Lyapunov Exponents for Discrete Time-Varying Systems

Czornik, Adam; Nawrat, Aleksander; Niezabitowski, Michał

doi:10.1007/978-3-642-31665-4_3

Cited by 13 publications

(10 citation statements)

References 36 publications

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“…There are several approaches to defining the exponential growth rate, all of which turn out to be equivalent. A trajectory based definition considers Lyapunov exponents [36][37][38][39][40][41][42][43][44][45][46][47][48] of individual trajectories, which are defined in the discrete-time case as…”

Section: Lyapunov Exponents Methodsmentioning

confidence: 99%

“…The quantity ln ρ(d) is known in the control theory literature as the greatest Lyapunov exponent. Properties of Lyapunov exponents were studied in [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51], where sufficient conditions so that Lyapunov exponents characterizes exponential stability for switched systems are shown. In particular, it was shown that the system (1) is absolutely stable if and only if sup…”

Section: Lyapunov Exponents Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Stability and controllability of switched systems

Klamka¹,

Czornik²,

Niezabitowski³

2013

Bulletin of the Polish Academy of Sciences: Technical Sciences

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Abstract. The study of properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. This paper aims to briefly survey recent results on stability and controllability of switched linear systems. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability. After that, we review the controllability results.

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Section: Lyapunov Exponents Methodsmentioning

confidence: 99%

Section: Lyapunov Exponents Methodsmentioning

confidence: 99%

Stability and controllability of switched systems

Klamka¹,

Czornik²,

Niezabitowski³

2013

Bulletin of the Polish Academy of Sciences: Technical Sciences

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“…Note that there are various essential generalizations of LCEs or LEs (see, e.g. [7,49,8,23,10]). See also some recent papers on LEs application [11,55,41,44].…”

Section: ∀Tmentioning

confidence: 99%

Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations

Kuznetsov¹,

Alexeeva²,

Леонов³

2016

Nonlinear Dyn

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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 singular values of the fundamental matrix of linearized system (Lyapunov exponents, LEs). In this work the relation between Lyapunov exponents and Lyapunov characteristic exponents is discussed. The invariance of Lyapunov exponents for regular and irregular linearizations under the change of coordinates is demonstrated.

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“…However, determining LEs remains the subject of many works and grown into a real software industry for modern nonlinear physics (see, e.g. [Hegger et al, 1999;Barreira & Pesin, 2001;Skokos, 2010;Czornik et al, 2013;Pikovsky & Politi, 2016;Vallejo & Sanjuan, 2017] and others). Nowadays there are two widely used definitions 1 , of the LEs: via the exponential growth rates of norms of the fundamental matrix columns [Lyapunov, 1892] and via the exponential growth rates of the sigular values of fundamental matrix [Oseledets, 1968].…”

Section: Introductionmentioning

confidence: 99%

Matlab Code for Lyapunov Exponents of Fractional-Order Systems

Danca

Kuznetsov

2018

Int. J. Bifurcation Chaos

142

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In this paper the Benettin-Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo's derivative and the corresponding Matlab code are presented. First it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. The underlying numerical method to solve the extended system of fractional order, composed of the initial value problem and the variational system, is the predictor-corrector Adams-Bashforth-Moulton for fractional differential equations. The Matlab program prints and plots the Lyapunov exponents as function of time. Also, the programs to obtain Lyapunov exponents as function of the bifurcation parameter and as function of the fractional order are described. The Matlab program for Lyapunov exponents is developed from an existing Matlab program for Lyapunov exponents of integer order. To decrease the computing time, a fast Matlab program which implements the Adams-Bashforth-Moulton method, is utilized. Four representative examples are considered.

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Lyapunov Exponents for Discrete Time-Varying Systems

Cited by 13 publications

References 36 publications

Stability and controllability of switched systems

Stability and controllability of switched systems

Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations

Matlab Code for Lyapunov Exponents of Fractional-Order Systems

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