The Lyapunov Characteristic Exponents and their computation

TL;DR

This paper surveys the theory and numerical methods for computing Lyapunov Characteristic Exponents (LCEs), essential for analyzing chaos in dynamical systems, covering algorithms, theoretical foundations, and applications to various systems.

Contribution

It provides a comprehensive review of the theoretical basis and numerical algorithms for calculating LCEs, including recent developments and their relation to chaos detection techniques.

Findings

Detailed analysis of the multiplicative ergodic theorem of Oseledec

Comparison of algorithms for computing maximal and multiple LCEs

Discussion of LCE computation in conservative and dissipative systems

Abstract

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.