Consistent Lyapunov exponent Estimation for one-dimensional dynamical systems

Summary  The author proves the consistency of a nearest neighbor estimator of the Lyapunov exponent for a general class of one-dimensional ergodic dynamical systems. The author shows that this estimator has good practical properties on a set of simulations.     

Estimation of the largest Lyapunov exponent in systems with impacts

The method of estimation of the largest Lyapunov exponent for mechanical systems with impacts using the properties of synchronization phenomenon is demonstrated. The presented method is based on the coupling of two identical dynamical systems and is tested on the classical Duffing oscillator with impacts.     

Analysis of bifurcations for non-linear stochastic non-smooth vibro-impact system via top Lyapunov exponent

Bifurcations are discussed by the criterion of top Lyapunov exponent. Based on the local map and Kaminski’s algorithms, a general formulation of the top Lyapunov exponents is proposed for non-linear vibro-impact oscillators with Gaussian white noise perturbation. The analytical results are verified by phase portraits and bifurcation diagrams for a classical stochastic Duffing vibro-impact oscillator. Both results are consistent.     

Positive Lyapunov exponent for generic one-parameter families of unimodal maps

Letf _{a a∈A} be a C² one-parameter family of non-flat unimodal maps of an interval into itself anda* a parameter value such that

1. f_a* satisfies the Misiurewicz Condition,
2. f_a* satisfies a backward Collet-Eckmann-like condition,
3. the partial derivatives with respect tox anda of f _a ⁿ (x), respectively at the critical value and ata*, are comparable for largen.

Thena* is a Lebesgue density point of the set of parameter valuesa such that the Lyapunov exponent of f_a at the critical value is positive, and f_a admits an invariant probability measure absolutely continuous with respect to the Lebesgue measure. We also show that given f_a* satisfying (a) and (b), condition (c) is satisfied for an open dense set of one-parameter families passing through f_a*.     

Switching systems with dwell time: Computing the maximal Lyapunov exponent

We study asymptotic stability of continuous-time systems with mode-dependent guaranteed dwell time. These systems are reformulated as special cases of a general class of mixed (discrete–continuous) linear switching systems on graphs, in which some modes correspond to discrete actions and some others correspond to continuous-time evolutions. Each discrete action has its own positive weight which accounts for its time-duration. We develop a theory of stability for the mixed systems; in particular, we prove the existence of an invariant Lyapunov norm for mixed systems on graphs and study its structure in various cases, including discrete-time systems for which discrete actions have inhomogeneous time durations. This allows us to adapt recent methods for the joint spectral radius computation (Gripenberg’s algorithm and the Invariant Polytope Algorithm) to compute the Lyapunov exponent of mixed systems on graphs.     

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 coupling approach to estimating the Lyapunov exponent of stochastic max-plus linear systems

This paper addresses the problem of approximately computing the Lyapunov exponent of stochastic max-plus linear systems. Our approach allows for an efficient simulation of bounds for the Lyapunov exponent. We provide sufficient conditions for the convergence of the bounds. In particular, a perfect sampling scheme for the Lyapunov exponent is established. We illustrate the effectiveness of our bounds with an application to (real-life) railway systems.     

Computable examples of the maximal Lyapunov exponent

Summary Some new examples are given of sequences of matrix valued random variables for which it is possible to compute the maximal Lyapunov exponent. The examples are constructed by using a sequence of stopping times to group the original sequence into commuting blocks. If the original sequence is the outcome of independent Bernoulli trials with success probability p, then the maximal Lyapunov exponent may be expressed in terms of power series in p, with explicit formulae for the coefficients. The convexity of the maximal Lyapunov exponent as a function of p is discussed, as is an application to branching processes in a random environment.     

Lower bounds for the maximal Lyapunov exponent

Upper bounds for the maximal Lyapunov exponent,E, of a sequence of matrix-valued random variables are easy to come by asE is the infimum of a real-valued sequence. We shall show that under irreducibility conditions similar to those needed to prove the Perron-Frobenius theorem, one can find sequences which increase toE. As a byproduct of the proof we shall see that we may replace the matrix norm with the spectral radius when computingE in such cases. Finally, a sufficient condition for transience of random walk in a random environment is given.     

On the positivity of the upper Lyapunov exponent in one-parameter families of linear differential systems

For a special class of linear differential systems affinely depending on a real parameter m, we obtain tests for the positivity of the upper characteristic exponent λ(A _μ ) for all μ in some set J of positive Lebesgue measure.     

Calculating the Lyapunov exponent for generalized linear systems with exponentially distributed elements of the transition matrix

A stochastic dynamic system of second order is considered. The system evolution is described by a dynamic equation with a stochastic transition matrix, which is linear in the idempotent algebra with operations of maximum and addition. It is assumed that some entries of the matrix are zero constants and all other entries are mutually independent and exponentially distributed. The problem considered is the computation of the Lyapunov exponent, which is defined as the average asymptotic rate of growth of the state vector of the system. The known results related to this problem are limited to systems whose matrices have zero off-diagonal entries. In the cases of matrices with a zero row, zero diagonal entries, or only one zero entry, the Lyapunov exponent is calculated using an approach which is based on constructing and analyzing a certain sequence of one-dimensional distribution functions. The value of the Lyapunov exponent is calculated as the average value of a random variable determined by the limiting distribution of this sequence.     

Lyapunov exponent of a stochastic SIRS model

We consider a SIRS (susceptible–infected–removed–susceptible) model influenced by random perturbations. We prove that the solutions are positive for positive initial conditions and are global, that is, there is no finite explosion time. We present necessary and sufficient conditions for the almost sure asymptotic stability of the steady state of the stochastic system.     

Bifurcations of non-smooth systems

Non-smooth systems (namely piecewise-smooth systems) have received much attention in the last decade. Many contributions in this area show that theory and applications (to electronic circuits, mechanical systems, …) are relevant to problems in science and engineering. Specially, new bifurcations have been reported in the literature, and this was the topic of this minisymposium. Thus both bifurcation theory and its applications were included. Several contributions from different fields show that non-smooth bifurcations are a hot topic in research. Thus in this paper the reader can find contributions from electronics, energy markets and population dynamics. Also, a carefully-written specific algebraic software tool is presented.     

Generalized synchronization of chaotic systems by pure error dynamics and elaborate Lyapunov function

The generalized synchronization is studied by applying pure error dynamics and elaborate Lyapunov function in this paper. Generalized synchronization can be obtained by pure error dynamics without auxiliary numerical simulation, instead of current mixed error dynamics in which master state variables and slave state variables are presented. The elaborate Lyapunov function is applied rather than the current plain square sum Lyapunov function, deeply weakening the power of Lyapunov direct method. The scheme is successfully applied to both autonomous and nonautonomous double Mathieu systems with numerical simulations.     

Approximate and efficient calculation of dominant Lyapunov exponents of high-dimensional nonlinear dynamic systems

This short communication presents an efficient method for calculating dominant Lyapunov exponents (LEs) of high-dimensional nonlinear dynamic systems based on their reduced-order models obtained from the linear model reduction theory. Mathematical derivation shows that the LEs of the reduced-order models correspond to the dominant LEs of the original systems. Two numerical examples are provided to demonstrate the effectiveness of the method.     

On the Lyapunov spectrum for rational maps

We study the dimension spectrum for Lyapunov exponents for rational maps on the Riemann sphere.