Lyapunov Exponents Of Dynamical Systems Research Articles

Lyapunov Exponent, Universality and Phase Transition for Products of Random Matrices

Products of M i.i.d. random matrices of size \(N \times N\) are related to classical limit theorems in probability theory (\(N=1\) and large M), to Lyapunov exponents in dynamical systems (finite N and large M), and to universality in random matrix theory (finite M and large N). Under the two different limits of \(M\rightarrow \infty \) and \(N\rightarrow \infty \), the local singular value statistics display Gaussian and random matrix theory universality, respectively. However, it is unclear what happens if both M and N go to infinity. This problem, proposed by Akemann et al. (J Phys A 47(39):395202, 2014) and Deift (SIGMA Symmetry Integr Geom Methods Appl 13, 2017), lies at the heart of understanding both kinds of universal limits. In the case of complex Gaussian random matrices, we prove that there exists a crossover phenomenon as the relative ratio of M and N changes from 0 to \(\infty \): sine and Airy kernels from the Gaussian Unitary Ensemble (GUE) when \(M/N \rightarrow 0\), Gaussian fluctuation when \(M/N \rightarrow \infty \), and new critical phenomena when \(M/N \rightarrow \gamma \in (0,\infty )\). Accordingly, we further prove that the largest singular value undergoes a phase transition between the Gaussian and GUE Tracy–Widom distributions.

An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps.

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.

A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks

The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.

Lyapunov exponents of stochastic systems—from micro to macro

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.

An application of the ergodic theorem of information theory to Lyapunov exponents of cellular automata

We prove a generalization of the individual ergodic theorem of the information theory and we apply it to give a new proof of the Shereshevsky inequality connecting the metric entropy and Lyapunov exponents of dynamical systems generated by cellular automata.

Control of the Lyapunov exponents of dynamical systems in the plane with incomplete observation

Control linear systems in the plane are studied under the assumption of incomplete observation and incomplete control. In this situation ordinary static output controls may fail to stabilize the system. That is why special dynamic output feedback controls with finitely many states (hybrid feedback controls) are applied. Necessary and sufficient conditions are offered that guarantee exponential convergence/divergence of the solutions at an arbitrary rate. It is also shown that the general case can be reduced to two particular cases which are treated in detail.

Evaluation of the largest Lyapunov exponent in dynamical systems with time delay

The method of estimation of the largest Lyapunov exponents for dynamical systems with time delay has been developed. This method can be applied both for flows and discrete maps. Our approach is based on the phenomenon of synchronization of identical systems coupled by linear negative feedback mechanism (flows) and exponential perturbation (maps). The existence of linear dependence of the largest Lyapunov exponent on the coupled parameter allows the precise estimation of this exponent.

Evaluation of the largest Lyapunov exponent in dynamical systems with time delay

The method of estimation of the largest Lyapunov exponents for dynamical systems with time delay has been developed. This method can be applied both for flows and discrete maps. Our approach is based on the phenomenon of synchronization of identical systems coupled by linear negative feedback mechanism (flows) and exponential perturbation (maps). The existence of linear dependence of the largest Lyapunov exponent on the coupled parameter allows the precise estimation of this exponent.

A numerical method for the computation of the Lyapunov exponents of nonlinear ordinary differential equations

In this paper, an alternative method to compute the Lyapunov exponents of dynamical systems described by ordinary differential equations (ODEs) is introduced. The Lyapunov exponents are computed in terms of the solutions of two piecewise linear ODEs that approximate, respectively, the solutions of the original ODE and its associated variational equation. This approach is strongly connected with the local linearization (LL) method for ODEs and its major advantage is that these piecewise linear ODEs might be exactly integrated in a non-simultaneous way. The performance of the method is illustrated with a numerical example.

Estimation of Lyapunov exponents of dynamical systems using a spatial average.

We present an unconventional method of estimating all of the Lyapunov exponents of a dynamical system from either a known map or a set of experimental data. Rather than averaging exponents along a single trajectory, we instead represent each exponent as an integral over all of phase space. The contribution to each exponent, calculated at each point in phase space, is averaged spatially by weighting areas of high density more heavily than areas of low density, according to the invariant measure of the system. Explicit formulas for approximating both the contributions to the exponents and the invariant measure are given, and convergence results stated. The techniques are illustrated in detail for the H\'enon system.

Estimation of Lyapunov Exponents Using a Semi-Discrete Formulation

Lyaponov exponents are a generalization of the eigenvalues of a dynamical system at an equilibrium point. They are used to determine the stability of any type of steady-state behavior, including chaotic solutions. More specifically, Lyapunov exponents measure the exponential rates of divergence or convergence associated with nearby trajectories. This paper presents an efficient method of estimating the Lyapunov spectrum of continuous dynamical systems. Based on the Lie series expansion of the flow, the technique can be readily implemented to estimate the Lyapunov exponents of dynamical systems governed by ordinary differential equations.

Transformation properties of the Lyapunov exponent of dynamical systems under diffeomorphisms

In this paper we present a geometric definition of the Lyapunov exponent on a differential manifold and investigate its transformation properties under changes of coordinates, or, more generally, under diffeomorphisms. The result is that not every diffeomorphism leaves the Lyapunov exponent invariant. A sufficient condition for invariance is the following: the tangent map of the diffeomorphism is bounded exponentially in the curve parameter for any curve in the manifold and any direction in the tangent bundle with basis restricted to this curve. At the end we show that for a free particle there are diffeomorphisms violating this condition, although they are even canonical maps.

Comparison of Different Methods for Computing Lyapunov Exponents

Different discrete and continuous methods for computing the Lyapunov exponents of dynamical systems are compared for their efficiency and accuracy. All methods are either based on the QR or the singular value decomposition. The relationship between the discrete methods is discussed in terms of the iteration algorithms and the decomposition procedures used. We give simple deriva­ tions of the differential equations for continuous methods proposed recently and show that they cannot be recommended because of their long computation time and numerical instabilities. The methods are tested with the damped and driven Toda chain and the driven van der Pol osciIlator.

Least squares approximation of Lyapunov exponents

Discrete least squares approximations are shown to converge for the Lyapunov exponents of dynamical systems. Numerical examples demonstrate the approximation’s utility.