The Lyapunov Characteristic Exponents and their computation

Details The Lyapunov Characteristic Exponents and their computation The Lyapunov Characteristic Exponents and their computation

2008-11-06

arxiv.org/abs/0811.0882v2

Lect. Notes Phys. 790, 63-135 (2010)

Nonlinear Sciences

Chaotic Dynamics

74 pages, 8 figures, accepted for publication in Lecture Notes in Physics

Scientific paper

10.1007/978-3-642-04458-8_2

We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec \cite{O_68}, which provides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been extensively used as an indicator of chaos, and the algorithm of the so--called `standard method', developed by Benettin et al. \cite{BGGS_80b}, for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although, we are mainly interested in finite--dimensional conservative systems, i. e. autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed.

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