego A
2016-06-13
The Loschmidt echo--also known as fidelity--is a very useful tool to study irreversibility in quantum mechanics due to perturbations or imperfections. Many different regimes, as a function of time and strength of the perturbation, have been identified. For chaotic systems, there is a range of perturbation strengths where the decay of the Loschmidt echo is perturbation independent, and given by the classical Lyapunov exponent. But observation of the Lyapunov decay depends strongly on the type of initial state upon which an average is carried out. This dependence can be removed by averaging the fidelity over the Haar measure, and the Lyapunov regime is recovered, as has been shown for quantum maps. In this work, we introduce an analogous quantity for systems with infinite dimensional Hilbert space, in particular the quantum stadium billiard, and we show clearly the universality of the Lyapunov regime.
Random Matrices and Lyapunov Coefficients Regularity
Gallavotti, Giovanni
2017-02-01
Analyticity and other properties of the largest or smallest Lyapunov exponent of a product of real matrices with a "cone property" are studied as functions of the matrices entries, as long as they vary without destroying the cone property. The result is applied to stability directions, Lyapunov coefficients and Lyapunov exponents of a class of products of random matrices and to dynamical systems. The results are not new and the method is the main point of this work: it is is based on the classical theory of the Mayer series in Statistical Mechanics of rarefied gases.
The Lyapunov dimension and its estimation via the Leonov method
Kuznetsov, N.V., E-mail: nkuznetsov239@gmail.com
2016-06-03
Highlights: • Survey on effective analytical approach for Lyapunov dimension estimation, proposed by Leonov, is presented. • Invariance of Lyapunov dimension under diffeomorphisms and its connection with Leonov method are demonstrated. • For discrete-time dynamical systems an analog of Leonov method is suggested. - Abstract: Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, proposed by G.A. Leonov in 1991. The Leonov method is based on the direct Lyapunov method with special Lyapunov-like functions. The advantage of the method is that it allows one to estimate the Lyapunov dimension of invariant sets without localization of the set in the phase space and, in many cases, to get effectively an exact Lyapunov dimension formula. In this work the invariance of the Lyapunov dimension with respect to diffeomorphisms and its connection with the Leonov method are discussed. For discrete-time dynamical systems an analog of Leonov method is suggested. In a simple but rigorous way, here it is presented the connection between the Leonov method and the key related works: Kaplan and Yorke (the concept of the Lyapunov dimension, 1979), Douady and Oesterlé (upper bounds of the Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foiaş, and Temam (upper bounds of the Hausdorff dimension via the Lyapunov exponents and Lyapunov dimension of dynamical systems, 1985–90), and the numerical calculation of the Lyapunov exponents and dimension.
M. Widi Triyatno
2015-03-01
Full Text Available Disturbances in the operation of the power system may cause disturbance in voltage stability. Therefore, dynamic voltage stability analysis before and after disturbance needs to be performed. This paper proposes dynamic voltage stability prediction using maximum Lyapunov exponent with Lampung’s electrical system as case study. Voltage stability simulation is performed with various types of disturbances that occur at line between of Baturaja substation and Bukit Kemuning substation. Time-series data of voltage measurement of simulation results at GI Baturaja is applied for voltage stability prediction analysis using maximum Lyapunov exponent. With the same number of data samples and the same time for circuit breakers to interrupt disturbances, the simulation results using maximum Lyapunov exponent show that the voltage can be stabilized at 1.7 seconds after the occurrence of the three-phase disturbance, at 1.2 seconds after the occurrence of the phase-to-ground disturbance, at 0,9 second after the occurrence of the disturbance between phase, at 1.2 seconds after the occurrence of the loss of line disturbance and 1.4 seconds after the occurrence of the loss of load disturbance. The amount of data samples used in analysis affect the time for the voltage reaches stability.
On the computation of quantum characteristic exponents
Vilela-Mendes, R; Coutinho, Ricardo
1998-01-01
A quantum characteristic exponent may be defined, with the same operational meaning as the classical Lyapunov exponent when the latter is expressed as a functional of densities. Existence conditions and supporting measure properties are discussed as well as the problems encountered in the numerical computation of the quantum exponents. Although an example of true quantum chaos may be exhibited, the taming effect of quantum mechanics on chaos is quite apparent in the computation of the quantum exponents. However, even when the exponents vanish, the functionals used for their definition may still provide a characterization of distinct complexity classes for quantum behavior.
Lyapunov instabilities of Lennard-Jones fluids.
Yang, Hong-liu; Radons, Günter
2005-03-01
Recent work on many-particle systems reveals the existence of regular collective perturbations corresponding to the smallest positive Lyapunov exponents (LEs), called hydrodynamic Lyapunov modes. Until now, however, these modes have been found only for hard-core systems. Here we report results on Lyapunov spectra and Lyapunov vectors (LVs) for Lennard-Jones fluids. By considering the Fourier transform of the coordinate fluctuation density u((alpha)) (x,t) , it is found that the LVs with lambda approximately equal to 0 are highly dominated by a few components with low wave numbers. These numerical results provide strong evidence that hydrodynamic Lyapunov modes do exist in soft-potential systems, although the collective Lyapunov modes are more vague than in hard-core systems. In studying the density and temperature dependence of these modes, it is found that, when the value of the Lyapunov exponent lambda((alpha)) is plotted as function of the dominant wave number k(max) of the corresponding LV, all data from simulations with different densities and temperatures collapse onto a single curve. This shows that the dispersion relation lambda((alpha)) vs k(max) for hydrodynamical Lyapunov modes appears to be universal for the low-density cases studied here. Despite the wavelike character of the LVs, no steplike structure exists in the Lyapunov spectrum of the systems studied here, in contrast to the hard-core case. Further numerical simulations show that the finite-time LEs fluctuate strongly. We have also investigated localization features of LVs and propose a length scale to characterize the Hamiltonian spatiotemporal chaotic states.
Comparison between covariant and orthogonal Lyapunov vectors.
Yang, Hong-liu; Radons, Günter
2010-10-01
Two sets of vectors, covariant Lyapunov vectors (CLVs) and orthogonal Lyapunov vectors (OLVs), are currently used to characterize the linear stability of chaotic systems. A comparison is made to show their similarity and difference, especially with respect to the influence on hydrodynamic Lyapunov modes (HLMs). Our numerical simulations show that in both Hamiltonian and dissipative systems HLMs formerly detected via OLVs survive if CLVs are used instead. Moreover, the previous classification of two universality classes works for CLVs as well, i.e., the dispersion relation is linear for Hamiltonian systems and quadratic for dissipative systems, respectively. The significance of HLMs changes in different ways for Hamiltonian and dissipative systems with the replacement of OLVs with CLVs. For general dissipative systems with nonhyperbolic dynamics the long-wavelength structure in Lyapunov vectors corresponding to near-zero Lyapunov exponents is strongly reduced if CLVs are used instead, whereas for highly hyperbolic dissipative systems the significance of HLMs is nearly identical for CLVs and OLVs. In contrast the HLM significance of Hamiltonian systems is always comparable for CLVs and OLVs irrespective of hyperbolicity. We also find that in Hamiltonian systems different symmetry relations between conjugate pairs are observed for CLVs and OLVs. Especially, CLVs in a conjugate pair are statistically indistinguishable in consequence of the microreversibility of Hamiltonian systems. Transformation properties of Lyapunov exponents, CLVs, and hyperbolicity under changes of coordinate are discussed in appendices.
Baire classes of Lyapunov invariants
Bykov, V. V.
2017-05-01
It is shown that no relations exist (apart from inherent ones) between Baire classes of Lyapunov transformation invariants in the compact- open and uniform topologies on the space of linear differential systems. It is established that if a functional on the space of linear differential systems with the compact-open topology is the repeated limit of a multisequence of continuous functionals, then these can be chosen to be determined by the values of system coefficients on a finite interval of the half-line (one for each functional). It is proved that the Lyapunov exponents cannot be represented as the limit of a sequence of (not necessarily continuous) functionals such that each of these depends only on the restriction of the system to a finite interval of the half-line. Bibliography: 28 titles.
Covariant Lyapunov vectors for rigid disk systems.
Bosetti, Hadrien; Posch, Harald A
2010-10-05
We carry out extensive computer simulations to study the Lyapunov instability of a two-dimensional hard-disk system in a rectangular box with periodic boundary conditions. The system is large enough to allow the formation of Lyapunov modes parallel to the x-axis of the box. The Oseledec splitting into covariant subspaces of the tangent space is considered by computing the full set of covariant perturbation vectors co-moving with the flow in tangent space. These vectors are shown to be transversal, but generally not orthogonal to each other. Only the angle between covariant vectors associated with immediate adjacent Lyapunov exponents in the Lyapunov spectrum may become small, but the probability of this angle to vanish approaches zero. The stable and unstable manifolds are transverse to each other and the system is hyperbolic.
A finite-time exponent for random Ehrenfest gas
Moudgalya, Sanjay; Chandra, Sarthak [Indian Institute of Technology, Kanpur 208016 (India); Jain, Sudhir R., E-mail: srjain@barc.gov.in [Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085 (India)
2015-10-15
We consider the motion of a system of free particles moving on a plane with regular hard polygonal scatterers arranged in a random manner. Calling this the Ehrenfest gas, which is known to have a zero Lyapunov exponent, we propose a finite-time exponent to characterize its dynamics. As the number of sides of the polygon goes to infinity, when polygon tends to a circle, we recover the usual Lyapunov exponent for the Lorentz gas from the exponent proposed here. To obtain this result, we generalize the reflection law of a beam of rays incident on a polygonal scatterer in a way that the formula for the circular scatterer is recovered in the limit of infinite number of vertices. Thus, chaos emerges from pseudochaos in an appropriate limit. - Highlights: • We present a finite-time exponent for particles moving in a plane containing polygonal scatterers. • The exponent found recovers the Lyapunov exponent in the limit of the polygon becoming a circle. • Our findings unify pseudointegrable and chaotic scattering via a generalized collision rule. • Stretch and fold:shuffle and cut :: Lyapunov:finite-time exponent :: fluid:granular mixing.
Kolyada, Sergiy; Rybak, Oleksandr
2013-01-01
We introduce and study the Lyapunov numbers -- quantitative measures of the sensitivity of a dynamical system $(X,f)$ given by a compact metric space $X$ and a continuous map $f:X \\to X$. In particular, we prove that for a minimal topologically weakly mixing system all Lyapunov numbers are the same.
Relative Lyapunov Center Bifurcations
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....
Lyapunov spectra of Coulombic and gravitational periodic systems
Kumar, Pankaj
2016-01-01
We compute Lyapunov spectra for Coulombic and gravitational versions of the one-dimensional systems of parallel sheets with periodic boundary conditions. Exact time evolution of tangent-space vectors are derived and are utilized toward computing Lypaunov characteristic exponents using an event-driven algorithm. The results indicate that the energy dependence of the largest Lyapunov exponent emulates that of Kolmogorov-entropy density for each system at different degrees of freedom. Our approach forms an effective and approximation-free tool toward studying the dynamical properties exhibited by the Coulombic and gravitational systems and finds applications in investigating indications of thermodynamic transitions in large versions of the spatially periodic systems.
贺国光; 万兴义
2003-01-01
利用交通流模型产生交通流时间序列.再利用Lyapunov指数的矩形阵算法,计算出交通流时间序列的最大Lyapunov指数.由于Lyapunov指数是定量描述混沌吸引子的重要指标,可根据Lyapunov指数对混沌序列的辨别原理,进而识别该由基于模型的动力系统是否处于混沌状态.
Hydrodynamic Lyapunov modes and strong stochasticity threshold in Fermi-Pasta-Ulam models.
Yang, Hong-Liu; Radons, Günter
2006-06-01
The existence of a strong stochasticity threshold (SST) has been detected in many Hamiltonian lattice systems, including the Fermi-Pasta-Ulam (FPU) model, which is characterized by a crossover of the system dynamics from weak to strong chaos with increasing energy density epsilon. Correspondingly, the relaxation time to energy equipartition and the largest Lyapunov exponent exhibit different scaling behavior in the regimes below and beyond the threshold value. In this paper, we attempt to go one step further in this direction to explore further changes in the energy density dependence of other Lyapunov exponents and of hydrodynamic Lyapunov modes (HLMs). In particular, we find that for the FPU-beta and FPU-alpha(beta) models the scalings of the energy density dependence of all Lyapunov exponents experience a similar change at the SST as that of the largest Lyapunov exponent. In addition, the threshold values of the crossover of all Lyapunov exponents are nearly identical. These facts lend support to the point of view that the crossover in the system dynamics at the SST manifests a global change in the geometric structure of phase space. They also partially answer the question of why the simple assumption that the ambient manifold representing the system dynamics is quasi-isotropic works quite well in the analytical calculation of the largest Lyapunov exponent. Furthermore, the FPU-beta model is used as an example to show that HLMs exist in Hamiltonian lattice models with continuous symmetries. Some measures are defined to indicate the significance of HLMs. Numerical simulations demonstrate that there is a smooth transition in the energy density dependence of these variables corresponding to the crossover in Lyapunov exponents at the SST. In particular, our numerical results indicate that strong chaos is essential for the appearance of HLMs and those modes become more significant with increasing degree of chaoticity.
On the recurrence and Lyapunov time scales of the motion near the chaos border
Shevchenko, Ivan I
2016-01-01
Conditions for the emergence of a statistical relationship between $T_r$, the chaotic transport (recurrence) time, and $T_L$, the local Lyapunov time (the inverse of the numerically measured largest Lyapunov characteristic exponent), are considered for the motion inside the chaotic layer around the separatrix of a nonlinear resonance. When numerical values of the Lyapunov exponents are measured on a time interval not greater than $T_r$, the relationship is shown to resemble the quadratic one. This tentatively explains numerical results presented in the literature.
Bosetti, Hadrien; Posch, Harald A; Dellago, Christoph; Hoover, William G
2010-10-01
Recently, a new algorithm for the computation of covariant Lyapunov vectors and of corresponding local Lyapunov exponents has become available. Here we study the properties of these still unfamiliar quantities for a simple model representing a harmonic oscillator coupled to a thermal gradient with a two-stage thermostat, which leaves the system ergodic and fully time reversible. We explicitly demonstrate how time-reversal invariance affects the perturbation vectors in tangent space and the associated local Lyapunov exponents. We also find that the local covariant exponents vary discontinuously along directions transverse to the phase flow.
A statistical approach to estimate the LYAPUNOV spectrum in disc brake squeal
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.
Critical exponents in the transition to chaos in one-dimensional discrete systems
G Ambika; N V Sujatha
2002-07-01
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 Lyapunov exponents (LE) in the cross over region is also studied for a complete characterization.
Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization
Christiansen, F; Christiansen, Freddy; Rugh, Hans Henrik
1996-01-01
We present a straightforward and reliable continuous method for computing the full or a partial Lyapunov spectrum associated with a dynamical system specified by a set of differential equations. We do this by introducing a stability parameter beta>0 and augmenting the dynamical system with an orthonormal k-dimensional frame and a Lyapunov vector such that the frame is continuously Gram-Schmidt orthonormalized and at most linear growth of the dynamical variables is involved. We prove that the method is strongly stable when beta > -lambda_k where lambda_k is the k'th Lyapunov exponent in descending order and we show through examples how the method is implemented. It extends many previous results.
Detecting Epileptic Seizure from Scalp EEG Using Lyapunov Spectrum
Truong Quang Dang Khoa
2012-01-01
Full Text Available One of the inherent weaknesses of the EEG signal processing is noises and artifacts. To overcome it, some methods for prediction of epilepsy recently reported in the literature are based on the evaluation of chaotic behavior of intracranial electroencephalographic (EEG recordings. These methods reduced noises, but they were hazardous to patients. In this study, we propose using Lyapunov spectrum to filter noise and detect epilepsy on scalp EEG signals only. We determined that the Lyapunov spectrum can be considered as the most expected method to evaluate chaotic behavior of scalp EEG recordings and to be robust within noises. Obtained results are compared to the independent component analysis (ICA and largest Lyapunov exponent. The results of detecting epilepsy are compared to diagnosis from medical doctors in case of typical general epilepsy.
Detecting epileptic seizure from scalp EEG using Lyapunov spectrum.
Khoa, Truong Quang Dang; Huong, Nguyen Thi Minh; Toi, Vo Van
2012-01-01
One of the inherent weaknesses of the EEG signal processing is noises and artifacts. To overcome it, some methods for prediction of epilepsy recently reported in the literature are based on the evaluation of chaotic behavior of intracranial electroencephalographic (EEG) recordings. These methods reduced noises, but they were hazardous to patients. In this study, we propose using Lyapunov spectrum to filter noise and detect epilepsy on scalp EEG signals only. We determined that the Lyapunov spectrum can be considered as the most expected method to evaluate chaotic behavior of scalp EEG recordings and to be robust within noises. Obtained results are compared to the independent component analysis (ICA) and largest Lyapunov exponent. The results of detecting epilepsy are compared to diagnosis from medical doctors in case of typical general epilepsy.
Lyapunov analysis: from dynamical systems theory to applications
Cencini, Massimo; Ginelli, Francesco
2013-06-01
The study of deterministic laws of evolution has characterized the development of science since Newton's times. Chaos, namely the manifestation of irregular and unpredictable dynamics (not random but look random [1]), entered the debate on determinism at the end of the 19th century with the discovery of sensitivity to initial conditions, meaning that small infinitesimal differences in the initial state might lead to dramatic differences at later times. Poincaré [2, 3] was the first to realize that solutions of the three-body problem are generically highly sensitive to initial conditions. At about the same time, this property was recognized in geodesic flows with negative curvature by Hadamard [4]. One of the first experimental observations of chaos, as understood much later, was when irregular noise was heard by Van der Pol in 1927 [5] while studying a periodically forced nonlinear oscillator. Nevertheless, it was only with the advent of digital computing that chaos started to attract the interest of the wider scientific community. After the pioneering investigation of ergodicity in a chain of nonlinear oscillators by Fermi, Pasta and Ulam in 1955 [6], it was in the early 1960s that the numerical studies of Lorenz [7] and Hénon and Heiles [8] revealed that irregular and unpredictable motions are a generic feature of low-dimensional nonlinear deterministic systems. The existence and onset of chaos was then rigorously analyzed in several systems. While an exhaustive list of such mathematical proofs is beyond the scope of this preface, one should mention the contributions of Kolmogorov [9, 10], Chirikov [11], Smale [12], Ruelle and Takens [13], Li and Yorke [14] and Feigenbaum [15]. The characteristic Lyapunov exponents introduced by Oseledets in 1968 [16] are the fundamental quantities for measuring the sensitivity to initial conditions. Oseledets' work generalized the concept of Lyapunov stability to irregular trajectories building upon earlier studies of Birkhoff
Estimating the Lyapunov spectrum of time delay feedback systems from scalar time series.
Hegger, R
1999-08-01
On the basis of a recently developed method for modeling time delay systems, we propose a procedure to estimate the spectrum of Lyapunov exponents from a scalar time series. It turns out that the spectrum is approximated very well and allows for good estimates of the Lyapunov dimension even if the sampling rate of the time series is so low that the infinite dimensional tangent space is spanned quite sparsely.
Lyapunov, Floquet, and singular vectors for baroclinic waves
R. M. Samelson
2001-01-01
Full Text Available The dynamics of the growth of linear disturbances to a chaotic basic state is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave-mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal flow correction. The leading Lyapunov vector is nearly parallel to the leading Floquet vector f1 of the lowest-order unstable periodic orbit over most of the attractor. Departures of the Lyapunov vector from this orientation are primarily rotations of the vector in an approximate tangent plane to the large-scale attractor structure. Exponential growth and decay rates of the Lyapunov vector during individual Poincaré section returns are an order of magnitude larger than the Lyapunov exponent l ≈ 0.016. Relatively large deviations of the Lyapunov vector from parallel to f1 are generally associated with relatively large transient decays. The transient growth and decay of the Lyapunov vector is well described by the transient growth and decay of the leading Floquet vectors of the set of unstable periodic orbits associated with the attractor. Each of these vectors is also nearly parallel to f1. The dynamical splitting of the complete sets of Floquet vectors for the higher-order cycles follows the previous results on the lowest-order cycle, with the vectors divided into wave-dynamical and decaying zonal flow modes. Singular vectors and singular values also generally follow this split. The primary difference between the leading Lyapunov and singular vectors is the contribution of decaying, inviscidly-damped wave-dynamical structures to the singular vectors.
Jumping property of Lyapunov values
毛锐; 王铎
1996-01-01
A sufficient condition for fcth Lyapunov value to be zero for planar polynomial vector fields is given, which extends the result of "jumping property’ of Lyapunov values obtained by Wang Duo to more general cases. A concrete example that the origin cannot be weak focus of order 1, 2, 4, 5, 8 is presented.
Broucke, R.
1982-01-01
It is pointed out that the Lyapunov Characteristic Numbers constitute a new tool for determining stability of trajectories of dynamical systems, or, even more generally, of solutions of systems of ordinary differential equations. In contrast with the characteristic exponents, which apply only to periodic solutions, the Lyapunov Characteristic Numbers apply to arbitrary nonperiodic solutions as well. A description is presented of the numerical experiments which have been made in order to investigate the practical value of the Lyapunov Characteristic Number and the Kolmogorov Entropy for the purpose of estimating the stability of trajectories and/or numerical integration methods in celestial mechanics. It is found that the Lyapunov Characteristic Numbers are extremely useful for the classification of the solutions of nonintegrable dynamical systems, especially in order to distinguish between quasi-periodic and chaotic solutions. However, the Lyapunov Characteristics Numbers do not appear to be useful for the purpose of evaluating numerical integration methods.
