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The Lyapunov Characteristic Exponents and Their Computation

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Dynamics of Small Solar System Bodies and Exoplanets

Part of the book series: Lecture Notes in Physics ((LNP,volume 790))

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 [102], 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. [14], 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.

For want of a nail the shoe was lost.

For want of a shoe the horse was lost.

For want of a horse the rider was lost.

For want of a rider the battle was lost.

For want of a battle the kingdom was lost.

And all for the want of a horseshoe nail.

For Want of a Nail (proverbial rhyme)

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  1. Astronomie et Systèmes Dynamiques, IMCCE, Observatoire de Paris, 77 avenue Denfert–Rochereau, F-75014, Paris, France

    Ch. Skokos

  2. Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187, Dresden, Germany

    Ch. Skokos

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  2. Astronomisches Institut, Universität Wien, Türkenschanzstr. 17, Wien, 1180, Austria

    Rudolf Dvorak

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Skokos, C. (2010). The Lyapunov Characteristic Exponents and Their Computation. In: Souchay, J., Dvorak, R. (eds) Dynamics of Small Solar System Bodies and Exoplanets. Lecture Notes in Physics, vol 790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04458-8_2

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