A hybrid method for computing Lyapunov exponents
type: Item
instance of: scholarly article
title: A hybrid method for computing Lyapunov exponents
zbMATH Open document ID: 1178.37117
author: Wolf-Jürgen Beyn
author: Alexander Lust
published in: Numerische Mathematik
publication date: 2009-09-14T00:00:00Z
review text: The authors propose a numerical method based on the discrete QR-method combined with spatial integration to compute all the Lyapunov exponents of a dynamical system. This is a combination of the usual time averages in the computation of Lyapunov exponents (in this case a common QR discrete method) and the method of spatial averages introduced by \textit{P. J. Alston} and \textit{M. Dellnitz} [Comput. Methods Appl. Mech. Eng. 170, No.~3--4, 223--237 (1999; Zbl 0945.65142)] using the software GAIO. As explained in the work, it is based on the computation of the sequence of integrals converging to the Lyapunov exponents from the following theorem: Denoting by \(\lambda_1 \geq \ldots \geq \lambda_d\) the Lyapunov exponents (including multiplicities) of a discrete dynamical system \(g: M \rightarrow M\) (\(C^1\)-diffeomorphism on a \(d\)-dimensional submanifold \(M\) of \(\mathbb{R}^k\)), and further assuming that, \[ \lambda_j=\lim_{n \to \infty} \frac{1}{n} \ln R_{jj}(Dg^n(x)) \quad \mu \text{ a.e.}, \quad j=1,\ldots,d. \] Then one has, \[ \lambda_j=\lim_{n \to \infty} \frac{1}{n} \int \ln R_{jj}(Dg^n(x)) d\mu(x), \quad j=1,\ldots,d, \] where \(R_{jj}(A)\) stands for the \(j\)th diagonal element of the \(R\)-factor in the QR-decomposition of a nonsingular matrix \(A\) and \(Dg^n\) denotes the derivative of the \(n\)th iterate \(g^n\). In the paper, the methodology is applied to the Hénon system and to the time-1-map Lorenz system and the atmospheric model of \textit{R. V. Bekryaev} [Atmospheric and Oceanic Physics 31, No.~4, 455--464 (1996)]. An analysis of the errors under the conditions of the Oseledets theorem [\textit{V. I. Oseledets}, Trans. Mosc. Math. Soc. 19, 197--231 (1968); translation from Tr. Mosk. Mat. Obshch. 19, 179--210 (1968; Zbl 0236.93034); ] is presented (but will be proved in a forthcoming paper) and the extrapolation results to accelerate the convergence are implemented in these examples.
reviewed by: Josep J. Masdemont
Mathematics Subject Classification ID: 37M25
Mathematics Subject Classification ID: 65P40
zbMATH DE Number: 5603874
zbMATH Keywords: Lyapunov exponents
describes a project that uses: GAIO
MaRDI profile type: Publication
full work available at URL: s00211-009-0236-4
OpenAlex ID: W2043871925
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DOI: 10.1007/S00211-009-0236-4
description: scientific article; zbMATH DE number 5603874