Broucke, R.
1982-01-01
It is pointed out that the Lyapunov Characteristic Numbers constitute a new tool for determining stability of trajectories of dynamical systems, or, even more generally, of solutions of systems of ordinary differential equations. In contrast with the characteristic exponents, which apply only to periodic solutions, the Lyapunov Characteristic Numbers apply to arbitrary nonperiodic solutions as well. A description is presented of the numerical experiments which have been made in order to investigate the practical value of the Lyapunov Characteristic Number and the Kolmogorov Entropy for the purpose of estimating the stability of trajectories and/or numerical integration methods in celestial mechanics. It is found that the Lyapunov Characteristic Numbers are extremely useful for the classification of the solutions of nonintegrable dynamical systems, especially in order to distinguish between quasi-periodic and chaotic solutions. However, the Lyapunov Characteristics Numbers do not appear to be useful for the purpose of evaluating numerical integration methods.
Yang, Hong-Liu; Radons, Günter
2008-01-01
Crossover from weak to strong chaos in high-dimensional Hamiltonian systems at the strong stochasticity threshold (SST) was anticipated to indicate a global transition in the geometric structure of phase space. Our recent study of Fermi-Pasta-Ulam models showed that corresponding to this transition the energy density dependence of all Lyapunov exponents is identical apart from a scaling factor. The current investigation of the dynamic XY model discovers an alternative scenario for the energy dependence of the system dynamics at SSTs. Though similar in tendency, the Lyapunov exponents now show individually different energy dependencies except in the near-harmonic regime. Such a finding restricts the use of indices such as the largest Lyapunov exponent and the Ricci curvatures to characterize the global transition in the dynamics of high-dimensional Hamiltonian systems. These observations are consistent with our conjecture that the quasi-isotropy assumption works well only when parametric resonances are the dominant sources of dynamical instabilities. Moreover, numerical simulations demonstrate the existence of hydrodynamical Lyapunov modes (HLMs) in the dynamic XY model and show that corresponding to the crossover in the Lyapunov exponents there is also a smooth transition in the energy density dependence of significance measures of HLMs. In particular, our numerical results confirm that strong chaos is essential for the appearance of HLMs.
Chaotic Griffiths Phase with Anomalous Lyapunov Spectra in Coupled Map Networks.
Shinoda, Kenji; Kaneko, Kunihiko
2016-12-16
Dynamics of coupled chaotic oscillators on a network are studied using coupled maps. Within a broad range of parameter values representing the coupling strength or the degree of elements, the system repeats formation and split of coherent clusters. The distribution of the cluster size follows a power law with the exponent α, which changes with the parameter values. The number of positive Lyapunov exponents and their spectra are scaled anomalously with the power of the system size with the exponent β, which also changes with the parameters. The scaling relation α∼2(β+1) is uncovered, which is universal independent of parameters and among random networks.
Chaotic Griffiths Phase with Anomalous Lyapunov Spectra in Coupled Map Networks
Shinoda, Kenji; Kaneko, Kunihiko
2016-12-01
Dynamics of coupled chaotic oscillators on a network are studied using coupled maps. Within a broad range of parameter values representing the coupling strength or the degree of elements, the system repeats formation and split of coherent clusters. The distribution of the cluster size follows a power law with the exponent α , which changes with the parameter values. The number of positive Lyapunov exponents and their spectra are scaled anomalously with the power of the system size with the exponent β , which also changes with the parameters. The scaling relation α ˜2 (β +1 ) is uncovered, which is universal independent of parameters and among random networks.
Lyapunov spectra and conjugate-pairing rule for confined atomic fluids
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 confined...... the spectrum reflects the presence of two different dynamics in the system: one for the unthermostatted fluid atoms and the other one for the thermostatted and tethered wall atoms. In particular the Lyapunov spectrum of the whole system does not satisfy the conjugate-pairing rule. Two regions are instead...... distinguishable, one with negative pairs' sum and one with a sum close to zero. To locate the different contributions to the spectrum of the system, we computed "approximate" Lyapunov exponents belonging to the phase space generated by the thermostatted area and the unthermostatted area alone. To achieve this, we...
Persistence probabilities \\& exponents
Aurzada, Frank
2012-01-01
This article deals with the asymptotic behaviour as $t\\to +\\infty$ of the survival function $P[T > t],$ where $T$ is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behaviour is of the type $P[T > t]=t^{-\\theta + o(1)}$ for a known or unknown positive parameter $\\theta$ which is called a persistence exponent. The problem is well understood for random walks or L\\'evy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.
The isentropic exponent in plasmas
K.T.A.L. Burm,; W. J. Goedheer,; D.C. Schram,
1999-01-01
The isentropic exponent for gases is a physical quantity that can ease significantly the hydrodynamic modeling effort. In gas dynamics the isentropic exponent depends only on the number of degrees of freedom of the considered gas. The isentropic exponent for a plasma is lower due to an extra degree
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.
Allometric Exponent and Randomness
Yi, Su Do; Minnhagen, Petter; 10.1088/1367-2630/15/4/043001
2013-01-01
An allometric height-mass exponent $\\gamma$ gives an approximative power-law relation $ \\propto H^\\gamma$ between the average mass $$ and the height $H$, for a sample of individuals. The individuals in the present study are humans but could be any biological organism. The sampling can be for a specific age of the individuals or for an age-interval. The body-mass index (BMI) is often used for practical purposes when characterizing humans and it is based on the allometric exponent $\\gamma=2$. It is here shown that the actual value of $\\gamma$ is to large extent determined by the degree of correlation between mass and height within the sample studied: no correlation between mass and height means $\\gamma=0$, whereas if there was a precise relation between mass and height such that all individuals had the same shape and density then $\\gamma=3$. The connection is demonstrated by showing that the value of $\\gamma$ can be obtained directly from three numbers characterizing the spreads of the relevant random Gaussian ...
何岱海; 徐健学; 陈永红; 谭宁
2000-01-01
本文研究条件Lyapunov指数与τ-条件Lyapunov指数的定义、求解技术及其应用.两种指数从不同角度对系统本质特性进行刻划.条件Lyapunov指数在混沌同步中有重要应用,近来它还被用来进行相空间重构问题的研究.时间τ-条件Lyapunov指数是一类利用状态变量的离散采样作驱动信号的脉冲方式同步的重要定量指标.本文提出一种简便的求解技术,在Wolf求解Lyapunov指数谱程序的基础上,稍加改动即可使其适用于Lyapunov指数、条件Lyapunov指数和时间τ-条件Lyapunov指数的计算,对其正确性进行了验证.研究发现对时间τ-条件Lyapunov指数的计算,可以准确估计脉冲方式同步的最大间隔τ和最优区间,对于实际工作具有重要意义.
Taniguchi, Tooru; Morriss, Gary P
2005-01-01
The time-dependent mode structure of the Lyapunov vectors associated with the stepwise structure of the Lyapunov spectra and its relation to the momentum autocorrelation function are discussed in quasi-one-dimensional many-hard-disk systems. We obtain the complete mode structures (Lyapunov modes) for all components of the Lyapunov vectors, including the longitudinal and transverse components of both the spatial and momentum parts, and their phase relations. These mode structures are suggested by the form of the Lyapunov vectors for the zero-Lyapunov exponents. The spatial node structures of these modes are explained by the reflection properties of the hard walls used in the models. Our main result is that the largest time-oscillating period of the Lyapunov modes is twice as long as the time-oscillating period of the longitudinal momentum autocorrelation function. This relation is satisfied irrespective of the number of particles and the boundary conditions. A simple explanation for this relation is given based on the form of the time-dependent Lyapunov mode.
Regeneration cycle and the covariant Lyapunov vectors in a minimal wall turbulence.
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.
Construction of the Lyapunov Spectrum in a Chaotic System Displaying Phase Synchronization
Carlo, Leonardo De, E-mail: neoleodeo@gmail.com [Gran Sasso Science Institute (GSSI) (Italy); Gentile, Guido, E-mail: gentile@mat.uniroma3.it; Giuliani, Alessandro, E-mail: giuliani@mat.uniroma3.it [Università degli Studi Roma Tre, Dipartimento di Matematica e Fisica (Italy)
2016-06-15
We consider a three-dimensional chaotic system consisting of the suspension of Arnold’s cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relative phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.
Guastello, Stephen J; Nathan, Dominic E; Johnson, Michelle J
2009-01-01
The principles of attractors and Lyapunov exponents were used to develop a reaching-to-grasp model for use in a robotic therapy system for stroke patients. Previously known models for these movements, the fifth order minimum jerk and the seventh order polynomial, do not account for the change in grasp aperture of the hand. The Lyapunov model was tested with reaching-to-grasp movements performed by five neurologically intact subjects and produced an average R-square = .97 over 15 replications for 41 different task events, reflecting a notable advantage over the fifth order (average R-square = .58) and seventh order (average R-square = .67) models. A similar level of success was obtained for the Lyapunov model that was specific to grasp aperture. The results indicated that intentional movements can be accurately characterized as attractor trajectories, and as functions of position along two Cartesian coordinates rather than as functions of time. The Lyapunov exponent model requires fewer parameters and provides an efficient platform for real-time implementation.
Lyapunov spectra and conjugate-pairing rule for confined atomic fluids.
Bernardi, Stefano; Todd, B D; Hansen, J S; Searles, Debra J; Frascoli, Federico
2010-06-28
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 confined system have been considered. Two different types of flow have been simulated: planar Couette flow and planar Poiseuille flow. Several studies exist on the former for homogeneous flows, so a direct comparison with previous results is performed. An important outcome of this work is the demonstration of how the spectrum reflects the presence of two different dynamics in the system: one for the unthermostatted fluid atoms and the other one for the thermostatted and tethered wall atoms. In particular the Lyapunov spectrum of the whole system does not satisfy the conjugate-pairing rule. Two regions are instead distinguishable, one with negative pairs' sum and one with a sum close to zero. To locate the different contributions to the spectrum of the system, we computed "approximate" Lyapunov exponents belonging to the phase space generated by the thermostatted area and the unthermostatted area alone. To achieve this, we 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.
Lyapunov Function Synthesis - Algorithm and Software
Leth, Tobias; Wisniewski, Rafal; Sloth, Christoffer
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...
Rank-one LMIs and Lyapunov's inequality
Henrion, D.; Meinsma, Gjerrit
2001-01-01
We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semi-definite programming. Links are established between the Lyapunov matrix,
Rank-one LMIs and Lyapunov's inequality
Henrion, D.; Meinsma, G.
2001-01-01
We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semi-definite programming. Links are established between the Lyapunov matrix, rank-on
Lyapunov Function Synthesis - Algorithm and Software
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...
Predictability of chaotic dynamics a finite-time Lyapunov exponents approach
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...
Robust lyapunov controller for uncertain systems
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.
Characteristic exponents of complex networks
Nicosia, Vincenzo; Latora, Vito
2013-01-01
We propose a method to characterize and classify complex networks based on the time series generated by random walks and different node properties. The analysis of the fluctuations of the time series reveals the presence of long-range correlations, and allows to define, for each network, a set of characteristic exponents that capture its essential structural properties. By considering a large data set of real-world networks, we show that the characteristic exponents can be used to classify complex networks according to their function, and are able to discriminate social from biological and technological systems.
Infinitesimal Lyapunov functions and singular-hyperbolicity
Araujo, Vitor
2012-01-01
We present an extension of the notion of infinitesimal Lyapunov function to singular flows on three-dimensional manifolds, and show how this technique provides a characterization of partially hyperbolic structures for invariant sets for such flows, and also of singular-hyperbolicity. In the absence of singularities, we can also rephrase uniform hyperbolicity with the language of infinitesimal Lyapunov functions. These conditions are expressed using the vector field X and its space derivative DX together with an infinitesimal Lyapunov function only and are reduced to checking that a certain symmetric operator is positive definite on the trapping region: we show how to express partial hyperbolicity using only the interplay between the infinitesimal generator X of the flow X_t, its derivative DX and the infinitesimal Lyapunov function.
Coordinate-invariant incremental Lyapunov functions
Zamani, Majid
2011-01-01
The notion of incremental stability was proposed by several researchers as a strong property of dynamical and control systems. In this type of stability, the focus is on the convergence of trajectories with respect to themselves, rather than with respect to an equilibrium point or a particular trajectory. Similarly to stability, Lyapunov functions play an important role in the study of incremental stability. In this paper, we propose coordinate-invariant notions of incremental Lyapunov function and provide the description of incremental stability in terms of existence of the proposed Lyapunov functions. Moreover, we develop a backstepping design approach providing a recursive way of constructing controllers as well as incremental Lyapunov functions. The effectiveness of our method is illustrated by synthesizing a controller rendering a single-machine infinite-bus electrical power system incrementally stable.
Localization properties of covariant Lyapunov vectors for quasi-one-dimensional hard disks.
Morriss, G P
2012-05-01
The Lyapunov exponent spectrum and covariant Lyapunov vectors are studied for a quasi-one-dimensional system of hard disks as a function of density and system size. We characterize the system using the angle distributions between covariant vectors and the localization properties of both Gram-Schmidt and covariant vectors. At low density there is a kinetic regime that has simple scaling properties for the Lyapunov exponents and the average localization for part of the spectrum. This regime shows strong localization in a proportion of the first Gram-Schmidt and covariant vectors and this can be understood as highly localized configurations dominating the vector. The distribution of angles between neighboring covariant vectors has characteristic shapes depending upon the difference in vector number, which vary over the continuous region of the spectrum. At dense gas- or liquid-like densities the behavior of the covariant vectors are quite different. The possibility of tangencies between different components of the unstable manifold and between the stable and unstable manifolds is explored but it appears that exact tangencies do not occur for a generic chaotic trajectory.
A survey of quantum Lyapunov control methods.
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.
Critical exponents from cluster coefficients
Rotman, Z.; Eisenberg, E.
2009-09-01
For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal real symmetric matrix Rmn , whose elements converge to two constants. This allows for an effective extrapolation of the equation of state for these models. Due to a nearby (nonphysical) singularity on the negative real z axis, standard methods (e.g., Padé approximants based on the cluster integrals expansion) fail to capture the behavior of these models near the ordering transition, and, in particular, do not detect the critical point. A recent work [E. Eisenberg and A. Baram, Proc. Natl. Acad. Sci. U.S.A. 104, 5755 (2007)] has shown that the critical exponents σ and σ' , characterizing the singularity of the density as a function of the activity, can be exactly calculated if the decay of the R matrix elements to their asymptotic constant follows a 1/n2 law. Here we employ renormalization group (RG) arguments to extend this result and analyze cases for which the asymptotic approach of the R matrix elements toward their limiting value is of a more general form. The relevant asymptotic correction terms (in RG sense) are identified, and we then present a corrected exact formula for the critical exponents. We identify the limits of usage of the formula and demonstrate one physical model, which is beyond its range of validity. The formula is validated numerically and then applied to analyze a number of concrete physical models.
A.B. Demchyshyn
2012-03-01
Full Text Available Differences between critical exponents of this model and the continuous percolation model indicate that the dependence of the modified structure area on the dose and the angle related with the correlation between individual tracks. It results in next effect: angular dependence of the surface area of the branched structure has maximum value at certain «critical» angle of ions incidence. Differences between critical exponents of this model and the continuous percolation model indicate that the dependence of the modified structure area on the dose and the angle related with the correlation between individual tracks. It results in next effect: angular dependence of the surface area of the branched structure has maximum value at certain «critical» angle of ions incidence. Differences between critical exponents of this model and the continuous percolation model indicate that the dependence of the modified structure area on the dose and the angle related with the correlation between individual tracks. It results in next effect: angular dependence of the surface area of the branched structure has maximum value at certain «critical» angle of ions incidence.
Lyapunov functions for fractional order systems
Aguila-Camacho, Norelys; Duarte-Mermoud, Manuel A.; Gallegos, Javier A.
2014-09-01
A new lemma for the Caputo fractional derivatives, when 0<α<1, is proposed in this paper. This result has proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying.
Inertia theorems for operator Lyapunov inequalities
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, bu
Lyapunov Function Synthesis - Infeasibility and Farkas' Lemma
Leth, Tobias; Wisniewski, Rafal; Sloth, Christoffer
2017-01-01
In this paper we prove the convergence of an algorithm synthesising continuous piecewise-polynomial Lyapunov functions for polynomial vector elds dened on simplices. We subsequently modify the algorithm to sub-divide locally by utilizing information from infeasible linear problems. We prove...
Inertia theorems for operator Lyapunov inequalities
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,
Controllability of semilinear matrix Lyapunov systems
Bhaskar Dubey
2013-02-01
Full Text Available In this article, we establish some sufficient conditions for the complete controllability of semilinear matrix Lyapunov systems involving Lipschitzian and non-Lipschitzian nonlinearities. In case of non-Lipschitzian nonlinearities, we assume that nonlinearities are of monotone type.
On stability of discontinuous systems via vector Lyapunov functions
无
2007-01-01
This paper deals with the stability of systems with discontinuous righthand side (with solutions in Filippov's sense) via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of "set-valued derivative" of vector Lyapunov functions is introduced, some generalized comparison principles on dis(c)ontinuous systems are shown. Furthermore, Lyapunov stability theory is developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.
Hurst Exponent Analysis of Financial Time Series
无
2001-01-01
Statistical properties of stock market time series and the implication of their Hurst exponents are discussed. Hurst exponents of DJ1A (Dow Jones Industrial Average) components are tested using re-scaled range analysis. In addition to the original stock return series, the linear prediction errors of the daily returns are also tested. Numerical results show that the Hurst exponent analysis can provide some information about the statistical properties of the financial time series.
Lyapunov vectors and assimilation in the unstable subspace: theory and applications
Palatella, Luigi; Carrassi, Alberto; Trevisan, Anna
2013-06-01
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’.
Control of acrobot based on Lyapunov function
赖旭芝; 吴敏; 佘锦华
2004-01-01
Fuzzy control based on Lyapunov function was employed to control the posture and the energy of an acrobot to make the transition from upswing control to balance control smoothly and stably. First, a control law based on Lyapunov function was used to control the angle and the angular velocity of the second link towards zero when the energy of the acrobot reaches the potential energy at the unstable straight-up equilibrium position in the upswing process. The controller based on Lyapunov function makes the second link straighten nature relatively to the first link. At the same time, a fuzzy controller was designed to regulate the parameters of the upper control law to keep the change of the energy of the acrobot to a minimum, so that the switching from upswing to balance can be properly carried out and the acrobot can enter the balance quickly. The results of simulation show that the switching from upswing to balance can be completed smoothly, and the control effect of the acrobot is improved greatly.
Diophantine exponents for mildly restricted approximation
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...
Developing Students' Understanding of Exponents and Logarithms.
Weber, Keith
In this paper, we describe instruction designed to teach students about exponents and logarithms and report a pilot study to test the effectiveness of this instruction. Based on the theoretical work of Dubinsky and Sfard, we postulate a set of mental constructions that a student could make to understand the concepts of exponents and logarithms. We…
Bol loops of odd prime exponent
Foguel, Tuval
2009-01-01
Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.
Diophantine exponents for mildly restricted approximation
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...
Hydrodynamische Lyapunov-Moden in mehrkomponentigen Lennard-Jones-Flüssigkeiten
Drobniewski, Christian
2011-01-01
Die Charakterisierung hochdimensionaler Systeme mit Lyapunov-Instabilität wird durch das Lyapunov-Spektrum und die zugehörigen Lyapunov-Vektoren ermöglicht. Für eine Vielzahl von derartigen Systemen (Coupled-Map-Lattices, Hartkugel-Systeme, Systeme mit ausgedehnten Potentialen ...) konnte durch die Untersuchung der Lyapunov-Vektoren die Existenz von hydrodynamischen Lyapunov-Moden nachgewiesen werden. Diese kollektiven Anregungen zeigen sich in Lyapunov-Vektoren, deren Lyapunov-Exponenten de...
Beaudette, Shawn M; Howarth, Samuel J; Graham, Ryan B; Brown, Stephen H M
2016-10-01
Several different state-space reconstruction methods have been employed to assess the local dynamic stability (LDS) of a 3D kinematic system. One common method is to use a Euclidean norm (N) transformation of three orthogonal x, y, and z time-series' followed by the calculation of the maximum finite-time Lyapunov exponent (λmax) from the resultant N waveform (using a time-delayed state space reconstruction technique). By essentially acting as a weighted average, N has been suggested to account for simultaneous expansion and contraction along separate degrees of freedom within a 3D system (e.g. the coupling of dynamic movements between orthogonal planes). However, when estimating LDS using N, non-linear transformations inherent within the calculation of N should be accounted for. Results demonstrate that the use of N on 3D time-series data with arbitrary magnitudes of relative bias and zero-crossings cause the introduction of error in estimates of λmax obtained through N. To develop a standard for the analysis of 3D dynamic kinematic waveforms, we suggest that all dimensions of a 3D signal be independently shifted to avoid the incidence of zero-crossings prior to the calculation of N and subsequent estimation of LDS through the use of λmax.
Universal scalings of universal scaling exponents
Llave, Rafael de la [Department of Mathematics, University of Texas, Austin, TX 78712 (United States); Olvera, Arturo [IIMAS-UNAM, FENOMEC, Apdo. Postal 20-726, Mexico DF 01000 (Mexico); Petrov, Nikola P [Department of Mathematics, University of Oklahoma, Norman, OK 73019 (United States)
2007-06-08
In the last decades, renormalization group (RG) ideas have been applied to describe universal properties of different routes to chaos (quasi-periodic, period doubling or tripling, Siegel disc boundaries, etc). Each of the RG theories leads to universal scaling exponents which are related to the action of certain RG operators. The goal of this announcement is to show that there is a principle that organizes many of these scaling exponents. We give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations. These relations are determined by combinatorial properties of the route and become exact in an appropriate limit. (fast track communication)
Nonuniform exponential dichotomies and Lyapunov functions
Barreira, Luis; Dragičević, Davor; Valls, Claudia
2017-05-01
For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.
Lyapunov Functions and Solutions of the Lyapunov Matrix Equation for Marginally Stable Systems
Kliem, Wolfhard; Pommer, Christian
2000-01-01
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...
Dynamic exponents for potts model cluster algorithms
Coddington, Paul D.; Baillie, Clive F.
We have studied the Swendsen-Wang and Wolff cluster update algorithms for the Ising model in 2, 3 and 4 dimensions. The data indicate simple relations between the specific heat and the Wolff autocorrelations, and between the magnetization and the Swendsen-Wang autocorrelations. This implies that the dynamic critical exponents are related to the static exponents of the Ising model. We also investigate the possibility of similar relationships for the Q-state Potts model.
Universality of Tail Exponents of Price Changes?
Huang, Luwen; Farmer, Doyne
2007-03-01
We study the tail exponents of the distribution of logarithmic price changes in financial markets, and investigate the conjecture that they are universal with an exponent near three. Using data from the London Stock Exchange, we construct the empirical distributions of price returns on several different time scales and study their variation as a function of parameters such as trading volume and tick size (the minimal unit of price variation).
2009-08-01
setting in Duc & Siegmund [28]: Definition A.10 (Dynamic partition of IR2). Consider the extended phase space, IR2 × I, associated with the flow...Fluid Dynamics, Cambridge University Press, Cam- bridge, 1967. [8] A. Berger, D. T. Son, and S. Siegmund , Nonautonomous finite-time dynamics, Discrete...28] L. H. Duc and S. Siegmund , Hyperbolicity and invariant manifolds for planar nonau- tonomous systems on finite time intervals, Int. J. Bif. Chaos
Lyapunov指数与混沌同步的计算研究%Computation of Lyapunov exponents and chaos synchronization
谌龙; 王德石
2003-01-01
条件Lyapunov指数是混沌系统同步的重要指标.文中以已知方程的Lyapunov指数谱计算方法为基础,通过数值计算考察了Lyapunov指数随矢量长度、演化时间、置换次数的变化规律,为在现有各种算法中选择参数提供参考.同时,用其计算了混沌同步系统的条件Lyapunov指数,并研究了混沌同步系统的稳定性.
A calculation method of Lyapunov exponent and its realization%一种Lyapunov指数算法及其实现
冯明库; 丘水生; 晋建秀
2007-01-01
提出了一种利用周期轨道不同权重计算Lyapunov指数的算法.对混沌序列的周期轨道进行统计,并计算不同的周期轨道的Lyapunov指数,依据周期轨道的权重加权求和得到整个混沌吸引子的平均Lyapunov指数.深入讨论了初始值等对平均Lyapunov指数的影响.该算法不用舍去开始迭代点,适用于复杂混沌系统.
Research on Lyapunov Exponents Algorithm and its Application%Lyapunov指数计算研究及应用
廖德玮; 朱伟强
2008-01-01
Lyapunov指数是判定系统是否处于混沌状态的简捷方法之一,但计算Lyapunov指数的诸多方法普遍存在精度不高、受噪声影响大且计算量大等问题而使应用受到限制.借助计算机代数系统Maple建立基于Wolf算法的Lyapunov指数的机械化算法,可以方便地计算Lyapunov指数,从而可以迅速判定系统的混沌性.
Stability of time-delay systems via Lyapunov functions
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.
Construction of Lyapunov functions by the localization method
Krishchenko, A. P.; Kanatnikov, A. N.
2017-07-01
In this paper, we examine the problem of construction of Lyapunov functions for asymptotically stable equilibrium points. We exploit conditions of asymptotic stability in terms of compact invariant sets and positively invariant sets. Our results are methods of verification of these conditions and construction of Lyapunov functions by the localization method of compact invariant sets. These results are illustrated by an example.
Analysis of stability problems via matrix Lyapunov functions
Anatoly A. Martynyuk
1990-01-01
Full Text Available The stability of nonlinear systems is analyzed by the direct Lyapunov's method in terms of Lyapunov matrix functions. The given paper surveys the main theorems on stability, asymptotic stability and nonstability. They are applied to systems of nonlinear equations, singularly-perturbed systems and hybrid systems. The results are demonstrated by an example of a two-component system.
Preparing entangled states by Lyapunov control
Shi, Z. C.; Wang, L. C.; Yi, X. X.
2016-09-01
By Lyapunov control, we present a protocol to prepare entangled states such as Bell states in the context of cavity QED system. The advantage of our method is of threefold. Firstly, we can only control the phase of classical fields to complete the preparation process. Secondly, the evolution time is sharply shortened when compared to adiabatic control. Thirdly, the final state is steady after removing control fields. The influence of decoherence caused by the atomic spontaneous emission and the cavity decay is discussed. The numerical results show that the control scheme is immune to decoherence, especially for the atomic spontaneous emission from |2rangle to |1rangle . This can be understood as the state staying in an invariant subspace. Finally, we generalize this method in preparation of W state.
Preparation of topological modes by Lyapunov control.
Shi, Z C; Zhao, X L; Yi, X X
2015-09-08
By Lyapunov control, we present a proposal to drive quasi-particles into a topological mode in quantum systems described by a quadratic Hamiltonian. The merit of this control is the individual manipulations on the boundary sites. We take the Kitaev's chain as an illustration for Fermi systems and show that an arbitrary excitation mode can be steered into the Majorana zero mode by manipulating the chemical potential of the boundary sites. For Bose systems, taking the noninteracting Su-Schrieffer-Heeger (SSH) model as an example, we illustrate how to drive the system into the edge mode. The sensitivity of the fidelity to perturbations and uncertainties in the control fields and initial modes is also examined. The experimental feasibility of the proposal and the possibility to replace the continuous control field with square wave pulses is finally discussed.
Preparing entangled states by Lyapunov control
Shi, Z. C.; Wang, L. C.; Yi, X. X.
2016-12-01
By Lyapunov control, we present a protocol to prepare entangled states such as Bell states in the context of cavity QED system. The advantage of our method is of threefold. Firstly, we can only control the phase of classical fields to complete the preparation process. Secondly, the evolution time is sharply shortened when compared to adiabatic control. Thirdly, the final state is steady after removing control fields. The influence of decoherence caused by the atomic spontaneous emission and the cavity decay is discussed. The numerical results show that the control scheme is immune to decoherence, especially for the atomic spontaneous emission from |2rangle to |1rangle . This can be understood as the state staying in an invariant subspace. Finally, we generalize this method in preparation of W state.
Numerical solution of large Lyapunov equations
Saad, Youcef
1989-01-01
A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems.
Fuzzy Lyapunov Reinforcement Learning for Non Linear Systems.
Kumar, Abhishek; Sharma, Rajneesh
2017-03-01
We propose a fuzzy reinforcement learning (RL) based controller that generates a stable control action by lyapunov constraining fuzzy linguistic rules. In particular, we attempt at lyapunov constraining the consequent part of fuzzy rules in a fuzzy RL setup. Ours is a first attempt at designing a linguistic RL controller with lyapunov constrained fuzzy consequents to progressively learn a stable optimal policy. The proposed controller does not need system model or desired response and can effectively handle disturbances in continuous state-action space problems. Proposed controller has been employed on the benchmark Inverted Pendulum (IP) and Rotational/Translational Proof-Mass Actuator (RTAC) control problems (with and without disturbances). Simulation results and comparison against a) baseline fuzzy Q learning, b) Lyapunov theory based Actor-Critic, and c) Lyapunov theory based Markov game controller, elucidate stability and viability of the proposed control scheme.
Experimentally realizable control fields in quantum Lyapunov control
Yi, X X; Wu, Chunfeng; Feng, X L; Oh, C H
2011-01-01
As a hybrid of techniques from open-loop and feedback control, Lyapunov control has the advantage that it is free from the measurement-induced decoherence but it includes the system's instantaneous message in the control loop. Often, the Lyapunov control is 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 controlled system. These results suggest that the Lyapunov control is robust gainst time delay, easy to realize and effective for high-dimensional quantum systems.
Lyapunov functionals and stability of stochastic functional differential equations
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...
Determination of the Turbulent Decay Exponent
Perot, J.; Zusi, Chris
2011-11-01
All theories concerning the decay of isotropic turbulence agree that the turbulent kinetic energy has a power law dependence on time. However, there is significant disagreement about what the value of the exponent should be for this power law. The primary theories, proposed by researchers such as Batchelor, Townsend, and Kolmogorov, have the decay exponent at values of 1, 6/5, 10/7, 3/2, 2, and 5/2. The debate over the decay exponent has remained unresolved for many decades because the decay exponent is an extremely sensitive quantity. Experiments have decay times which are too short to be able to accurately differentiate between the various theoretical possibilities, and all prior numerical simulations of decaying turbulence impose the decay rate a priori via the choice of initial conditions. In this work, direct numerical simulation is used to achieve very long decay times, and the initial turbulence is generated by the Navier-Stokes equations and is not imposed. The initial turbulence is created by the stirring action of the flow past 768 small randomly placed cubes. Stirring occurs at 1/30th of the simulation domain size so that the low wavenumber and large scale behavior of the turbulent spectrum which dictates the decay rate is generated by the fluid and is not imposed. It is shown that in all 16 simulations the decay exponent closely matches the theoretical predictions of Saffman at both high and low Reynolds numbers. Perot, AIP Advances 1, 022104 (2011).
Wu, Jing; Wang, Yu; Zhang, Weiwei; Nie, Zhenhua; Lin, Rong; Ma, Hongwei
2017-01-01
This study proposes a novel small defect detection approach for steel pipes using the Lyapunov dimension (D) of the Duffing chaotic system based on ultrasonic guided waves. In this paper, inspection model is constructed by inputting the measured guided wave signal into the Duffing equation as the external turbulent driving force term and then Dis calculated. The properties of the Duffing system's noise immunity are first demonstrated theoretically based on the Lyapunov exponents. By comparing Dof the Duffing inspection system between the conditions of the inputted pure noise and the guided wave signal, the amplitude of the periodic force (F), the important parameter of the Duffing inspection system, could be determined. The values of other parameters of the Duffing inspection system are subsequently determined according to the numerical investigation. Furthermore, a time-moving window function is constructed to scan along the measured signal to locate the defect. And the small defect echo signal polluted by the noise is illustrated to prove the availability of the proposed method. Both numerical and experimental results show that the proposed approach can be used to improve the sensitivity of small defect detection and locate the small defect in pipes.
Interaction of Lyapunov vectors in the formulation of the nonlinear extension of the Kalman filter.
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.
The random phase property and the Lyapunov spectrum for disordered multi-channel systems
Roemer, Rudolf A
2009-01-01
A random phase property establishing a link between quasi-one-dimensional random Schroedinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a Markov process with a unique invariant measure which is of geometric nature. On the elliptic part of the transfer matrices, this measure is invariant under the full hermitian symplectic group of the universality class under study. While the random phase property can up to now only be proved in special models or in a restricted sense, we provide strong numerical evidence that it holds in the Anderson model of localization. A main outcome of the random phase property is a perturbative calculation of the Lyapunov exponents which shows that the Lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary, orthogonal and symplectic ensemble differ by a factor 2 each. In an Anderson-And...
Non-trivial exponents in coarsening phenomena
Derrida, B.
1997-02-01
One of the simplest examples of stochastic automata is the Glauber dynamics of ferromagnetic spin models such as Ising or Potts models. At zero temperature, if the initial condition is random, one observes a pattern of growing domains with a characteristic size which increases with time like t {1}/{2}. In this self-similar regime, the fraction of spins which never flip up to time t decreases like t- θ where the exponent θ is non-trivial and depends both on the number q of states of the Potts model and on the dimension of space. This exponent can be calculated exactly in one dimension. Similar non-trivial exponents are also present in even simpler models of coarsening, where the dynamical rule is deterministic.
Critical exponent of the fractional Langevin equation.
Burov, S; Barkai, E
2008-02-22
We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent alpha(c)=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase. The critical exponent alpha(R)=0.441... marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the underdamped, the overdamped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.
Are Bred Vectors The Same As Lyapunov Vectors?
Kalnay, E.; Corazza, M.; Cai, M.
in the local dimen- sion starts to occur at about 6 BVs, and is essentially complete when the number of vectors is about 10-15 (Corazza et al, 2001a). This should be contrasted with the re- sults of Snyder and Joly (1998) and Palmer et al (1998) who showed that hundreds of Lyapunov vectors with positive Lyapunov exponents are needed to represent the attractor of the system in quasi-geostrophic models. 4) Since only a few bred vectors are needed, and background errors project strongly in the subspace of bred vectors, Corazza et al (2001b) were able to develop cost-efficient methods to improve the 3D-Var data assimilation by adding to the background error covariance terms proportional to the outer product of the bred vectors, thus represent- ing the "errors of the day". This approach led to a reduction of analysis error variance of about 40% at very low cost. 5) The fact that BVs have finite amplitude provides a natural way to filter out instabil- ities present in the system that have fast growth, but saturate nonlinearly at such small amplitudes that they are irrelevant for ensemble perturbations. As shown by Lorenz (1996) Lyapunov vectors (and singular vectors) of models including these physical phenomena would be dominated by the fast but small amplitude instabilities, unless they are explicitly excluded from the linearized models. Bred vectors, on the other 2 hand, through the choice of an appropriate size for the perturbation, provide a natural filter based on nonlinear saturation of fast but irrelevant instabilities. 6) Every bred vector is qualitatively similar to the *leading* LV. LVs beyond the leading LV are obtained by orthogonalization after each time step with respect to the previous LVs subspace. The orthogonalization requires the introduction of a norm. With an enstrophy norm, the successive LVs have larger and larger horizontal scales, and a choice of a stream function norm would lead to successively smaller scales in the LVs. Beyond the first few LVs
Critical exponents from large mass expansion
Yamada, Hirofumi
2014-01-01
We perform estimation of critical exponents via large mass expansion under crucial help of delta-expansion. We address to the three dimensional Ising model at high temperature and estimate omega, the correction-to-scaling exponent, nu, eta and gamma in unbiased and self-contained manner. The results read at the highest 25th order expansion omega=0.8002, nu=0.6295, eta=0.0369 and gamma=1.2357. Estimation biased by omega=0.84(4) is also performed and proved to be in agreement with the summary of recent literatures.
Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection.
Xu, M; Paul, M R
2016-06-01
We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.
Large-deviation joint statistics of the finite-time Lyapunov spectrum in isotropic turbulence
Johnson, Perry L., E-mail: pjohns86@jhu.edu; Meneveau, Charles [Department of Mechanical Engineering and Center for Environmental and Applied Fluid Mechanics, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218 (United States)
2015-08-15
One of the hallmarks of turbulent flows is the chaotic behavior of fluid particle paths with exponentially growing separation among them while their distance does not exceed the viscous range. The maximal (positive) Lyapunov exponent represents the average strength of the exponential growth rate, while fluctuations in the rate of growth are characterized by the finite-time Lyapunov exponents (FTLEs). In the last decade or so, the notion of Lagrangian coherent structures (which are often computed using FTLEs) has gained attention as a tool for visualizing coherent trajectory patterns in a flow and distinguishing regions of the flow with different mixing properties. A quantitative statistical characterization of FTLEs can be accomplished using the statistical theory of large deviations, based on the so-called Cramér function. To obtain the Cramér function from data, we use both the method based on measuring moments and measuring histograms and introduce a finite-size correction to the histogram-based method. We generalize the existing univariate formalism to the joint distributions of the two FTLEs needed to fully specify the Lyapunov spectrum in 3D flows. The joint Cramér function of turbulence is measured from two direct numerical simulation datasets of isotropic turbulence. Results are compared with joint statistics of FTLEs computed using only the symmetric part of the velocity gradient tensor, as well as with joint statistics of instantaneous strain-rate eigenvalues. When using only the strain contribution of the velocity gradient, the maximal FTLE nearly doubles in magnitude, highlighting the role of rotation in de-correlating the fluid deformations along particle paths. We also extend the large-deviation theory to study the statistics of the ratio of FTLEs. The most likely ratio of the FTLEs λ{sub 1} : λ{sub 2} : λ{sub 3} is shown to be about 4:1:−5, compared to about 8:3:−11 when using only the strain-rate tensor for calculating fluid volume
Huang, Jr-Chuan; Lee, Tsung-Yu; Teng, Tse-Yang; Chen, Yi-Chin; Huang, Cho-Ying; Lee, Cheing-Tung
2014-01-01
The exponent decay in landslide frequency-area distribution is widely used for assessing the consequences of landslides and with some studies arguing that the slope of the exponent decay is universal and independent of mechanisms and environmental settings. However, the documented exponent slopes are diverse and hence data processing is hypothesized for this inconsistency. An elaborated statistical experiment and two actual landslide inventories were used here to demonstrate the influences of the data processing on the determination of the exponent. Seven categories with different landslide numbers were generated from the predefined inverse-gamma distribution and then analyzed by three data processing procedures (logarithmic binning, LB, normalized logarithmic binning, NLB and cumulative distribution function, CDF). Five different bin widths were also considered while applying LB and NLB. Following that, the maximum likelihood estimation was used to estimate the exponent slopes. The results showed that the exponents estimated by CDF were unbiased while LB and NLB performed poorly. Two binning-based methods led to considerable biases that increased with the increase of landslide number and bin width. The standard deviations of the estimated exponents were dependent not just on the landslide number but also on binning method and bin width. Both extremely few and plentiful landslide numbers reduced the confidence of the estimated exponents, which could be attributed to limited landslide numbers and considerable operational bias, respectively. The diverse documented exponents in literature should therefore be adjusted accordingly. Our study strongly suggests that the considerable bias due to data processing and the data quality should be constrained in order to advance the understanding of landslide processes.
Jr-Chuan Huang
Full Text Available The exponent decay in landslide frequency-area distribution is widely used for assessing the consequences of landslides and with some studies arguing that the slope of the exponent decay is universal and independent of mechanisms and environmental settings. However, the documented exponent slopes are diverse and hence data processing is hypothesized for this inconsistency. An elaborated statistical experiment and two actual landslide inventories were used here to demonstrate the influences of the data processing on the determination of the exponent. Seven categories with different landslide numbers were generated from the predefined inverse-gamma distribution and then analyzed by three data processing procedures (logarithmic binning, LB, normalized logarithmic binning, NLB and cumulative distribution function, CDF. Five different bin widths were also considered while applying LB and NLB. Following that, the maximum likelihood estimation was used to estimate the exponent slopes. The results showed that the exponents estimated by CDF were unbiased while LB and NLB performed poorly. Two binning-based methods led to considerable biases that increased with the increase of landslide number and bin width. The standard deviations of the estimated exponents were dependent not just on the landslide number but also on binning method and bin width. Both extremely few and plentiful landslide numbers reduced the confidence of the estimated exponents, which could be attributed to limited landslide numbers and considerable operational bias, respectively. The diverse documented exponents in literature should therefore be adjusted accordingly. Our study strongly suggests that the considerable bias due to data processing and the data quality should be constrained in order to advance the understanding of landslide processes.
Bergman kernel and complex singularity exponent
LEE; HanJin
2009-01-01
We give a precise estimate of the Bergman kernel for the model domain defined by Ω F={(z,w) ∈ C n+1:Im w |F (z)| 2 > 0},where F=(f 1,...,f m) is a holomorphic map from C n to C m,in terms of the complex singularity exponent of F.
Bergman kernel and complex singularity exponent
CHEN BoYong; LEE HanJin
2009-01-01
We give a precise estimate of the Bergman kernel for the model domain defined by Ω_F = {(z,w) ∈ C~(n+1) : Imw - |F(z)|~2 > 0},where F = (f_1,... ,f_m) is a holomorphic map from C~n to C~m,in terms of the complex singularity exponent of F.
Diophantine exponents for mildly restricted approximation
Bugeaud, Yann; Kristensen, S.
2009-01-01
We are studying the Diophantine exponent μ n,l defined for integers 1≤l
Circular orbits, Lyapunov stability and Manev-type forces
Blaga, Cristina
2016-01-01
In this article we study the stability in the sense of Lyapunov of the circular orbits in the generalized Manev two bodies problem. First, we explore the existence of the circular orbits and determine their radius. Then, using the first integrals of motion we build a positive definite function, known as a Lyapunov function. It's existence proves that the circular orbit is stable in the sense of Lyapunov. In the end, we consider several real systems of two bodies and compare the characteristics of the circular orbits in Newtonian and modified Manev gravitational field, arguing about our possibilities to observe the differences between the motion in these two fields.
Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems
Liu, Jiang; Zhao, Hengjun
2011-01-01
The notion of Lyapunov function plays a key role in design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives of certain functions along the system's vector field. Furthermore, we present a complete method to automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is complete in the sense that it is able to discover all polynomial RLFs by enumerating all polynomial templates for any PDS.
Stabilization of nonlinear systems based on robust control Lyapunov function
CAI Xiu-shan; HAN Zheng-zhi; LU Gan-yun
2007-01-01
This paper deals with the robust stabilization problem for a class of nonlinear systems with structural uncertainty. Based on robust control Lyapunov function, a sufficient and necessary condition for a function to be a robust control Lyapunov function is given. From this condition, simply sufficient condition for the robust stabilization (robust practical stabilization) is deduced. Moreover, if the equilibrium of the closed-loop system is unique, the existence of such a robust control Lyapunov function will also imply robustly globally asymptotical stabilization. Then a continuous state feedback law can be constructed explicitly. The simulation shows the effectiveness of the method.
Controller design for TS models using delayed nonquadratic Lyapunov functions.
Lendek, Zsofia; Guerra, Thierry-Marie; Lauber, Jimmy
2015-03-01
In the last few years, nonquadratic Lyapunov functions have been more and more frequently used in the analysis and controller design for Takagi-Sugeno fuzzy models. In this paper, we developed relaxed conditions for controller design using nonquadratic Lyapunov functions and delayed controllers and give a general framework for the use of such Lyapunov functions. The two controller design methods developed in this framework outperform and generalize current state-of-the-art methods. The proposed methods are extended to robust and H∞ control and α -sample variation.
A Lyapunov approach to strong stability of semigroups
Paunonen, L.T.; Zwart, Heiko J.
2013-01-01
In this paper we present Lyapunov based proofs for the well-known Arendt–Batty–Lyubich–Vu Theorem for strongly continuous and discrete semigroups. We also study the spectral properties of the limit isometric groups used in the proofs.
Lyapunov functionals and stability of stochastic difference equations
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.
Stability, Resonance and Lyapunov Inequalities for Periodic Conservative Systems
Canada, Antonio
2010-01-01
This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.
Lyapunov Computational Method for Two-Dimensional Boussinesq Equation
Mabrouk, Anouar Ben
2010-01-01
A numerical method is developed leading to Lyapunov operators to approximate the solution of two-dimensional Boussinesq equation. It consists of an order reduction method and a finite difference discretization. It is proved to be uniquely solvable and analyzed for local truncation error for consistency. The stability is checked by using Lyapunov criterion and the convergence is studied. Some numerical implementations are provided at the end of the paper to validate the theoretical results.
A Spectral Lyapunov Function for Exponentially Stable LTV Systems
Zhu, J. Jim; Liu, Yong; Hang, Rui
2010-01-01
This paper presents the formulation of a Lyapunov function for an exponentially stable linear timevarying (LTV) system using a well-defined PD-spectrum and the associated PD-eigenvectors. It provides a bridge between the first and second methods of Lyapunov for stability assessment, and will find significant applications in the analysis and control law design for LTV systems and linearizable nonlinear time-varying systems.
Stabilization of discrete nonlinear systems based on control Lyapunov functions
无
2008-01-01
The stabilization of discrete nonlinear systems is studied.Based on control Lyapunov functions,asufficient and necessary condition for a quadratic function to be a control Lyapunov function is given.From this condition,a continuous state feedback law is constructed explicitly.It can globally asymptotically stabilize the equilibrium of the closed-loop system.A simulation example shows the effectiveness of the proposed method.
Lyapunov control of quantum systems with impulsive control fields.
Yang, Wei; Sun, Jitao
2013-01-01
We investigate the Lyapunov control of finite-dimensional quantum systems with impulsive control fields, where the studied quantum systems are governed by the Schrödinger equation. By three different Lyapunov functions and the invariant principle of impulsive systems, we study the convergence of quantum systems with impulsive control fields and propose new results for the mentioned quantum systems in the form of sufficient conditions. Two numerical simulations are presented to illustrate the effectiveness of the proposed control method.
Fuwape, Ibiyinka A.; Ogunjo, Samuel T.
2016-12-01
Radio refractivity index is used to quantify the effect of atmospheric parameters in communication systems. Scaling and dynamical complexities of radio refractivity across different climatic zones of Nigeria have been studied. Scaling property of the radio refractivity across Nigeria was estimated from the Hurst Exponent obtained using two different scaling methods namely: The Rescaled Range (R/S) and the detrended fluctuation analysis(DFA). The delay vector variance (DVV), Largest Lyapunov Exponent (λ1) and Correlation Dimension (D2) methods were used to investigate nonlinearity and the results confirm the presence of deterministic nonlinear profile in the radio refractivity time series. The recurrence quantification analysis (RQA) was used to quantify the degree of chaoticity in the radio refractivity across the different climatic zones. RQA was found to be a good measure for identifying unique fingerprint and signature of chaotic time series data. Microwave radio refractivity was found to be persistent and chaotic in all the study locations. The dynamics of radio refractivity increases in complexity and chaoticity from the Coastal region towards the Sahelian climate. The design, development and deployment of robust and reliable microwave communication link in the region will be greatly affected by the chaotic nature of radio refractivity in the region.
Universality of modulation length and time exponents.
Chakrabarty, Saurish; Dobrosavljević, Vladimir; Seidel, Alexander; Nussinov, Zohar
2012-10-01
We study systems with a crossover parameter λ, such as the temperature T, which has a threshold value λ(*) across which the correlation function changes from exhibiting fixed wavelength (or time period) modulations to continuously varying modulation lengths (or times). We introduce a hitherto unknown exponent ν(L) characterizing the universal nature of this crossover and compute its value in general instances. This exponent, similar to standard correlation length exponents, is obtained from motion of the poles of the momentum (or frequency) space correlation functions in the complex k-plane (or ω-plane) as the parameter λ is varied. Near the crossover (i.e., for λ→λ(*)), the characteristic modulation wave vector K(R) in the variable modulation length "phase" is related to that in the fixed modulation length "phase" q via |K(R)-q|[proportionality]|T-T(*)|(νL). We find, in general, that ν(L)=1/2. In some special instances, ν(L) may attain other rational values. We extend this result to general problems in which the eigenvalue of an operator or a pole characterizing general response functions may attain a constant real (or imaginary) part beyond a particular threshold value λ(*). We discuss extensions of this result to multiple other arenas. These include the axial next-nearest-neighbor Ising (ANNNI) model. By extending our considerations, we comment on relations pertaining not only to the modulation lengths (or times), but also to the standard correlation lengths (or times). We introduce the notion of a Josephson time scale. We comment on the presence of aperiodic "chaotic" modulations in "soft-spin" and other systems. These relate to glass-type features. We discuss applications to Fermi systems, with particular application to metal to band insulator transitions, change of Fermi surface topology, divergent effective masses, Dirac systems, and topological insulators. Both regular periodic and glassy (and spatially chaotic behavior) may be found in
Critical exponents of the classical Heisenberg ferromagnet
Holm, C; Holm, Christian; Janke, Wolfhard
1997-01-01
In a recent letter, R.G. Brown and M. Ciftan (Phys. Rev. Lett. 76, 1352, 1996) reported high precision Monte Carlo (MC) estimates of the static critical exponents of the classical 3D Heisenberg model, which stand in sharp contrast to values obtained by four independent approaches, namely by other recent high statistics MC simulations, high-temperature series analyses, field theoretical methods, and experimental studies. In reply to the above cited work we submitted this paper as a comment to Phys. Rev. Lett.
On Bruck Loops of 2-power Exponent
Baumeister, Barbara; Stroth, Gernot
2009-01-01
We classify "nice" loop envelopes to Bruck loops of 2-power exponent under the assumption that every nonabelian simple section of $G$ is either passive or isomorphic to $\\PSL_2(q)$, $q-1 \\ge 4$ a 2-power. The hypothesis is verified in a separate paper. This paper is a continuation of the work by Aschbacher, Kinyon and Phillips on finite Bruck loops [AKP]. In [BS3] we applied these results and get a neat description of the structure of the finite Bruck loops.
Schubert, Sebastian; Lucarini, Valerio
2016-04-01
One of the most relevant weather regimes in the mid latitudes atmosphere is the persistent deviation from the approximately zonally symmetric jet stream to the emergence of so-called blocking patterns. Such configurations are usually connected to exceptional local stability properties of the flow which come along with an improved local forecast skills during the phenomenon. It is instead extremely hard to predict onset and decay of blockings. Covariant Lyapunov Vectors (CLVs) offer a suitable characterization of the linear stability of a chaotic flow, since they represent the full tangent linear dynamics by a covariant basis which explores linear perturbations at all time scales. Therefore, we will test whether CLVs feature a signature of the blockings. We examine the CLVs for a quasi-geostrophic beta-plane two-layer model in a periodic channel baroclinically driven by a meridional temperature gradient ΔT. An orographic forcing enhances the emergence of localized blocked regimes. We detect the blocking events of the channel flow with a Tibaldi-Molteni scheme adapted to the periodic channel. When blocking occurs, the global growth rates of the fastest growing CLVs are significantly higher. Hence against intuition, globally the circulation is more unstable in blocked phases. Such an increase in the finite time Lyapunov exponents with respect to the long term average is attributed to stronger barotropic and baroclinic conversion in the case of high temperature gradients, while for low values of ΔT, the effect is only due to stronger barotropic instability. For the localization of the CLVs, we compare the meridionally averaged variance of the CLVs during blocked and unblocked phases. We find that on average the variance of the CLVs is clustered around the center of blocking. These results show that the blocked flow affects all time scales and processes described by the CLVs.
On monochromatic arm exponents for 2D critical percolation
Beffara, Vincent
2009-01-01
We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well-understood) polychromatic exponents.
Lipschitz Properties in Variable Exponent Problems via Relative Rearrangement
Jean-Michel RAKOTOSON
2010-01-01
The author first studies the Lipschitz properties of the monotone and relative rearrangement mappings in variable exponent Lebesgue spaces completing the result given in[9].This paper is ended by establishing the Lipschitz properties for quasilinear problems with variable exponent when the right-hand side is in some dual spaces of a suitable Sobolev space associated to variable exponent.
THE SECOND EXPONENT SET OF PRIMITIVE DIGRAPHS
无
2000-01-01
Let D = (V,E) be a primitive digraph. The exponent of D, denoted by γ(D), is the least integer k such that for any u, v ∈ V there is a directed walk of length k from u to v. The local exponent of D at a vertex u ∈ V, denoted by expD (u), is the least integer k such that there is a directed walk of length k from u to v for each v ∈ V. Let V = {1,2,... ,n}. Following [1], the vertices of V are ordered so that exPD (1) ≤expD (2)≤…≤expD (n) ＝γ(D). Let En(i) :＝ {expD (i) | D ∈ PDn}, where PDn is the set of all primitive digraphs of order n. It is known that En(n) = {γ(D) | D ∈ PDn} has been completely settled by [7]. In 1998, En(1)was characterized by [5]. In this paper, the authors describe En(2) for all n≥2.
Loops with exponent three in all isotopes
Kinyon, Michael
2011-01-01
It was shown by van Rees \\cite{vR} that a latin square of order $n$ cannot have more than $n^2(n-1)/18$ latin subsquares of order 3. He conjectured that this bound is only achieved if $n$ is a power of 3. We show that it can only be achieved if $n\\equiv3\\bmod6$. We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van Rees bound if and only if every loop isotope has exponent 3. We call such loops \\emph{van Rees loops} and show that they form an equationally defined variety. We also show that (1) In a van Rees loop, any subloop of index 3 is normal, (2) There are exactly 6 nonassociative van Rees loops of order 27 with a non-trivial nucleus, (3) There is a Steiner quasigroup associated with every van Rees loop and (4) Every Bol loop of exponent 3 is a van Rees loop.
Some comments on scaling exponents of turbulence
Baudet, C.; Ciliberto, S.; Nhan Tien, Phan
1993-03-01
Several authors have reported that in turbulence the scaling exponent of the first order velocity structure function increases when the Reynolds number Re decreases. From this result some important consequences on the transition to turbulence could be obtained. However we report experiemntal evidence that this result is coming only from an improper definition of the inertial range. Our data clearly show that the scaling exponents remain constant as a function of Re which is consistent with the selfsimilarity of spectra. Quelques auteurs ont prétendu observer dans les écoulements turbulents un accroissement de l'exposant de la loi de puissance pour la fonction de structure du premier ordre de la vitesse lorsque le nombre de Reynolds décroît. D'importantes déductions relatives à la transition vers la turbulence pourraient être tirées de ce résultat. Cependant, il ressort des résultats expérimentaux que nous avons obtenus récemment que ce résultat ne provient que d'une définition incorrecte du domaine inertiel. Nos données expérimentales prouvent clairement que les exposants des fonctions de structrue de la vitesse ne varient pas avec le nombre de Reynolds, ce qui est cohérent avec le caractère auto-similaire des fluctuations de vitesse en turbulence.
High-resolution satellite image segmentation using Hölder exponents
Debasish Chakraborty; Gautam Kumar Sen; Sugata Hazra
2009-10-01
Texture in high-resolution satellite images requires substantial amendment in the conventional segmentation algorithms. A measure is proposed to compute the Hölder exponent (HE) to assess the roughness or smoothness around each pixel of the image. The localized singularity information is incorporated in computing the HE. An optimum window size is evaluated so that HE reacts to localized singularity. A two-step iterative procedure for clustering the transformed HE image is adapted to identify the range of HE, densely occupied in the kernel and to partition Hölder exponents into a cluster that matches with the range. Hölder exponent values (noise or not associated with the other cluster) are clubbed to a nearest possible cluster using the local maximum likelihood analysis.
On Controllability and Observability of Fuzzy Dynamical Matrix Lyapunov Systems
M. S. N. Murty
2008-04-01
Full Text Available We provide a way to combine matrix Lyapunov systems with fuzzy rules to form a new fuzzy system called fuzzy dynamical matrix Lyapunov system, which can be regarded as a new approach to intelligent control. First, we study the controllability property of the fuzzy dynamical matrix Lyapunov system and provide a sufficient condition for its controllability with the use of fuzzy rule base. The significance of our result is that given a deterministic system and a fuzzy state with rule base, we can determine the rule base for the control. Further, we discuss the concept of observability and give a sufficient condition for the system to be observable. The advantage of our result is that we can determine the rule base for the initial value without solving the system.
Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues
Canada, Antonio
2011-01-01
This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \\ 1 \\leq p \\leq +\\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in $\\real^{N}$. It is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.
Nonlinear Direct Robust Adaptive Control Using Lyapunov Method
Chunbo Xiu
2013-07-01
Full Text Available The problem of robust adaptive stabilization of a class of multi-input nonlinear systems with arbitrary unknown parameters and unknown structure of bounded variation have been considered. By employing the direct adaptive and control Lyapunov function method, a robust adaptive controller is designed to complete the globally adaptive stability of the system states. By employing our result, a kind of nonlinear system is analyzed, the concrete form of the control law is given and the meaningful quadratic control Lyapunov function for the system is constructed. Simulation of parallel manipulator is provided to illustrate the effectiveness of the proposed method.
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
An iterative decoupling solution method for large scale Lyapunov equations
Athay, T. M.; Sandell, N. R., Jr.
1976-01-01
A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
An iterative decoupling solution method for large scale Lyapunov equations
Athay, T. M.; Sandell, N. R., Jr.
1976-01-01
A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.
Magnetic entropy change and critical exponents in double perovskite Y2NiMnO6
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.
Kinkhabwala, Ali
2013-01-01
The most fundamental problem in statistics is the inference of an unknown probability distribution from a finite number of samples. For a specific observed data set, answers to the following questions would be desirable: (1) Estimation: Which candidate distribution provides the best fit to the observed data?, (2) Goodness-of-fit: How concordant is this distribution with the observed data?, and (3) Uncertainty: How concordant are other candidate distributions with the observed data? A simple unified approach for univariate data that addresses these traditionally distinct statistical notions is presented called "maximum fidelity". Maximum fidelity is a strict frequentist approach that is fundamentally based on model concordance with the observed data. The fidelity statistic is a general information measure based on the coordinate-independent cumulative distribution and critical yet previously neglected symmetry considerations. An approximation for the null distribution of the fidelity allows its direct conversi...
Critical exponents for diluted resistor networks.
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.
Ising exponents from the functional renormalisation group
Litim, Daniel F
2010-01-01
We study the 3d Ising universality class using the functional renormalisation group. With the help of background fields and a derivative expansion up to fourth order we compute the leading index, the subleading symmetric and anti-symmetric corrections to scaling, the anomalous dimension, the scaling solution, and the eigenperturbations at criticality. We also study the cross-correlations of scaling exponents, and their dependence on dimensionality. We find a very good numerical convergence of the derivative expansion, also in comparison with earlier findings. Evaluating the data from all functional renormalisation group studies to date, we estimate the systematic error which is found to be small and in good agreement with findings from Monte Carlo simulations, \\epsilon-expansion techniques, and resummed perturbation theory.
A conjecture on the norm of Lyapunov mapping
Daizhan CHENG; Yahong ZHU; Hongsheng QI
2009-01-01
A conjecture that the norm of Lyapunov mapping LA equals to its restriction to the symmetric set,S,i.e.,‖LA‖ = ‖LA |s‖ was proposed in [1].In this paper,a method for numerical testing is provided first.Then,some recent progress on this conjecture is presented.
Construction of Lyapunov Function for Dissipative Gyroscopic System
XU Wei; YUAN Bo; AO Ping
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.%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.
On the Computation of Lyapunov Functions for Interconnected Systems
Sloth, Christoffer
2016-01-01
This paper addresses the computation of additively separable Lyapunov functions for interconnected systems. The presented results can be applied to reduce the complexity of the computations associated with stability analysis of large scale systems. We provide a necessary and sufficient condition...
Quadratic Lyapunov Function and Exponential Dichotomy on Time Scales
ZHANG JI; LIU ZHEN-XIN
2011-01-01
In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△ ＝ A(t)x on time scales.Moreover, for the nonlinear perturbed equation x△ ＝ A(t)x + f(t,x) we give the instability of the zero solution when f is sufficiently small.
Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?
Beims, Marcus W; Gallas, Jason A C
2016-11-15
We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.
Control Lyapunov Stabilization of Nonlinear Systems with Structural Uncertainty
CAI Xiu-shan; HAN Zheng-zhi; TANG Hou-jun
2005-01-01
This paper deals with global stabilization problem for the nonlinear systems with structural uncertainty.Based on control Lyapunov function, a sufficient and necessary condition for the globally and asymptotically stabilizing the equailibrium of the closed system is given. Moreovery, an almost smooth state feedback control law is constructed. The simulation shows the effectiveness of the method.
Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?
Beims, Marcus W.; Gallas, Jason A. C.
2016-11-01
We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.
STABILIZATION OF NONLINEAR TIME-VARYING SYSTEMS: A CONTROL LYAPUNOV FUNCTION APPROACH
Zhongping JIANG; Yuandan LIN; Yuan WANG
2009-01-01
This paper presents a control Lyapunov function approach to the global stabilization problem for general nonlinear and time-varying systems. Explicit stabilizing feedback control laws are proposed based on the method of control Lyapunov functions and Sontag's universal formula.
Lyapunov matrices approach to the parametric optimization of time-delay systems
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 Matrices Approach to the Parametric Optimization of a System with Two Delays
Duda Jozef
2016-09-01
Full Text Available In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with two commensurate delays and a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of the Lyapunov functional for the initial function of time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix.
The Hurst exponent in energy futures prices
Serletis, Apostolos; Rosenberg, Aryeh Adam
2007-07-01
This paper extends the work in Elder and Serletis [Long memory in energy futures prices, Rev. Financial Econ., forthcoming, 2007] and Serletis et al. [Detrended fluctuation analysis of the US stock market, Int. J. Bifurcation Chaos, forthcoming, 2007] by re-examining the empirical evidence for random walk type behavior in energy futures prices. In doing so, it uses daily data on energy futures traded on the New York Mercantile Exchange, over the period from July 2, 1990 to November 1, 2006, and a statistical physics approach-the ‘detrending moving average’ technique-providing a reliable framework for testing the information efficiency in financial markets as shown by Alessio et al. [Second-order moving average and scaling of stochastic time series, Eur. Phys. J. B 27 (2002) 197-200] and Carbone et al. [Time-dependent hurst exponent in financial time series. Physica A 344 (2004) 267-271; Analysis of clusters formed by the moving average of a long-range correlated time series. Phys. Rev. E 69 (2004) 026105]. The results show that energy futures returns display long memory and that the particular form of long memory is anti-persistence.
Hurst exponents for short time series
Qi, Jingchao; Yang, Huijie
2011-12-01
A concept called balanced estimator of diffusion entropy is proposed to detect quantitatively scalings in short time series. The effectiveness is verified by detecting successfully scaling properties for a large number of artificial fractional Brownian motions. Calculations show that this method can give reliable scalings for short time series with length ˜102. It is also used to detect scalings in the Shanghai Stock Index, five stock catalogs, and a total of 134 stocks collected from the Shanghai Stock Exchange Market. The scaling exponent for each catalog is significantly larger compared with that for the stocks included in the catalog. Selecting a window with size 650, the evolution of scaling for the Shanghai Stock Index is obtained by the window's sliding along the series. Global patterns in the evolutionary process are captured from the smoothed evolutionary curve. By comparing the patterns with the important event list in the history of the considered stock market, the evolution of scaling is matched with the stock index series. We can find that the important events fit very well with global transitions of the scaling behaviors.
Complementarity Properties of the Lyapunov Transformation over Symmetric Cones
Yuan Min LI; Xing Tao WANG; De Yun WEI
2012-01-01
The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA* is positive definite.In this paper,we extend this theorem to the setting of any Euclidean Jordan algebra V.Given any element a ∈ V,we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the Ro-property for La and show that La has the R0-property if and only if a is invertible.Finally,we provide La with some characterizatious of the E0-property and the nondegeneracy property.
Quantum synchronization in an optomechanical system based on Lyapunov control.
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.
Modified periodogram method for estimating the Hurst exponent of fractional Gaussian noise.
Liu, Yingjun; Liu, Yong; Wang, Kun; Jiang, Tianzi; Yang, Lihua
2009-12-01
Fractional Gaussian noise (fGn) is an important and widely used self-similar process, which is mainly parametrized by its Hurst exponent (H) . Many researchers have proposed methods for estimating the Hurst exponent of fGn. In this paper we put forward a modified periodogram method for estimating the Hurst exponent based on a refined approximation of the spectral density function. Generalizing the spectral exponent from a linear function to a piecewise polynomial, we obtained a closer approximation of the fGn's spectral density function. This procedure is significant because it reduced the bias in the estimation of H . Furthermore, the averaging technique that we used markedly reduced the variance of estimates. We also considered the asymptotical unbiasedness of the method and derived the upper bound of its variance and confidence interval. Monte Carlo simulations showed that the proposed estimator was superior to a wavelet maximum likelihood estimator in terms of mean-squared error and was comparable to Whittle's estimator. In addition, a real data set of Nile river minima was employed to evaluate the efficiency of our proposed method. These tests confirmed that our proposed method was computationally simpler and faster than Whittle's estimator.
PHYSIOLOGICAL RESPONSES DURING MATCHES AND PROFILE OF ELITE PENCAK SILAT EXPONENTS
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
Prasath, V B Surya; Vorotnikov, Dmitry; Pelapur, Rengarajan; Jose, Shani; Seetharaman, Guna; Palaniappan, Kannappan
2015-12-01
Edge preserving regularization using partial differential equation (PDE)-based methods although extensively studied and widely used for image restoration, still have limitations in adapting to local structures. We propose a spatially adaptive multiscale variable exponent-based anisotropic variational PDE method that overcomes current shortcomings, such as over smoothing and staircasing artifacts, while still retaining and enhancing edge structures across scale. Our innovative model automatically balances between Tikhonov and total variation (TV) regularization effects using scene content information by incorporating a spatially varying edge coherence exponent map constructed using the eigenvalues of the filtered structure tensor. The multiscale exponent model we develop leads to a novel restoration method that preserves edges better and provides selective denoising without generating artifacts for both additive and multiplicative noise models. Mathematical analysis of our proposed method in variable exponent space establishes the existence of a minimizer and its properties. The discretization method we use satisfies the maximum-minimum principle which guarantees that artificial edge regions are not created. Extensive experimental results using synthetic, and natural images indicate that the proposed multiscale Tikhonov-TV (MTTV) and dynamical MTTV methods perform better than many contemporary denoising algorithms in terms of several metrics, including signal-to-noise ratio improvement and structure preservation. Promising extensions to handle multiplicative noise models and multichannel imagery are also discussed.
Modified periodogram method for estimating the Hurst exponent of fractional Gaussian noise
Liu, Yingjun; Liu, Yong; Wang, Kun; Jiang, Tianzi; Yang, Lihua
2009-12-01
Fractional Gaussian noise (fGn) is an important and widely used self-similar process, which is mainly parametrized by its Hurst exponent (H) . Many researchers have proposed methods for estimating the Hurst exponent of fGn. In this paper we put forward a modified periodogram method for estimating the Hurst exponent based on a refined approximation of the spectral density function. Generalizing the spectral exponent from a linear function to a piecewise polynomial, we obtained a closer approximation of the fGn’s spectral density function. This procedure is significant because it reduced the bias in the estimation of H . Furthermore, the averaging technique that we used markedly reduced the variance of estimates. We also considered the asymptotical unbiasedness of the method and derived the upper bound of its variance and confidence interval. Monte Carlo simulations showed that the proposed estimator was superior to a wavelet maximum likelihood estimator in terms of mean-squared error and was comparable to Whittle’s estimator. In addition, a real data set of Nile river minima was employed to evaluate the efficiency of our proposed method. These tests confirmed that our proposed method was computationally simpler and faster than Whittle’s estimator.
Kolmogorov complexity, Lovasz local lemma and critical exponents
Rumyantsev, Andrey
2010-01-01
D. Krieger and J. Shallit have proved that every real number greater than 1 is a critical exponent of some sequence. We show how this result can be derived from some general statements about sequences whose subsequences have (almost) maximal Kolmogorov complexity. In this way one can also construct a sequence that has no "approximate" fractional powers with exponent that exceeds a given value.
The Exponent Set of Central Symmetric Primitive Matrices
陈佘喜; 胡亚辉
2004-01-01
This paper first establishes a distance inequality of the associated diagraph of a central symmetric primitive matrix, then characters the exponent set of central symmetric primitive matrices, and proves that the exponent set of central symmetric primitive matrices of order n is {1, 2,… ,n-1}. There is no gap in it.
Scaling Exponents for Lattice Quantum Gravity in Four Dimensions
Hamber, Herbert W
2015-01-01
In this work nonperturbative aspects of quantum gravity are investigated using the lattice formulation, and some new results are presented for critical exponents, amplitudes and invariant correlation functions. Values for the universal scaling dimensions are compared with other nonperturbative approaches to gravity in four dimensions, and specifically to the conjectured value for the universal critical exponent $\
Lyapunov functions for a dengue disease transmission model
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.
Analisis Kestabilan Model Matematika Penyakit Leukimia dengan Fungsi Lyapunov
2015-01-01
This study aims to analyze the stability of the equilibrium point of the mathematical model of leukemia before and after undergoing chemotherapy. Analysis of the stability of the model is done by analyzing the model by using a Lyapunov function. By using MATLAB program will be described stability of the model before chemotherapy and after chemotherapy. The results showed that the equilibrium point of stem cell compartment model is asymptotically stable for certain parameter values. This is be...
Analysis of Lyapunov Method for Control of Quantum States
Wang, Xiaoting; Schirmer, Sonia
2009-01-01
The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalle's invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the...
Using Lyapunov function to design optimal controller for AQM routers
ZHANG Peng; YE Cheng-qing; MA Xue-ying; CHEN Yan-hua; LI Xin
2007-01-01
It was shown that active queue management schemes implemented in the routers of communication networks supporting transmission control protocol (TCP) flows can be modelled as a feedback control system. In this paper based on Lyapunov function we developed an optimal controller to improve active queue management (AQM) router's stability and response time,which are often in conflict with each other in system performance. Ns-2 simulations showed that optimal controller outperforms PI controller significantly.
Lyapunov Criteria for Structural Stability of Supply Chain System
LU Ying-jin; TANG Xiao-wo; ZHOU Zong-fang
2004-01-01
In this paper, based on Cobb-Douglas production function, the structural stability of the supply chain system are analyzed by employing Lyapunov criteria. That the supply chain system structure,with the variance of the rate of re-production input funding, becomes unstable is proved. Noticeably, the solutions shows that when the optimal combination of input parameter element, the qualitative properties of supply chain system change and the supply chain system becomes unstable.
ON THE UPPER GENERALIZED EXPONENTS OF MINISTRONG DIGRAPHS
周波
2001-01-01
@@ 1. Introduction A digraph G is called primitive if there exists a positive integer k such that there is a walk of length k from u to v for each ordered pair of not necessarily distinct vertices u and v. The smallest such k is called the exponent of G, denoted by γ(G). Exponents for primitive digraphs have been studied extensively due to their intrinsic importance in graph theory, combinatorics, matrix theory, and their applications in communication problems. As a generalization of exponents, Brualdi and Liu[1] introduced the concept of upper generalized exponents for primitive digraphs. Recently, Shao, Wu and Hwang[2'3] extended this concept of upper generalized exponents from primitive digraphs to general digraphs which are not necessarily primitive.
Fitting Power-laws in empirical data with estimators that work for all exponents
Hanel, Rudolf; Liu, Bo; Thurner, Stefan
2016-01-01
It has been repeatedly stated that maximum likelihood (ML) estimates of exponents of power-law distributions can only be reliably obtained for exponents smaller than minus one. The main argument that power laws are otherwise not normalizable, depends on the underlying sample space the data is drawn from, and is true only for sample spaces that are unbounded from above. Here we show that power-laws obtained from bounded sample spaces (as is the case for practically all data related problems) are always free of such limitations and maximum likelihood estimates can be obtained for arbitrary powers without restrictions. Here we first derive the appropriate ML estimator for arbitrary exponents of power-law distributions on bounded discrete sample spaces. We then show that an almost identical estimator also works perfectly for continuous data. We implemented this ML estimator and discuss its performance with previous attempts. We present a general recipe of how to use these estimators and present the associated com...
Universality of persistence exponents in two-dimensional Ostwald ripening.
Soriano, Jordi; Braslavsky, Ido; Xu, Di; Krichevsky, Oleg; Stavans, Joel
2009-11-27
We measured persistence exponents theta(phi) of Ostwald ripening in two dimensions, as a function of the area fraction phi occupied by coarsening domains. The values of theta(phi) in two systems, succinonitrile and brine, quenched to their liquid-solid coexistence region, compare well with one another, providing compelling evidence for the universality of the one-parameter family of exponents. For small phi, theta(phi) approximately = 0.39phi, as predicted by a model that assumes no correlations between evolving domains. These constitute the first measurements of persistence exponents in the case of phase transitions with a conserved order parameter.
Intrinsic modulation of ENSO predictability viewed through a local Lyapunov lens
Karamperidou, Christina; Cane, Mark A.; Lall, Upmanu; Wittenberg, Andrew T.
2014-01-01
The presence of rich ENSO variability in the long unforced simulation of GFDL's CM2.1 motivates the use of tools from dynamical systems theory to study variability in ENSO predictability, and its connections to ENSO magnitude, frequency, and physical evolution. Local Lyapunov exponents (LLEs) estimated from the monthly NINO3 SSTa model output are used to characterize periods of increased or decreased predictability. The LLEs describe the growth of infinitesimal perturbations due to internal variability, and are a measure of the immediate predictive uncertainty at any given point in the system phase-space. The LLE-derived predictability estimates are compared with those obtained from the error growth in a set of re-forecast experiments with CM2.1. It is shown that the LLEs underestimate the error growth for short forecast lead times (less than 8 months), while they overestimate it for longer lead times. The departure of LLE-derived error growth rates from the re-forecast rates is a linear function of forecast lead time, and is also sensitive to the length of the time series used for the LLE calculation. The LLE-derived error growth rate is closer to that estimated from the re-forecasts for a lead time of 4 months. In the 2,000-year long simulation, the LLE-derived predictability at the 4-month lead time varies (multi)decadally only by 9-18 %. Active ENSO periods are more predictable than inactive ones, while epochs with regular periodicity and moderate magnitude are classified as the most predictable by the LLEs. Events with a deeper thermocline in the west Pacific up to five years prior to their peak, along with an earlier deepening of the thermocline in the east Pacific in the months preceding the peak, are classified as more predictable. Also, the GCM is found to be less predictable than nature under this measure of predictability.
Asymptotic expansions of Feynman integrals of exponentials with polynomial exponent
Kravtseva, A. K.; Smolyanov, O. G.; Shavgulidze, E. T.
2016-10-01
In the paper, an asymptotic expansion of path integrals of functionals having exponential form with polynomials in the exponent is constructed. The definition of the path integral in the sense of analytic continuation is considered.
Critical exponents of O(N) models in fractional dimensions
Codello, A; D'Odorico, G
2014-01-01
We compute critical exponents of O(N) models in fractal dimensions between two and four, and for continuos values of the number of field components N, in this way completing the RG classification of universality classes for these models. In d=2 the N-dependence of the correlation length critical exponent gives us the last piece of information needed to establish a RG derivation of the Mermin-Wagner theorem. We also report critical exponents for multi-critical universality classes in the cases N>1 and N=0. Finally, in the large-N limit our critical exponents correctly approach those of the spherical model, allowing us to set N~100 as threshold for the quantitative validity of leading order large-N estimates.
A MONTE-CARLO METHOD FOR ESTIMATING THE CORRELATION EXPONENT
MIKOSCH, T; WANG, QA
1995-01-01
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.
A MONTE-CARLO METHOD FOR ESTIMATING THE CORRELATION EXPONENT
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.
ACCURATE ESTIMATES OF CHARACTERISTIC EXPONENTS FOR SECOND ORDER DIFFERENTIAL EQUATION
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.
Lyapunov Orbits in the Jupiter System Using Electrodynamic Tethers
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.
Continuation of probability density functions using a generalized Lyapunov approach
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.
Abstraction of Continuous Dynamical Systems Utilizing Lyapunov Functions
Sloth, Christoffer; Wisniewski, Rafal
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....
Suppressing chaos via Lyapunov-Krasovskii's method
Kuang, J.L. [Faculty of Science and Engineering, City University of Hong Kong, Hong Kong (China)] e-mail: kuangjinlu@hotmail.com; Meehan, P.A. [Department of Mechanical Engineering, University of Queensland, Brisbane, Qld 4072 (Australia)] e-mail: meehan@uq.edu.au; Leung, A.Y.T. [Faculty of Science and Engineering, City University of Hong Kong, Hong Kong (China)] e-mail: bcaleung@cityu.edu.hk
2006-03-01
An algorithm for suppressing the chaotic oscillations in non-linear dynamical systems with singular Jacobian matrices is developed using a linear feedback control law based upon the Lyapunov-Krasovskii (LK) method. It appears that the LK method can serve effectively as a generalised method for the suppression of chaotic oscillations for a wide range of systems. Based on this method, the resulting conditions for undisturbed motions to be locally or globally stable are sufficient and conservative. The generalized Lorenz system and disturbed gyrostat equations are exemplified for the validation of the proposed feedback control rule.
Lyapunov Orbits in the Jupiter System Using Electrodynamic Tethers
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.
Design of Connectivity Preserving Flocking Using Control Lyapunov Function
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.
Avrami exponent under transient and heterogeneous nucleation transformation conditions
Sinha, I; Mandal, R. K.
2010-01-01
The Kolmogorov-Johnson-Mehl-Avrami model for isothermal transformation kinetics is universal under specific assumptions. However, the experimental Avrami exponent deviates from the universal value. In this context, we study the effect of transient heterogeneous nucleation on the Avrami exponent for bulk materials and also for transformations leading to nanostructured materials. All transformations are assumed to be polymorphic. A discrete version of the KJMA model is modified for this purpose...
Control design and comprehensive stability analysis of acrobots based on Lyapunov functions
LAI Xu-zhi; WU Yun-xin; SHE Jin-hua; WU Min
2005-01-01
A design method for controllers and a comprehensive stability analysis for an acrobat based on Lyapunov functions are presented. Three control laws based on three Lyapunov functions are designed to increase the energy so as to move the acrobot into the unstable inverted equilibrium position, and solve the problem of posture and energy. The concept of a non-smooth Lyapunov function is employed to analyze the stability of the whole system. The validity of this strategy is demonstrated by simulations.
Construction of a Smooth Lyapunov Function for the Robust and Exact Second-Order Differentiator
2016-01-01
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...
Lyapunov functions for a class of nonlinear systems using Caputo derivative
Fernandez-Anaya, G.; Nava-Antonio, G.; Jamous-Galante, J.; Muñoz-Vega, R.; Hernández-Martínez, E. G.
2017-02-01
This paper presents an extension of recent results that allow proving the stability of Caputo nonlinear and time-varying systems, by means of the fractional order Lyapunov direct method, using quadratic Lyapunov functions. This article introduces a new way of building polynomial Lyapunov functions of any positive integer order as a way of determining the stability of a greater variety of systems when the order of the derivative is 0 < α < 1. Some examples are given to validate these results.
2013-06-01
STABILITY BY COMPUTING A SINGLE QUADRATIC LYAPUNOV FUNCTION Mehrdad Pakmehr∗ PhD Candidate Decision and Control Laboratory (DCL) School of Aerospace...linearization and linear matrix inequality (LMI) techniques. Using convex optimization tools, a single quadratic Lyapunov function is computed for multiple...Scheduling Control of Gas Turbine Engines: Stability by Computing a Single Quadratic Lyapunov Function 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c
Covariant hydrodynamic Lyapunov modes and strong stochasticity threshold in Hamiltonian lattices.
Romero-Bastida, M; Pazó, Diego; López, Juan M
2012-02-01
We scrutinize the reliability of covariant and Gram-Schmidt Lyapunov vectors for capturing hydrodynamic Lyapunov modes (HLMs) in one-dimensional Hamiltonian lattices. We show that, in contrast with previous claims, HLMs do exist for any energy density, so that strong chaos is not essential for the appearance of genuine (covariant) HLMs. In contrast, Gram-Schmidt Lyapunov vectors lead to misleading results concerning the existence of HLMs in the case of weak chaos.
Time-Delay Systems Lyapunov Functionals and Matrices
Kharitonov, Vladimir L
2013-01-01
Stability is one of the most studied issues in the theory of time-delay systems, but the corresponding chapters of published volumes on time-delay systems do not include a comprehensive study of a counterpart of classical Lyapunov theory for linear delay free systems. The principal goal of the book is to fill this gap, and to provide readers with a systematic and exhaustive treatment of the basic concepts of the Lyapunov-Krasovskii approach to the stability analysis of linear time-delay systems. The book is organized into two parts. The first part is dedicated to the case of retarded type time-delay systems; it consists of four chapters, which respectively deal with results concerning the existence and uniqueness of the solutions of an initial value problem, the class of linear systems with one delay, the case of systems with several delays, and the case of systems with distributed delays. The second part of the book studies the case of neutral type time-delay systems, containing three chapters that e...
丁丹平; 田立新
2000-01-01
自1990年,美国马里兰大学的Ott,Grebogi和Yorke三人首先从理论上提出控制混沌的方法,即OGY方法,混沌控制已成了非线性理论及应用中重要的组成部分.但混沌控制(OGY)方法在数学理论上还有许多工作需要完善,从数学理论上对OGY方法进一步论证和探讨,对混沌控制理论的建立和体系化有很重要的意义.而笔者利用Lyapunov指数讨论了混沌控制(OGY方法)有效的充分条件,获得了具体的表示式.并将此方法用于讨论具体的控制参数的选择及控制参数所须满足的条件.最后对二维Henon映射的轨道稳定化控制的有效性给出了解释.
Lyapunov特性指数谱的研究及其应用%The Study of Lyapunov Characteristic Exponents Spectrum and Its Applications
虞文锦; 蔡艳岭; 高英台
2002-01-01
本文对于已有的运动方程的Lyapunov特性指数谱计算公式,从易于数值实现的角度作了简化,并通过Henon映射和Lorena方程对Lyapunov特性指数与倍周期分岔的关系作了分析和比较,最后通过对受特定参数下系统周期轨道控制时的混沌Lorenz系统的Lyapunov特性指数谱的计算和分析,进一步验证了Lyapunov特性指数与倍周期分岔的内在关系.
Research of Judging the Chaotic Characteristics with the Lyapunov Exponents%Lyapunov指数混沌特性判定研究
郁俊莉; 王其文
2004-01-01
探讨了Lyapunov指数混沌特性判据原理,分析了时间序列Lyapunov指数的计算过程,通过计算我国上证综合指数收益率时间序列的Lyapunov指数谱系,分析了我国资本市场的混沌特性.
Empirical relations between static and dynamic exponents for Ising model cluster algorithms
Coddington, Paul D.; Baillie, Clive F.
1992-02-01
We have measured the autocorrelations for the Swendsen-Wang and the Wolff cluster update algorithms for the Ising model in two, three, and four dimensions. The data for the Wolff algorithm suggest that the autocorrelations are linearly related to the specific heat, in which case the dynamic critical exponent is zint,EW=α/ν. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that zint,ESW=β/ν for the Ising model.
Empirical relations between static and dynamic exponents for Ising model cluster algorithms
Coddington, P.D. (Department of Physics, Syracuse University, Syracuse, New York 13244 (United States)); Baillie, C.F. (Department of Physics, University of Colorado, Boulder, Colorado 80309 (United States))
1992-02-17
We have measured the autocorrelations for the Swendsen-Wang and the Wolff cluster update algorithms for the Ising model in two, three, and four dimensions. The data for the Wolff algorithm suggest that the autocorrelations are linearly related to the specific heat, in which case the dynamic critical exponent is {ital z}{sub int,}{ital E}{sup W}={alpha}/{nu}. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that {ital z}{sub int,}{ital E}{sup SW}={beta}/{nu} for the Ising model.
A new theoretical interpretation of Archie's saturation exponent
Glover, Paul W. J.
2017-07-01
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.
Groups of order p^8 and exponent p
Michael Vaughan-Lee
2015-12-01
Full Text Available We prove that for p>7 there are p^4 +2p^3 +20p^2 +147p+(3p+29gcd(p−1,3+5gcd(p−1,4+1246 groups of order p^8 with exponent p. If P is a group of order p^8 and exponent p, and if P has class c>1 then P is a descendant of P/γ c (P. For each group of exponent p with order less than p^8 we calculate the number of descendants of order p^8 with exponent p. In all but one case we are able to obtain a complete and irredundant list of the descendants. But in the case of the three generator class two group of order p^6 and exponent p (p>3 , while we are able to calculate the number of descendants of order p^8, we have not been able to obtain a list of the descendants.
Universal construction of control Lyapunov functions for a class of nonlinear systems
无
2007-01-01
A method is developed by which control Lyapunov functions of a class of nonlinear systems can be constructed systematically.Based on the control Lyapunov function,a feedback control is obtained to stabilize the closed-loop system.In addition,this method is applied to stabilize the Benchmark system.A simulation shows the effectiveness of the method.
Average Transient Lifetime and Lyapunov Dimension for Transient Chaos in a High-Dimensional System
陈洪; 汤建新; 唐少炎; 向红; 陈新
2001-01-01
The average transient lifetime of a chaotic transient versus the Lyapunov dimension of a chaotic saddle is studied for high-dimensional nonlinear dynamical systems. Typically the average lifetime depends upon not only the system parameter but also the Lyapunov dimension of the chaotic saddle. The numerical example uses the delayed feedback differential equation.
Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility
Korobeinikov, Andrei
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.
Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility.
Melnik, Andrey V; Korobeinikov, Andrei
2013-04-01
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.
Testing Universality in Critical Exponents: the Case of Rainfall
Deluca, Anna; Corral, Alvaro
2015-01-01
One of the key clues to consider rainfall as a self-organized critical phenomenon is the existence of power-law distributions for rain-event sizes. We have studied the problem of universality in the exponents of these distributions by means of a suitable statistic whose distribution is inferred by several variations of a permutational test. In contrast to more common approaches, our procedure does not suffer from the difficulties of multiple testing and does not require the precise knowledge of the uncertainties associated to the power-law exponents. When applied to seven sites monitored by the Atmospheric Radiation Measurement Program the test lead to the rejection of the universality hypothesis, despite the fact that the exponents are rather close to each other.
On the average exponent of elliptic curves modulo $p$
Freiberg, Tristan
2012-01-01
Given an elliptic curve $E$ defined over $\\mathbb{Q}$ and a prime $p$ of good reduction, let $\\tilde{E}(\\mathbb{F}_p)$ denote the group of $\\mathbb{F}_p$-points of the reduction of $E$ modulo $p$, and let $e_p$ denote the exponent of said group. Assuming a certain form of the Generalized Riemann Hypothesis (GRH), we study the average of $e_p$ as $p \\le X$ ranges over primes of good reduction, and find that the average exponent essentially equals $p\\cdot c_{E}$, where the constant $c_{E} > 0$ depends on $E$. For $E$ without complex multiplication (CM), $c_{E}$ can be written as a rational number (depending on $E$) times a universal constant. Without assuming GRH, we can determine the average exponent when $E$ has CM, as well as give an upper bound on the average in the non-CM case.
Time-varying Hurst exponent for US stock markets
Alvarez-Ramirez, Jose; Alvarez, Jesus; Rodriguez, Eduardo; Fernandez-Anaya, Guillermo
2008-10-01
In this work, the dynamical behavior of the US stock markets is characterized on the basis of the temporal variations of the Hurst exponent estimated with detrended fluctuation analysis (DFA) over moving windows for the historical Dow Jones (1928-2007) and the S&P-500 (1950-2007) daily indices. According to the results drawn: (i) the Hurst exponent displays an erratic dynamics with some episodes alternating low and high persistent behavior, (ii) the major breakthrough of the long-term trend of the scaling behavior occurred in 1972, at the end of the Bretton Woods system, when the Hurst exponent shifted form a positive to a negative long-term trend. Other effects, such as the 1987 crisis and the emergence of anti-correlated behavior in the recent two years, are also discussed.
Adaptive Fuzzy-Lyapunov Controller Using Biologically Inspired Swarm Intelligence
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.
A Lyapunov theory based UPFC controller for power flow control
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)
Time-delay effects and simplified control fields in quantum Lyapunov control
Yi, X X; Wu, S L [School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024 (China); Wu, Chunfeng; Feng, X L; Oh, C H, E-mail: yixx@dlut.edu.cn, E-mail: phyohch@nus.edu.sg [Centre for Quantum Technologies and Department of Physics, National University of Singapore, 117543 (Singapore)
2011-10-14
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.
Construction of a Smooth Lyapunov Function for the Robust and Exact Second-Order Differentiator
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.
Parameter-dependent Lyapunov functional for systems with multiple time delays
Min WU; Yong HE
2004-01-01
The separation of the Lyapunov matrices and system matrices plays an important role when one uses parameter-dependent Lyapunov functional handling systems with polytopic type uncertainties.The delay-dependent robust stability problem for systems with polytopic type uncertainties is discussed by using parameter-dependent Lyapunov functional.The derivative term in the derivative of Lyapunov functional is reserved and the free weighting matrices are employed to express the relationship between the terms in the system equation such that the Lyapunov matrices are not involved in any product terms with the system matrices.In addition,the relationships between the terms in the Leibniz Newton formula are also described by some free weighting matrices and some delay-dependent stability conditions are derived.Numerical examples demonstrate that the proposed criteria are more effective than the previous results.
Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks
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......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 potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have...
Vannitsem, Stephane
2015-01-01
We study a simplified coupled atmosphere-ocean model using the formalism of covariant Lyapunov vectors (CLVs), which link physically-based directions of perturbations to growth/decay rates. The model is obtained via a severe truncation of quasi-geostrophic equations for the two fluids, and includes a simple yet physically meaningful representation of their dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and the parameters are chosen so that a chaotic behaviour is observed. One finds two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen near-zero LEs. The presence of many near-zero LEs results from the vast time-scale separation between the characteristic time scales of the two fluids, and leads to nontrivial error growth properties in the tangent space spanned by the corresponding CLVs, which are geometrically very degenerate. Such CLVs correspond to two different classes of ocean/atmosphere coupled modes. The tangent space spanned by the CLVs corresponding to the ...
Gaps and the exponent of convergence of an integer sequence
Grekos, Georges; Sleziak, Martin
2012-01-01
Professor Tibor \\v{S}al\\'at, at one of his seminars at Comenius University, Bratislava, asked to study the influence of gaps of an integer sequence A={a_1
Lattice Based Attack on Common Private Exponent RSA
Santosh Kumar Ravva
2012-03-01
Full Text Available Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Lattice reduction has been successfully utilizing in Number Theory, Linear algebra and Cryptology. Not only the existence of lattice based cryptosystems of hard in nature, but also has vulnerabilities by lattice reduction techniques. In this paper, we show that Wieners small private exponent attack, when viewed as a heuristic lattice based attack, is extended to attack many instances of RSA when they have the same small private exponent.
Critical exponents of a three dimensional O(4) spin model
Kanaya, K; Kanaya, K; Kaya, S
1995-01-01
By Monte Carlo simulation we study the critical exponents governing the transition of the three-dimensional classical O(4) Heisenberg model, which is considered to be in the same universality class as the finite-temperature QCD with massless two flavors. We use the single cluster algorithm and the histogram reweighting technique to obtain observables at the critical temperature. After estimating an accurate value of the inverse critical temperature \\Kc=0.9360(1) we make non-perturbative estimates for various critical exponents by finite-size scaling analysis. They are in excellent agreement with those obtained with the 4-\\epsilon expansion method with errors reduced to about halves of them.
Critical Exponents for Fast Diffusion Equations with Nonlinear Boundary Sources
WANG LU-SHENG; WANG ZE-JIA
2011-01-01
In this paper, we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball. We are interested in the critical global exponent q0 and the critical Fujita exponent qc for the problem considered, and show that q0 ＝ qc for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources, which is quite different from the known results that q0 ＜ qc for the onedimensional case; moreover, the value is different from the slow case.
Second-order relative exponent of isotropic turbulence
无
2011-01-01
Theoretical results on the scaling properties of turbulent velocity fields are reported in this letter.Based on the Kolmogorov equation and typical models of the second-order statistical moments (energy spectrum and the second-order structure function),we have studied the relative scaling using the ESS method.It is found that the relative EES scaling exponent S_2 is greater than the real or theoretical inertial range scaling exponentξ_2,which is attributed to an evident bump in the ESS range.
Construction of Control Lyapunov Functions for a Class of Nonlinear Systems%一类非线性系统控制Lyapunov函数的构造
蔡秀珊; 韩正之; 汪晓东
2006-01-01
The construction of control Lyapunov functions for a class of nonlinear systems is considered. We develop a method by which a control Lyapunov function for the feedback linearizable part can be constructed systematically via Lyapunov equation. Moreover, by a control Lyapunov function of the feedback linearizable part and a Lyapunov function of the zero dynamics, a control Lyapunov function for the overall nonlinear system is established.
Analytical Computation of Critical Exponents in Several Holographic Superconductors
Zeng, Hua-Bi; Jiang, Yu; Zong, Hong-Shi
2010-01-01
It is very interesting that all holographic superconductors like $s$-wave, $p$-wave and $d$-wave holographic superconductors shows the universal mean-field critical exponent $1/2$ at the critical temperature just like Gindzburg-Landau (G-L) theory for second order phase transitions. Now it is believed that the universal critical exponents appear since the dual gravity theory is classic in the large $N$ limit. However, there is an exception called "non-mean-field theory" even in the large $N$ limit: An extension of the $s$-wave model with a cubic term of the charged scalar field provides a different critical exponent $1$. In this paper, we try to use analytical calculation to get the critical exponents for these models to see how these properties of the gravity action decides the appearance of the mean-field or "non-mean-field" behaviors. It will be seen that like the G-L theory, it is the fundamental symmetries rather than the detail parameters of the bulk theory result in the universal properties of the holo...
QUASILINEAR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT AND HARDY TERM
无
2008-01-01
This paper is concerned with a p-Laplacian elliptic problem with critical Sobolev-Hardy exponent and Hardy term. By variational methods and genus theory, we guarantee that this problem has at least one positive solution and admits many solutions with negative energy under sufficient conditions.
Inverted rank distributions: Macroscopic statistics, universality classes, and critical exponents
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.
Finite Element Error Estimates for Critical Exponent Semilinear Problems without Angle Conditions
Bank, Randolph E; Szypowski, Ryan; Zhu, Yunrong
2011-01-01
In this article we consider a priori error estimates for semilinear problems with critical and subcritical polynomial nonlinearity in d space dimensions. When d=2 and d=3, it is well-understood how mesh geometry impacts finite element interpolant quality. However, much more restrictive conditions on angles are needed to derive basic a priori quasi-optimal error estimates as well as a priori pointwise estimates for Galerkin approximations. In this article, we show how to derive these types of a priori estimates without requiring the discrete maximum principle, hence eliminating the need for restrictive angle conditions that are difficult to satisfy in three dimensions or adaptive settings. We first describe a class of semilinear problems with critical exponents. The solution theory for this class of problems is then reviewed, including generalized maximum principles and the construction of a priori L-infinity bounds using cutoff functions and the De Giorgi iterative method (or Stampacchia truncation method). W...
Stability of Cellular Automata Trajectories Revisited: Branching Walks and Lyapunov Profiles
Baetens, Jan M.; Gravner, Janko
2016-10-01
We study non-equilibrium defect accumulation dynamics on a cellular automaton trajectory: a branching walk process in which a defect creates a successor on any neighborhood site whose update it affects. On an infinite lattice, defects accumulate at different exponential rates in different directions, giving rise to the Lyapunov profile. This profile quantifies instability of a cellular automaton evolution and is connected to the theory of large deviations. We rigorously and empirically study Lyapunov profiles generated from random initial states. We also introduce explicit and computationally feasible variational methods to compute the Lyapunov profiles for periodic configurations, thus developing an analog of Floquet theory for cellular automata.
Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks.
Anderson, David F; Craciun, Gheorghe; Gopalkrishnan, Manoj; Wiuf, Carsten
2015-09-01
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 potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.
Global stabilization of nonlinear systems based on vector control lyapunov functions
Karafyllis, Iasson
2012-01-01
This paper studies the use of vector Lyapunov functions for the design of globally stabilizing feedback laws for nonlinear systems. Recent results on vector Lyapunov functions are utilized. The main result of the paper shows that the existence of a vector control Lyapunov function is a necessary and sufficient condition for the existence of a smooth globally stabilizing feedback. Applications to nonlinear systems are provided: simple and easily checkable sufficient conditions are proposed to guarantee the existence of a smooth globally stabilizing feedback law. The obtained results are applied to the problem of the stabilization of an equilibrium point of a reaction network taking place in a continuous stirred tank reactor.
Kajiwara, Tsuyoshi; Sasaki, Toru; Takeuchi, Yasuhiro
2015-02-01
We present a constructive method for Lyapunov functions for ordinary differential equation models of infectious diseases in vivo. We consider models derived from the Nowak-Bangham models. We construct Lyapunov functions for complex models using those of simpler models. Especially, we construct Lyapunov functions for models with an immune variable from those for models without an immune variable, a Lyapunov functions of a model with absorption effect from that for a model without absorption effect. We make the construction clear for Lyapunov functions proposed previously, and present new results with our method.
Flexible Lyapunov Functions and Applications to Fast Mechatronic Systems
Lazar, M
2010-01-01
The property that every control system should posses is stability, which translates into safety in real-life applications. A central tool in systems theory for synthesizing control laws that achieve stability are control Lyapunov functions (CLFs). Classically, a CLF enforces that the resulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, and which is centered at and converges to a desired converging point. However, such a requirement often proves to be overconservative, which is why most of the real-time controllers do not have a stability guarantee. Recently, a novel idea that improves the design of CLFs in terms of flexibility was proposed. The focus of this new approach is on the design of optimization problems that allow certain parameters that define a cone associated with a standard CLF to be decision variables. In this way non-monotonicity of the CLF is explicitly linked with a decision variable that can be optimized on-line. Conservativeness is significantly ...
Flexible Lyapunov Functions and Applications to Fast Mechatronic Systems
Mircea Lazar
2010-03-01
Full Text Available The property that every control system should posses is stability, which translates into safety in real-life applications. A central tool in systems theory for synthesizing control laws that achieve stability are control Lyapunov functions (CLFs. Classically, a CLF enforces that the resulting closed-loop state trajectory is contained within a cone with a fixed, predefined shape, and which is centered at and converges to a desired converging point. However, such a requirement often proves to be overconservative, which is why most of the real-time controllers do not have a stability guarantee. Recently, a novel idea that improves the design of CLFs in terms of flexibility was proposed. The focus of this new approach is on the design of optimization problems that allow certain parameters that define a cone associated with a standard CLF to be decision variables. In this way non-monotonicity of the CLF is explicitly linked with a decision variable that can be optimized on-line. Conservativeness is significantly reduced compared to classical CLFs, which makes flexible CLFs more suitable for stabilization of constrained discrete-time nonlinear systems and real-time control. The purpose of this overview is to highlight the potential of flexible CLFs for real-time control of fast mechatronic systems, with sampling periods below one millisecond, which are widely employed in aerospace and automotive applications.
Estimation of Hurst Exponent for the Financial Time Series
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.
Determination of the decay exponent in mechanically stirred isotropic turbulence
J. Blair Perot
2011-06-01
Full Text Available Direct numerical simulation is used to investigate the decay exponent of isotropic homogeneous turbulence over a range of Reynolds numbers sufficient to display both high and low Re number decay behavior. The initial turbulence is generated by the stirring action of the flow past many small randomly placed cubes. Stirring occurs at 1/30th of the simulation domain size so that the low-wavenumber and large scale behavior of the turbulent spectrum is generated by the fluid and is not imposed. It is shown that the decay exponent in the resulting turbulence matches the theoretical predictions for a k2 low-wavenumber spectrum at both high and low Reynolds numbers. The transition from high Reynolds number behavior to low Reynolds number behavior occurs relatively abruptly at a turbulent Reynolds number of around 250 ( Re λ≈41.
Determination of critical exponents of inhomogeneous Gd films
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.
Error Exponents of Optimum Decoding for the Interference Channel
Etkin, Raul; Ordentlich, Erik
2008-01-01
Exponential error bounds for the finite-alphabet interference channel (IFC) with two transmitter-receiver pairs, are investigated under the random coding regime. Our focus is on optimum decoding, as opposed to heuristic decoding rules that have been used in previous works, like joint typicality decoding, decoding based on interference cancellation, and decoding that considers the interference as additional noise. Indeed, the fact that the actual interfering signal is a codeword and not an i.i.d. noise process complicates the application of conventional techniques to the performance analysis of the optimum decoder. Using analytical tools rooted in statistical physics, we derive a single letter expression for error exponents achievable under optimum decoding and demonstrate strict improvement over error exponents obtainable using suboptimal decoding rules, but which are amenable to more conventional analysis.
Critical dynamics and global persistence exponent on Taiwan financial market
Chen, I C; Li, P C; Chen, H J; Tseng, Hsen-Che; Li, Ping-Cheng; Chen, Hung-Jung
2006-01-01
We investigated the critical dynamics on the daily Taiwan stock exchange index (TSE) from 1971 to 2005, and the 5-min intraday data from 1996 to 2005. A global persistence exponent $\\theta_{p}$ was defined for non-equilibrium critical phenomena \\cite{Janssen,Majumdar}, and describing dynamic behavior in an economic index \\cite{Zheng}. In recent numerical analysis studies of literatures, it is illustrated that the persistence probability has a universal scaling form $P(t) \\sim t^{-\\theta_{p}}$ \\cite{Zheng1}. In this work, we analyzed persistence properties of universal scaling behavior on Taiwan financial market, and also calculated the global persistence exponent $\\theta_{p}$. We found our analytical results in good agreement with the same universality.
Nonlinear analysis of anesthesia dynamics by Fractal Scaling Exponent.
Gifani, P; Rabiee, H R; Hashemi, M R; Taslimi, P; Ghanbari, M
2006-01-01
The depth of anesthesia estimation has been one of the most research interests in the field of EEG signal processing in recent decades. In this paper we present a new methodology to quantify the depth of anesthesia by quantifying the dynamic fluctuation of the EEG signal. Extraction of useful information about the nonlinear dynamic of the brain during anesthesia has been proposed with the optimum Fractal Scaling Exponent. This optimum solution is based on the best box sizes in the Detrended Fluctuation Analysis (DFA) algorithm which have meaningful changes at different depth of anesthesia. The Fractal Scaling Exponent (FSE) Index as a new criterion has been proposed. The experimental results confirm that our new Index can clearly discriminate between aware to moderate and deep anesthesia levels. Moreover, it significantly reduces the computational complexity and results in a faster reaction to the transients in patients' consciousness levels in relations with the other algorithms.
Scaling exponents for fracture surfaces in opal glass
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.
无
2010-01-01
The asymptotic Lyapunov stability of one quasi-integrable Hamiltonian system with time-delayed feedback control is studied by using Lyapunov functions and stochastic averaging method.First,a quasi-integrable Hamiltonian system with time-delayed feedback control subjected to Gaussian white noise excitation is approximated by a quasi-integrable Hamiltonian system without time delay.Then,stochastic averaging method for quasi-integrable Hamiltonian system is used to reduce the dimension of the original system,and after that the Lyapunov function of the averaged It? equation is taken as the optimal linear combination of the corresponding independent first integrals in involution.Finally,the stability of the system is determined by using the largest eigenvalue of the linearized system.Two examples are used to illustrate the proposed procedure and the effects of delayed time on the Lyapunov stability are discussed as well.
Lyapunov-type inequalities for quasilinear elliptic equations with Robin boundary condition.
Dinlemez Kantar, Ülkü; Özden, Tülay
2017-01-01
The aim of this study is to prove Lyapunov-type inequalities for a quasilinear elliptic equation in [Formula: see text]. Also the lower bound for the first positive eigenvalue of the boundary value problem is obtained.
Stabilization of the Ball on the Beam System by Means of the Inverse Lyapunov Approach
Carlos Aguilar-Ibañez
2012-01-01
Full Text Available A novel inverse Lyapunov approach in conjunction with the energy shaping technique is applied to derive a stabilizing controller for the ball on the beam system. The proposed strategy consists of shaping a candidate Lyapunov function as if it were an inverse stability problem. To this purpose, we fix a suitable dissipation function of the unknown energy function, with the property that the selected dissipation divides the corresponding time derivative of the candidate Lyapunov function. Afterwards, the stabilizing controller is directly obtained from the already shaped Lyapunov function. The stability analysis of the closed-loop system is carried out by using the invariance theorem of LaSalle. Simulation results to test the effectiveness of the obtained controller are presented.
MIMO Lyapunov Theory-Based RBF Neural Classifier for Traffic Sign Recognition
King Hann Lim
2012-01-01
Full Text Available Lyapunov theory-based radial basis function neural network (RBFNN is developed for traffic sign recognition in this paper to perform multiple inputs multiple outputs (MIMO classification. Multidimensional input is inserted into RBF nodes and these nodes are linked with multiple weights. An iterative weight adaptation scheme is hence designed with regards to the Lyapunov stability theory to obtain a set of optimum weights. In the design, the Lyapunov function has to be well selected to construct an energy space with a single global minimum. Weight gain is formed later to obey the Lyapunov stability theory. Detail analysis and discussion on the proposed classifier’s properties are included in the paper. The performance comparisons between the proposed classifier and some existing conventional techniques are evaluated using traffic sign patterns. Simulation results reveal that our proposed system achieved better performance with lower number of training iterations.
Robust Backstepping Control Based on a Lyapunov Redesign for Skid-Steered Wheeled Mobile Robots
Eun-Ju Hwang
2013-01-01
Full Text Available This paper represents a robust backstepping tracking control based on a Lyapunov redesign for Skid‐Steered Wheeled Mobile Robots (WMRs. We present kinematic and dynamic models that explicitly relate the perturbations to the skidding in order to improve the tracking performance during real running. A robust controller is synthesized in the backstepping approach and the Lyapunov redesign technique, which forces the error dynamics to stabilize to the reference trajectories. We design an additional feedback control ‐ a Lyapunov redesign ‐ such that the overall control stabilizes the actual system in the presence of uncertainty and perturbation with the knowledge of the Lyapunov function. Simulation results are provided to validate and analyse the performance and stability of the proposed controller.
Dynamical behaviors determined by the Lyapunov function in competitive Lotka-Volterra systems.
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.
One Form of Lyapunov Operator for Stochastic Dynamic System with Markov Parameters
Taras Lukashiv
2016-01-01
Full Text Available The form of weak infinitesimal operator of Lyapunov type on solutions of stochastic dynamic systems of random structure with constant delay which exist under the action of Markov perturbations is obtained.
Stability of dynamical systems on the role of monotonic and non-monotonic Lyapunov functions
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...
The multi-dimension RSA and its low exponent security
曹珍富
2000-01-01
Using a well-known result of polynomial over the finite field , we show that the Euler-Fermat theorem holds in N[ x]. We present a multi-dimension RSA cryptosystem and point out that low exponent algorithm of attacking RSA is not suitable for the multi-dimension RSA. Therefore, it is believed that the security of the new cryptosystem is mainly based on the factorization of large integers.
Clustering of Casablanca stock market based on hurst exponent estimates
Lahmiri, Salim
2016-08-01
This paper deals with the problem of Casablanca Stock Exchange (CSE) topology modeling as a complex network during three different market regimes: general trend characterized by ups and downs, increasing trend, and decreasing trend. In particular, a set of seven different Hurst exponent estimates are used to characterize long-range dependence in each industrial sector generating process. They are employed in conjunction with hierarchical clustering approach to examine the co-movements of the Casablanca Stock Exchange industrial sectors. The purpose is to investigate whether cluster structures are similar across variable, increasing and decreasing regimes. It is observed that the general structure of the CSE topology has been considerably changed over 2009 (variable regime), 2010 (increasing regime), and 2011 (decreasing regime) time periods. The most important findings follow. First, in general a high value of Hurst exponent is associated to a variable regime and a small one to a decreasing regime. In addition, Hurst estimates during increasing regime are higher than those of a decreasing regime. Second, correlations between estimated Hurst exponent vectors of industrial sectors increase when Casablanca stock exchange follows an upward regime, whilst they decrease when the overall market follows a downward regime.
Lifshitz black holes with arbitrary dynamical exponent in Horndeski theory
Bravo-Gaete, Moises
2013-01-01
In arbitrary dimensions, we consider a particular Horndeski action given by the Einstein-Hilbert Lagrangian with a cosmological constant term, while the source part is described by a real scalar field with its usual kinetic term together with a nonminimal kinetic coupling. For this model, whose field equations are of second-order, we report a class of Lifshitz black hole solutions with arbitrary dynamical exponents z. The solutions depend on a unique constant of integration with a scalar field that can not be switched off, and the signs of the coupling constants must be fixed in a precise way. In the second part, we show that this model also supports Lifshitz black hole solutions with a time-dependent scalar field only for a special value of the dynamical exponent z=1/3. In this case, the configuration has an additional constant of integration which allows to leave free the signs of the coupling constants. Remarkably, in three dimensions, there are no restrictions on the dynamical exponent, and Lifshitz black...
One Lyapunov control for quantum systems and its application to entanglement generation
Yang, Wei; Sun, Jitao
2013-05-01
In this Letter, we investigate the control of finite dimensional ideal quantum systems in which the quantum states are represented by the density operators. A new Lyapunov function based on the Hilbert-Schmidt distance and mechanical quantity of the quantum system is given. We present a theoretical convergence result using LaSalle invariance principle. Applying the proposed Lyapunov method, the generation of the maximally entangled quantum states of two qubits is obtained.
One Lyapunov control for quantum systems and its application to entanglement generation
Yang, Wei, E-mail: 09yw@tongji.edu.cn [Department of Mathematics, Tongji University, Shanghai, 200092 (China); Sun, Jitao, E-mail: sunjt@sh163.net [Department of Mathematics, Tongji University, Shanghai, 200092 (China)
2013-05-03
In this Letter, we investigate the control of finite dimensional ideal quantum systems in which the quantum states are represented by the density operators. A new Lyapunov function based on the Hilbert–Schmidt distance and mechanical quantity of the quantum system is given. We present a theoretical convergence result using LaSalle invariance principle. Applying the proposed Lyapunov method, the generation of the maximally entangled quantum states of two qubits is obtained.
Robust H∞ Control for Singular Time-Delay Systems via Parameterized Lyapunov Functional Approach
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 function-based obstacle avoidance scheme for a two-wheeled mobile robot
Mingcong DENG; Akira INOUE; Katsuya SEKIGUCHI
2008-01-01
An obstacle avoidance scheme of a two-wheeled mobile robot is shown by selecting an appropriate Lyapunov function.When considering the obstacle,the Lyapunov function may have some local minima.A method which erases the local minima is proposed by using a function which covers the minima with a plane surface.The effectiveness of the proposed method is verified by numerical simulations.
Lyapunov based nonlinear control of electrical and mechanical systems
Behal, Aman
This Ph.D. dissertation describes the design and implementation of various control strategies centered around the following applications: (i) an improved indirect field oriented controller for the induction motor, (ii) partial state feedback control of an induction motor with saturation effects, (iii) tracking control of an underactuated surface vessel, and (iv) an attitude tracking controller for an underactuated spacecraft. The theory found in each of these sections is demonstrated through simulation or experimental results. An introduction to each of these four primary chapters can be found in chapter one. In the second chapter, the previously published tracking control of [16] 1 is presented in the indirect field oriented control (IFOC) notation to achieve exponential rotor velocity/rotor flux tracking. Specifically, it is illustrated how the proposed IFOC controller can be rewritten in the manner of [16] to allow for a direct Lyapunov stability proof. Experimental results (implemented with the IFOC algorithm) are provided to corroborate the efficacy of the algorithm. In the third chapter, a singularity-free, rotor position tracking controller is presented for the full order, nonlinear dynamic model of the induction motor that includes the effects of magnetic saturation. Specifically, by utilizing the pi-equivalent saturation model, an observer/controller strategy is designed that achieves semi-global exponential rotor position tracking and only requires stator current, rotor velocity, and rotor position measurements. Simulation and experimental results are included to demonstrate the efficacy of the proposed algorithm. In the fourth chapter, a continuous, time-varying tracking controller is designed that globally exponentially forces the position/orientation tracking error of an under-actuated surface vessel to a neighborhood about zero that can be made arbitrarily small (i.e., global uniformly ultimately boundedness (GUUB)). The result is facilitated by
Polychromatic Arm Exponents for the Critical Planar FK-Ising model
Wu, Hao
2016-01-01
We derive the arm exponents of SLE$_{\\kappa}$ for $\\kappa\\in (4,8)$ and explain how to combine them with the convergence of the interface to obtain the arm exponents of critical FK-Ising model. We obtain six different patterns of boundary arm exponents and three different patterns of interior arm exponents of the critical planar FK-Ising model on the square lattice.
Critical Exponents of Ferromagnetic Ising Model on Fractal Lattices
Hsiao, Pai-Yi
2001-04-01
We review the value of the critical exponents ν-1, β/ν, and γ/ν of ferromagnetic Ising model on fractal lattices of Hausdorff dimension between one and three. They are obtained by Monte Carlo simulation with the help of Wolff algorithm. The results are accurate enough to show that the hyperscaling law df = 2β/ν + γ/ν is satisfied in non-integer dimension. Nevertheless, the discrepancy between the simulation results and the γ-expansion studies suggests that the strong universality should be adapted for the fractal lattices.
Critical exponent for damped wave equations with nonlinear memory
Fino, Ahmad
2010-01-01
We consider the Cauchy problem in $\\mathbb{R}^n,$ $n\\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\\to\\infty$ of small data solutions have been established in the case when $1\\leq n\\leq3.$ Moreover, we derive a blow-up result under some positive data for in any dimensional space. It turns out that the critical exponent indeed coincides with the one to the corresponding semilinear heat equation.
Maximum Autocorrelation Factorial Kriging
Nielsen, Allan Aasbjerg; Conradsen, Knut; Pedersen, John L.
2000-01-01
This paper describes maximum autocorrelation factor (MAF) analysis, maximum autocorrelation factorial kriging, and its application to irregularly sampled stream sediment geochemical data from South Greenland. Kriged MAF images are compared with kriged images of varimax rotated factors from...
Concerning the Nature of the Cosmic Ray Power Law Exponents
Widom, A; Srivastava, Y N
2014-01-01
We have recently shown that the cosmic ray energy distributions as detected on earthbound, low flying balloon or high flying satellite detectors can be computed by employing the heats of evaporation of high energy particles from astrophysical sources. In this manner, the experimentally well known power law exponents of the cosmic ray energy distribution have been theoretically computed as 2.701178 for the case of ideal Bose statistics, 3.000000 for the case of ideal Boltzmann statistics and 3.151374 for the case of ideal Fermi statistics. By "ideal" we mean virtually zero mass (i.e. ultra-relativistic) and noninteracting. These results are in excellent agreement with the experimental indices of 2.7 with a shift to 3.1 at the high energy ~ PeV "knee" in the energy distribution. Our purpose here is to discuss the nature of cosmic ray power law exponents obtained by employing conventional thermal quantum field theoretical models such as quantum chromodynamics to the cosmic ray sources in a thermodynamic scheme w...
Truncatable bootstrap equations in algebraic form and critical surface exponents
Gliozzi, Ferdinando
2016-10-01
We describe examples of drastic truncations of conformal bootstrap equations encoding much more information than that obtained by a direct numerical approach. A three-term truncation of the four point function of a free scalar in any space dimensions provides algebraic identities among conformal block derivatives which generate the exact spectrum of the infinitely many primary operators contributing to it. In boundary conformal field theories, we point out that the appearance of free parameters in the solutions of bootstrap equations is not an artifact of truncations, rather it reflects a physical property of permeable conformal interfaces which are described by the same equations. Surface transitions correspond to isolated points in the parameter space. We are able to locate them in the case of 3d Ising model, thanks to a useful algebraic form of 3d boundary bootstrap equations. It turns out that the low-lying spectra of the surface operators in the ordinary and the special transitions of 3d Ising model form two different solutions of the same polynomial equation. Their interplay yields an estimate of the surface renormalization group exponents, y h = 0 .72558(18) for the ordinary universality class and y h = 1 .646(2) for the special universality class, which compare well with the most recent Monte Carlo calculations. Estimates of other surface exponents as well as OPE coefficients are also obtained.
Truncatable bootstrap equations in algebraic form and critical surface exponents
Gliozzi, Ferdinando [Dipartimento di Fisica, Università di Torino andIstituto Nazionale di Fisica Nucleare - sezione di Torino,Via P. Giuria 1, Torino, I-10125 (Italy)
2016-10-10
We describe examples of drastic truncations of conformal bootstrap equations encoding much more information than that obtained by a direct numerical approach. A three-term truncation of the four point function of a free scalar in any space dimensions provides algebraic identities among conformal block derivatives which generate the exact spectrum of the infinitely many primary operators contributing to it. In boundary conformal field theories, we point out that the appearance of free parameters in the solutions of bootstrap equations is not an artifact of truncations, rather it reflects a physical property of permeable conformal interfaces which are described by the same equations. Surface transitions correspond to isolated points in the parameter space. We are able to locate them in the case of 3d Ising model, thanks to a useful algebraic form of 3d boundary bootstrap equations. It turns out that the low-lying spectra of the surface operators in the ordinary and the special transitions of 3d Ising model form two different solutions of the same polynomial equation. Their interplay yields an estimate of the surface renormalization group exponents, y{sub h}=0.72558(18) for the ordinary universality class and y{sub h}=1.646(2) for the special universality class, which compare well with the most recent Monte Carlo calculations. Estimates of other surface exponents as well as OPE coefficients are also obtained.
Complex Critical Exponents in Diluted Systems of Quantum Rotors
Fernandes, Rafael; Schmalian, Jörg
2011-03-01
In this work, we investigate the effects of the Berry phase 2 πρ on the critical properties of XY quantum-rotors that undergo a percolation transition. This model describes a variety of randomly-diluted quantum systems, such as interacting bosons coupled to a particle reservoir, quantum planar antiferromagnets under a perpendicular magnetic field, and Josephson-junction arrays with an external bias-voltage. Focusing on the quantum critical point at the percolation threshold, we find that, for rational ρ , one recovers the power-law behavior with the same critical exponents as in the case with no Berry phase. However, for irrational ρ , the low-energy excitations change completely and are given by emergent spinless fermions with fractal spectrum. As a result, critical properties that cannot be described by the usual Ginzburg-Landau-Wilson theory of phase transitions emerge, such as complex critical exponents, log-periodic oscillations, and dynamically-broken scale invariance. Research supported by the U.S. DOE, Office of BES, Materials Science and Engineering Division.
Exact computation of the critical exponents of the jamming transition
Zamponi, Francesco
2015-03-01
The jamming transition marks the emergence of rigidity in a system of amorphous and athermal grains. It is characterized by a divergent correlation length of the force-force correlation and non-trivial critical exponents that are independent of spatial dimension, suggesting that a mean field theory can correctly predict their values. I will discuss a mean field approach to the problem based on the exact solution of the hard sphere model in infinite dimension. An unexpected analogy with the Sherrington-Kirkpatrick spin glass model emerges in the solution: as in the SK model, the glassy states turn out to be marginally stable, and are described by a Parisi equation. Marginal stability has a deep impact on the critical properties of the jamming transition and allows one to obtain analytic predictions for the critical exponents. The predictions are consistent with a recently developed scaling theory of the jamming transition, and with numerical simulations. Finally, I will briefly discuss some possible extensions of this approach to other open issues in the theory of glasses.
Vannitsem, Stéphane; Lucarini, Valerio
2016-06-01
We study a simplified coupled atmosphere-ocean model using the formalism of covariant Lyapunov vectors (CLVs), which link physically-based directions of perturbations to growth/decay rates. The model is obtained via a severe truncation of quasi-geostrophic equations for the two fluids, and includes a simple yet physically meaningful representation of their dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and the parameters are chosen so that a chaotic behaviour is observed. There are two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen near-zero LEs. The presence of many near-zero LEs results from the vast time-scale separation between the characteristic time scales of the two fluids, and leads to nontrivial error growth properties in the tangent space spanned by the corresponding CLVs, which are geometrically very degenerate. Such CLVs correspond to two different classes of ocean/atmosphere coupled modes. The tangent space spanned by the CLVs corresponding to the positive and negative LEs has, instead, a non-pathological behaviour, and one can construct robust large deviations laws for the finite time LEs, thus providing a universal model for assessing predictability on long to ultra-long scales along such directions. Interestingly, the tangent space of the unstable manifold has substantial projection on both atmospheric and oceanic components. The results show the difficulties in using hyperbolicity as a conceptual framework for multiscale chaotic dynamical systems, whereas the framework of partial hyperbolicity seems better suited, possibly indicating an alternative definition for the chaotic hypothesis. They also suggest the need for an accurate analysis of error dynamics on different time scales and domains and for a careful set-up of assimilation schemes when looking at coupled atmosphere-ocean models.
Magnetic entropy change and critical exponents in double perovskite Y{sub 2}NiMnO{sub 6}
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.
Canada, Antonio
2011-01-01
Several different problems make the study of the so called Lyapunov type inequalities of great interest, both in pure and applied mathematics. Although the original historical motivation was the study of the stability properties of the Hill equation (which applies to many problems in physics and engineering), other questions that arise in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients, etc. lead to the study of $L_p$ Lyapunov inequalities ($1\\leq p\\leq \\infty$) for differential equations. In this work we review some recent results on these kinds of questions which can be formulated as optimal control problems. In the case of Ordinary Differential Equations, we consider periodic and antiperiodic boundary conditions at higher eigenvalues and by using a more accurate version of the Sturm separation theory, an explicit optimal result is obtained. Then, we establish Lyapunov inequalities for systems of equations. To this respect, a key point is the characterization of the be...
Barrier Lyapunov function-based model-free constraint position control for mechanical systems
Han, Seong Ik; Ha, Hyun Uk; Lee, Jang Myung [Pusan National University, Busan (Korea, Republic of)
2016-07-15
In this article, a motion constraint control scheme is presented for mechanical systems without a modeling process by introducing a barrier Lyapunov function technique and adaptive estimation laws. The transformed error and filtered error surfaces are defined to constrain the motion tracking error in the prescribed boundary layers. Unknown parameters of mechanical systems are estimated using adaptive laws derived from the Lyapunov function. Then, robust control used the conventional sliding mode control, which give rise to excessive chattering, is changed to finite time-based control to alleviate undesirable chattering in the control action and to ensure finite-time error convergence. Finally, the constraint controller from the barrier Lyapunov function is designed and applied to the constraint of the position tracking error of the mechanical system. Two experimental examples for the XY table and articulated manipulator are shown to evaluate the proposed control scheme.
Zhao, Xueyan; Deng, Feiqi
2016-07-01
In this paper, a particular property of Lyapunov functions for functional differential equations (FDEs) is developed, that is the direct dependence of the signs of the derivatives of the Lyapunov functions on the initial data. This property implies that the derivatives of the Lyapunov functions for FDEs cannot be guaranteed to be negative definite generally, and then makes the FDEs differ from the ordinary differential equations constitutionally. With this property, we give some enlightenments for the research methods for establishing stability theorems or criteria for FDEs, which may help us to form a common view about the choice of the investigation methods on the stability of FDEs. The conclusion is stated in both the deterministic and stochastic versions. Two illustrative examples are given to show and verify our conclusion through the paper.
GA and Lyapunov theory-based hybrid adaptive fuzzy controller for non-linear systems
Roy, Ananya; Das Sharma, Kaushik
2015-02-01
In this present article, a new hybrid methodology for designing stable adaptive fuzzy logic controllers (AFLCs) for a class of non-linear system is proposed. The proposed design strategy exploits the features of genetic algorithm (GA)-based stochastic evolutionary global search technique and Lyapunov theory-based local adaptation scheme. The objective is to develop a methodology for designing AFLCs with optimised free parameters and guaranteed closed-loop stability. Simultaneously, the proposed method introduces automation in the design process. The stand-alone Lyapunov theory-based design, GA-based design and proposed hybrid GA-Lyapunov design methodologies are implemented for two benchmark non-linear plants in simulation case studies with different reference signals and one experimental case study. The results demonstrate that the hybrid design methodology outperforms the other control strategies on the whole.
Lyapunov-based boundary feedback control in multi-reach canals
CEN LiHui; XI YuGeng
2009-01-01
This paper presents a Lyapunov-based approach to design the boundary feedback control for an open-channel network composed of a cascade of multi-reach canals, each described by a pair of Saint-Venant equations. The weighted sum of entropies of the multi-reaches is adopted to construct the Lyapunov function. The time derivative of the Lyapunov function is expressed by the water depth variations at the gate boundaries, based on which a class of boundary feedback controllers is presented to guarantee the local asymptotic closed-loop stability. The advantage of this approach is that only the water level depths at the gate boundaries are measured as the feedback.
A Converse Sum of Squares Lyapunov Result with a Degree Bound
Peet, Matthew M
2012-01-01
Sum of Squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems but several questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum-of-squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists an SOS Lyapunov function which is exponentially decreasing on that bounded set. The proof is constructive and uses the Picard iteration. A bound on the degree of this converse Lyapunov function is also given. This result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.
Ribard, Nicolas; Wisniewski, Rafael; Sloth, Christoffer
2016-01-01
In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use the represen......In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use...... the representation of the given control system in Bernstein basis. Subsequently, the control synthesis problem is reduced to finite number of evaluations of a polynomial on vertices of cubes in the space of parameters representing admissible controls and Lyapunov functions....
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.
Minimal Generalized Lyapunov Function Theorems for Nonlinear Systems%非线性系统的最小广义Lyapunov函数定理
玉强; 吴保卫
2014-01-01
利用Zorn引理和LaSalle不变原理，研究一般非线性系统最小广义 Lyapunov 函数的存在性，证明了最小(非负)广义 Lyapunov 函数和最小(非负)强广义 Lyapunov 函数的存在性定理。%Based on Zorn’s lemma and LaSalle’s invariance principle,the existence of the minimal generalized Lyapunov function for nonlinear systems was considered. We developed the minimal generalized Lyapunov function theorems and the minimal strong generalized Lyapunov function theorems for nonlinear dynamical systems.
Boundary trace embedding theorems for variable exponent Sobolev spaces
Fan, Xianling
2008-03-01
We study boundary trace embedding theorems for variable exponent Sobolev space W1,p([dot operator])([Omega]). Let [Omega] be an open (bounded or unbounded)[thin space]domain in satisfying strong local Lipschitz condition. Under the hypotheses that p[set membership, variant]L[infinity]([Omega]), 1[less-than-or-equals, slant]infp(x)[less-than-or-equals, slant]supp(x)N, we prove that there is a continuous boundary trace embedding W1,p([dot operator])([Omega])-->Lq([dot operator])([not partial differential][Omega]) provided q([dot operator]), a measurable function on [not partial differential][Omega], satisfies condition for x[set membership, variant][not partial differential][Omega].
Skeleton graph expansion of critical exponents in "cultural revolution" years
Hao, Bailin
Kenneth Wilson's Nobel Prize winning breakthrough in the renormalization group theory of phase transition and critical phenomena almost overlapped with the violent "cultural revolution" years (1966-1976) in China. An unexpected chance in 1972 brought the author of these lines close to the Wilson-Fisher є-expansion of critical exponents and eventually led to a joint paper with Lu Yu published entirely in Chinese without any English title and abstract. Even the original acknowledgment was deleted because of mentioning foreign names like Kenneth Wilson and Kerson Huang. In this article I will tell the 40-year old story as a much belated tribute to Kenneth Wilson and to reproduce the essence of our work in English. At the end, I give an elementary derivation of the Callan-Symanzik equation without referring to field theory.
Path Loss Exponent Estimation in Large Wireless Networks
Srinivasa, Sunil
2008-01-01
Even though the analyses in many wireless networking problems assume that the value of the path loss exponent (PLE) is known a priori, this is often not the case, and an accurate estimate is crucial for the study and design of wireless systems. In this paper, we address the problem of estimating the PLE in large wireless networks, which is relevant to several important issues in communications such as localization, energy-efficient routing, and channel access. We consider a large ad hoc network where nodes are distributed as a homogeneous Poisson point process on the plane, and the channels are subject to Nakagami-m fading. We propose and discuss three algorithms for PLE estimation under these settings that explicitly take into account the interference in the network. Simulation results are provided to demonstrate the performance of the algorithms and quantify the estimation errors.
The effect of the underlying distribution in Hurst exponent estimation.
Miguel Ángel Sánchez
Full Text Available In this paper, a heavy-tailed distribution approach is considered in order to explore the behavior of actual financial time series. We show that this kind of distribution allows to properly fit the empirical distribution of the stocks from S&P500 index. In addition to that, we explain in detail why the underlying distribution of the random process under study should be taken into account before using its self-similarity exponent as a reliable tool to state whether that financial series displays long-range dependence or not. Finally, we show that, under this model, no stocks from S&P500 index show persistent memory, whereas some of them do present anti-persistent memory and most of them present no memory at all.
Circular orbit spacecraft control at the L4 point using Lyapunov functions
Agrawal, Rachana
2015-01-01
The objective of this work is to demonstrate the utility of Lyapunov functions in control synthesis for the purpose of maintaining and stabilizing a spacecraft in a circular orbit around the L4 point in the circular restricted three body problem (CRTBP). Incorporating the requirements of a fixed radius orbit and a desired angular momentum, a Lyapunov function is constructed and the requisite analysis is performed to obtain a controller. Asymptotic stability is proved in a defined region around the L4 point using LaSalle's principle.
Blackwell, C. C.
1987-01-01
A relevant facet of the application of Lyapunov gradient-generated robust control to unstable linear autonomous plants is explored. It is demonstrated that if the plant, the output, and the nominal stabilizing control satisfy certain conditions, then the robust component alone stabilizes the nominal plant. An example characterized by two zero eigenvalues and two negative real value poles is presented. These results assure that the robust component will fulfill the role of nominal stabilization successfully so long as the possible magnitude of the robust component can overcome the contribution of the instability to positiveness of the Lyapunov rate.
Stabilisation of a class of 2-DOF underactuated mechanical systems via direct Lyapunov approach
Turker, Turker; Gorgun, Haluk; Cansever, Galip
2013-06-01
This paper represents an alternative stabilisation procedure for a class of two degree-of-freedom underactuated mechanical systems based on a set of transformations and a Lyapunov function. After simplifying dynamic equations of the system via partial feedback linearisation and coordinate changes, the stability of the system is provided with Lyapunov's direct method. Proposed control scheme is used on two different examples and asymptotic convergence for each system is proven by means of La Salle's invariance principle. The designed controller is successfully illustrated through numerical simulations for each example.
Lyapunov function and the basin of attraction for a single-joint muscle-skeletal model.
Giesl, Peter; Wagner, Heiko
2007-04-01
This paper provides an explicit Lyapunov function for a general single-joint muscle-skeletal model. Using this Lyapunov function one can determine analytically large subsets of the basin of attraction of an asymptotically stable equilibrium. Besides providing an analytical tool for the analysis of such a system we consider an elbow model and show that the theoretical predictions are in agreement with experimental results. Moreover, we can thus distinguish between regions where the self-stabilizing properties of the muscle-skeletal system guarantee stability and regions where nerval control and reflexes are necessary.
Blackwell, C. C.
1987-01-01
A relevant facet of the application of Lyapunov gradient-generated robust control to unstable linear autonomous plants is explored. It is demonstrated that if the plant, the output, and the nominal stabilizing control satisfy certain conditions, then the robust component alone stabilizes the nominal plant. An example characterized by two zero eigenvalues and two negative real value poles is presented. These results assure that the robust component will fulfill the role of nominal stabilization successfully so long as the possible magnitude of the robust component can overcome the contribution of the instability to positiveness of the Lyapunov rate.
Feng LI; Yin Lai JIN
2011-01-01
In this paper,center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated.With the help of computer algebra system MATHEMATICA,the first 8 quasi Lyapunov constants are deduced.As a result,the necessary and sufficient conditions to have a center are obtained.The fact that there exist 8 small amplitude limit cycles created from the three-order nilpotent critical point is also proved.Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems.
Nonlinearity exponent of ac conductivity in disordered systems
Nandi, U. N.; Sircar, S.; Karmakar, A.; Giri, S.
2012-07-01
We measured the real part of ac conductance Σ(x,f) or Σ(T,f) of iron-doped mixed-valent polycrystalline manganite oxides LaMn1-xFexO3 as a function of frequency f by varying initial conductance Σ0 by quenched disorder x at a fixed temperature T (room) and by temperature T at a fixed quenched disorder x. At a fixed temperature T, Σ(x,f) of a sample with fixed x remains almost constant at its zero-frequency dc value Σ0 at lower frequency. With increase in f, Σ(x,f) increases slowly from Σ0 and finally increases rapidly following a power law with an exponent s at high frequency. Scaled appropriately, the data for Σ(T,f) and Σ(x,f) fall on the same universal curve, indicating the existence of a general scaling formalism for the ac conductivity in disordered systems. The characteristic frequency fc at which Σ(x,f) or Σ(T,f) increases for the first time from Σ0 scales with initial conductance Σ0 as {f}_{{c}}\\sim {\\Sigma }_{0}^{{x}_{f}}, where xf is the onset exponent. The value of xf is nearly equal to one and is found to be independent of x and T. Further, an inverse relationship between xf and s provides a self-consistency check of the systematic description of Σ(x,f) or Σ(T,f). This apparent universal value of xf is discussed within the framework of existing theoretical models and scaling theories. The relevance to other similar disordered systems is also highlighted.
Cooney, Tom; Mosonyi, Milán; Wilde, Mark M.
2016-06-01
This paper studies the difficulty of discriminating between an arbitrary quantum channel and a "replacer" channel that discards its input and replaces it with a fixed state. The results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination. We show that, in this particular setting, the most general adaptive discrimination strategies provide no asymptotic advantage over non-adaptive tensor-power strategies. This conclusion follows by proving a quantum Stein's lemma for this channel discrimination setting, showing that a constant bound on the Type I error leads to the Type II error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels. The strong converse part of the lemma states that any attempt to make the Type II error decay to zero at a rate faster than the channel relative entropy implies that the Type I error necessarily converges to one. We then refine this latter result by identifying the optimal strong converse exponent for this task. As a consequence of these results, we can establish a strong converse theorem for the quantum-feedback-assisted capacity of a channel, sharpening a result due to Bowen. Furthermore, our channel discrimination result demonstrates the asymptotic optimality of a non-adaptive tensor-power strategy in the setting of quantum illumination, as was used in prior work on the topic. The sandwiched Rényi relative entropy is a key tool in our analysis. Finally, by combining our results with recent results of Hayashi and Tomamichel, we find a novel operational interpretation of the mutual information of a quantum channel {mathcal{N}} as the optimal Type II error exponent when discriminating between a large number of independent instances of {mathcal{N}} and an arbitrary "worst-case" replacer channel chosen from the set of all replacer channels.
Farivar, Faezeh; Shoorehdeli, Mahdi Aliyari
2012-01-01
In this paper, fault tolerant synchronization of chaotic gyroscope systems versus external disturbances via Lyapunov rule-based fuzzy control is investigated. Taking the general nature of faults in the slave system into account, a new synchronization scheme, namely, fault tolerant synchronization, is proposed, by which the synchronization can be achieved no matter whether the faults and disturbances occur or not. By making use of a slave observer and a Lyapunov rule-based fuzzy control, fault tolerant synchronization can be achieved. Two techniques are considered as control methods: classic Lyapunov-based control and Lyapunov rule-based fuzzy control. On the basis of Lyapunov stability theory and fuzzy rules, the nonlinear controller and some generic sufficient conditions for global asymptotic synchronization are obtained. The fuzzy rules are directly constructed subject to a common Lyapunov function such that the error dynamics of two identical chaotic motions of symmetric gyros satisfy stability in the Lyapunov sense. Two proposed methods are compared. The Lyapunov rule-based fuzzy control can compensate for the actuator faults and disturbances occurring in the slave system. Numerical simulation results demonstrate the validity and feasibility of the proposed method for fault tolerant synchronization.
Blanchini, F. [Universita di Udine (Italy); Carabelli, S. [Politecnico di Torino (Italy)
1994-12-31
We apply a technique recently proposed in literature for the robust stabilization of linear systems with time-varying uncertain parameters to a magnetic levitation system. This technique allows the construction of a polyhedral Lyapunov function and a linear variable-structure stabilizing controller.
Lyapunov vs. geometrical stability analysis of the Kepler and the restricted three body problems
Yahalom, A.; Levitan, J.; Lewkowicz, M.
2011-01-01
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...
Energy-Based Lyapunov Functions for Forced Hamiltonian Systems with Dissipation
Maschke, Bernhard; Ortega, Romeo; Schaft, Arjan J. van der
2000-01-01
In this paper, we propose a constructive procedure to modify the Hamiltonian function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for nonzero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network
A Lyapunov-Krasovskii methodology for asymptotic stability of discrete time delay systems
Stojanović Sreten B.
2007-01-01
Full Text Available This paper presents a Lyapunov-Krasovskii methodology for asymptotic stability of discrete time delay systems. Based on the methods, delay-independent stability condition is derived. A numerical example has been working out to show the applicability of results derived.
Robust Stabilization for Uncertain Control Systems Using Piecewise Quadratic Lyapunov Functions
无
2002-01-01
The sufficient condition based on piecewise quadratic simultaneous Lyapunov functions for robust stabilizationof uncertain control systems via a constant linear state feedback control law is obtained. The objective is to use a robuststability criterion that is less conservative than the usual quadratic stability criterion. Numerical example is given, show-ing the advanteges of the proposed method.
Energy-Based Lyapunov Functions for Forced Hamiltonian Systems with Dissipation
Maschke, Bernhard; Ortega, Romeo; Schaft, Arjan J. van der
2000-01-01
In this paper, we propose a constructive procedure to modify the Hamiltonian function of forced Hamiltonian systems with dissipation in order to generate Lyapunov functions for nonzero equilibria. A key step in the procedure, which is motivated from energy-balance considerations standard in network
Computation of non-monotonic Lyapunov functions for continuous-time systems
Li, Huijuan; Liu, AnPing
2017-09-01
In this paper, we propose two methods to compute non-monotonic Lyapunov functions for continuous-time systems which are asymptotically stable. The first method is to solve a linear optimization problem on a compact and bounded set. The proposed linear programming based algorithm delivers a CPA1
On Lyapunov-type inequalities for [Formula: see text]-Laplacian systems.
Jleli, Mohamed; Samet, Bessem
2017-01-01
We establish Lyapunov-type inequalities for a system involving one-dimensional [Formula: see text]-Laplacian operators ([Formula: see text]). Next, the obtained inequalities are used to derive some geometric properties of the generalized spectrum associated to the considered problem.
Chu, Chia-Chi; Tsai, Hung-Chi; Chang, Wei-Neng
A Lyapunov-based recurrent neural networks unified power flow controller (UPFC) is developed for improving transient stability of power systems. First, a simple UPFC dynamical model, composed of a controllable shunt susceptance on the shunt side and an ideal complex transformer on the series side, is utilized to analyze UPFC dynamical characteristics. Secondly, we study the control configuration of the UPFC with two major blocks: the primary control, and the supplementary control. The primary control is implemented by standard PI techniques when the power system is operated in a normal condition. The supplementary control will be effective only when the power system is subjected by large disturbances. We propose a new Lyapunov-based UPFC controller of the classical single-machine-infinite-bus system for damping enhancement. In order to consider more complicated detailed generator models, we also propose a Lyapunov-based adaptive recurrent neural network controller to deal with such model uncertainties. This controller can be treated as neural network approximations of Lyapunov control actions. In addition, this controller also provides online learning ability to adjust the corresponding weights with the back propagation algorithm built in the hidden layer. The proposed control scheme has been tested on two simple power systems. Simulation results demonstrate that the proposed control strategy is very effective for suppressing power swing even under severe system conditions.
Lyapunov-Based Control Scheme for Single-Phase Grid-Connected PV Central Inverters
Meza, C.; Biel, D.; Jeltsema, D.; Scherpen, J. M. A.
2012-01-01
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 p
UDU factored Lyapunov recursions solve optimal reduced-order LQG problems
Willigenburg, van L.G.; Koning, de W.L.
2004-01-01
A new algorithm is presented to solve both the finite-horizon time-varying and infinite-horizon time-invariant discrete-time optimal reduced-order LQG problem. In both cases the first order necessary optimality conditions can be represented by two non-linearly coupled discrete-time Lyapunov equation
Lyapunov-Based Control Scheme for Single-Phase Grid-Connected PV Central Inverters
Meza, C.; Biel, D.; Jeltsema, D.; Scherpen, J. M. A.
2012-01-01
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 p
Stability of quantized time-delay nonlinear systems : A Lyapunov-Krasowskii-functional approach
Persis, Claudio De; Mazenc, Frédéric
2009-01-01
Lyapunov-Krasowskii functionals are used to design quantized control laws for nonlinear continuous-time systems in the presence of time-invariant constant delays in the input. The quantized control law is implemented via hysteresis to avoid chattering. Under appropriate conditions, our analysis appl
Stabilization of Parametric Roll Resonance with Active U-Tanks via Lyapunov Control Design
Holden, Christian; Galeazzi, Roberto; Fossen, Thor Inge;
2009-01-01
design of an active u-tank stabilizer is carried out using Lyapunov theory. A nonlinear backstepping controller is developed to provide global exponential stability of roll. An extension of commonly used u-tank models is presented to account for large roll angles, and the control design is tested via...
Design of bounded feedback controls for linear dynamical systems by using common Lyapunov functions
Igor; Ananievskii; Nickolai; Anokhin; Alexander; Ovseevich
2011-01-01
For a linear dynamical system,we address the problem of devising a bounded feedback control,which brings the system to the origin in finite time.The construction is based on the notion of a common Lyapunov function.It is shown that the constructed control remains effective in the presence of small perturbations.
Toropov, A. V.; Toropova, L.V.
2014-01-01
Тhe problem of synthesis of nonlinear speed controller asynchronized switched motor is considered. To find the optimal control law by, the method of Bellman - Lyapunov by concept of "immersion" is used. Modeling and comparative analysis of the system with the standard PI - controller, as well as the synthesized regulators are made
Lyapunov based control of hybrid energy storage system in electric vehicles
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...
Maximum Autocorrelation Factorial Kriging
Nielsen, Allan Aasbjerg; Conradsen, Knut; Pedersen, John L.; Steenfelt, Agnete
2000-01-01
This paper describes maximum autocorrelation factor (MAF) analysis, maximum autocorrelation factorial kriging, and its application to irregularly sampled stream sediment geochemical data from South Greenland. Kriged MAF images are compared with kriged images of varimax rotated factors from an ordinary non-spatial factor analysis, and they are interpreted in a geological context. It is demonstrated that MAF analysis contrary to ordinary non-spatial factor analysis gives an objective discrimina...
On the Sobolev embedding theorem for variable exponent spaces in the critical range
Bonder, Julian Fernandez; Silva, Analia
2011-01-01
In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration--Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico.
Partial differential equations with variable exponents variational methods and qualitative analysis
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
The Scaling Exponent Distinguishes the Injured Sick Hearts Against Normal Healthy Hearts
Yazawa, Toru; Tanaka, Katsunori
2009-05-01
We analyzed heartbeat-intervals with our own program of detrended fluctuation analysis (DFA) to quantify the irregularity of the heartbeat. The present analysis revealed that normal healthy subjects have the scaling exponent of 1.0, and ischemic heart disease pushes the scaling exponent up to 1.2-1.5. We conclude that the scaling exponent, calculated by the DFA, reflects a risk for the "failing" heart. The scaling exponents could determine whether the subjects are under sick or in healthy conditions on the basis of cardiac physiology.
A generalization of the power law distribution with nonlinear exponent
Prieto, Faustino; Sarabia, José María
2017-01-01
The power law distribution is usually used to fit data in the upper tail of the distribution. However, commonly it is not valid to model data in all the range. In this paper, we present a new family of distributions, the so-called Generalized Power Law (GPL), which can be useful for modeling data in all the range and possess power law tails. To do that, we model the exponent of the power law using a non-linear function which depends on data and two parameters. Then, we provide some basic properties and some specific models of that new family of distributions. After that, we study a relevant model of the family, with special emphasis on the quantile and hazard functions, and the corresponding estimation and testing methods. Finally, as an empirical evidence, we study how the debt is distributed across municipalities in Spain. We check that power law model is only valid in the upper tail; we show analytically and graphically the competence of the new model with municipal debt data in the whole range; and we compare the new distribution with other well-known distributions including the Lognormal, the Generalized Pareto, the Fisk, the Burr type XII and the Dagum models.
Critical exponent for a damped wave system with fractional integral
Mijing Wu
2015-08-01
Full Text Available We shall present the critical exponent $$ F(p, q,\\alpha:=\\max\\big\\{\\alpha+\\frac{(\\alpha+1(p+1}{pq-1}, \\alpha+\\frac{(\\alpha+1(q+1}{pq-1}\\big\\}-\\frac{1}{2} $$ for the Cauchy problem $$\\displaylines{ u_{tt}-u_{xx}+u_t=J_{0|t}^{\\alpha}(|v|^{p}, \\quad (t,x\\in\\mathbb{R}^{+}\\times\\mathbb{R},\\cr v_{tt}-v_{xx}+v_t=J_{0|t}^{\\alpha}(|u|^{q}, \\quad (t, x\\in\\mathbb{R}^{+}\\times\\mathbb{R},\\cr (u(0,x, u_t(0,x=(u_0(x,u_1(x, \\quad x\\in \\mathbb{R},\\cr (v(0,x, v_t(0,x=(v_0(x,v_1(x, \\quad x\\in \\mathbb{R},\\cr }$$ where $p,q\\geq 1$, $pq>1$ and $0<\\alpha<1/2$; that is, the small data global existence of solutions can be derived to the problem above if $F(p, q, \\alpha<0$. Furthermore, in the case of $F(p, q, \\alpha\\geq 0$ the non-existence of global solution can be obtained with the initial data having positive average value.
Maximum likely scale estimation
Loog, Marco; Pedersen, Kim Steenstrup; Markussen, Bo
2005-01-01
A maximum likelihood local scale estimation principle is presented. An actual implementation of the estimation principle uses second order moments of multiple measurements at a fixed location in the image. These measurements consist of Gaussian derivatives possibly taken at several scales and/or ...
Nicola Dioguardi; Fabio Grizzi; Barbara Franceschini; Paola Bossi; Carlo Russo
2006-01-01
AIM: To provide the accurate alternative metrical means of monitoring the effects of new antiviral drugs on the reversal of newly formed collagen.METHODS: Digitized histological biopsy sections taken from 209 patients with chronic C virus hepatitis with different grade of fibrosis or cirrhosis, were measured by means of a new, rapid, user-friendly, fully computeraided method based on the international system meter rectified using fractal principles.RESULTS: The following were described: geometric perimeter, area and wrinkledness of fibrosis; the collation of the Knodell, Sheuer, Ishak and METAVIR scores with fractal-rectified metric measurements; the meaning of the physical composition of fibrosis in relation to the magnitude of collagen islets; the intra- and inter-biopsy sample variability of these parameters; the "staging"of biopsy sections indicating the pathway covered by fibrosis formation towards its maximum known value;the quantitative liver tissue architectural changes with the Hurst exponent.CONCLUSION: Our model provides the first metrical evaluations of the geometric properties of fibrosis and the quantitative architectural changes of the liver tissue.The representativeness of histological sections of the whole liver is also discussed in the light of the results obtained with the Hurst coefficient.
A Direct Calculation of Critical Exponents of Two-Dimensional Anisotropic Ising Model
XIONG Gang; WANG Xiang-Rong
2006-01-01
Using an exact solution of the one-dimensional quantum transverse-field Ising model, we calculate the critical exponents of the two-dimensional anisotropic classicalIsing model (IM). We verify that the exponents are the same as those of isotropic classical IM. Our approach provides an alternative means of obtaining and verifying these well-known results.
Conditional exponents, entropies and a measure of dynamical self-organization
Mendes, R V
1998-01-01
In dynamical systems composed of interacting parts, conditional exponents, conditional exponent entropies and cylindrical entropies are shown to be well defined ergodic invariants which characterize the dynamical selforganization and statitical independence of the constituent parts. An example of interacting Bernoulli units is used to illustrate the nature of these invariants.
Struzik, Z.R.
2000-01-01
The local effective H'older exponent has been applied to evaluate the variability of heart rate locally at an arbitrary position (time) and resolution (scale). The local effective H'older exponent [8, 9] used is effectively insensitive to local polynomial trends in heartbeat rate due to the use of t
Struzik, Z.R.
2001-01-01
The local effective Holder exponent has been applied to evaluate the variability of heart rate locally at an arbitrary position (time) and resolution (scale). The local effective Holder exponent [1,2] used is effectively insensitive to local polynomial trends in heartbeat rate due to the use of the
Direct embeddings of relatively hyperbolic groups with optimal $\\ell^p$ compression exponent
Hume, David
2011-01-01
We prove that for all $p>1$, every relatively hyperbolic group has $\\ell^p$ compression exponent equal to the minimum of the exponents of its maximal parabolic subgroups. Moreover, this embedding can be explicitly described in terms of given embeddings of the maximal parabolic subgroups.