What is a chaotic attractor?
type: Item
instance of: scholarly article
title: What is a chaotic attractor?
zbMATH Open document ID: 1175.37039
author: Clark Robinson
published in: Qualitative Theory of Dynamical Systems
publication date: 2009-04-14T00:00:00Z
review text: The author discusses the notion of chaos and chaotic attractors. After \textit{T.-Y. Li} and \textit{J. A. Yorke} [Am. Math. Mon. 82, 985--992 (1975; Zbl 0351.92021)] first used the word \textit{chaos} in deterministic systems, \textit{R. Devaney} [An introduction to chaotic dynamical systems. Redwood City, CA etc.: Addison-Wesley Publishing Company (1989; Zbl 0695.58002)] gave the first mathematical definition for a chaotic map. Since then, several different definitions of chaos have been proposed, see [\textit{M. Martelli, M. Dang}, and \textit{T. Seph}, Math. Mag. 71, No. 2, 112--122 (1998; Zbl 1008.37014)] for an overview. The author gives yet another definition (see the following Definition 3.3) of a chaotic diffeomorphism on an invariant set, and of a chaotic attractor, which is similar to the definition of \textit{M. Martelli} [Introduction to discrete dynamical systems and chaos. New York, NY: Wiley (1999; Zbl 1127.37300)] and \textit{M. Martelli, M. Dang}, and \textit{T. Seph} [loc. cit.]. Definition 3.3. A diffeomorphism \textbf{F} is said to be chaotic on an invariant set \textbf{A} provided that: (a) \textbf{F} has sensitive dependence on initial conditions when restricted to \textbf{A}. (b) \textbf{F} is topologically transitive. A set \textbf{A} is called a chaotic attractor for \textbf{F} if \textbf{A} is an attractor and \textbf{F} is chaotic on \textbf{A}. The author gives several examples to illustrate his definition, in particular an example of a chaotic attractor in his sense which has zero entropy and is not an attractor in the sense of \textit{K. T. Alligood, T. D. Sauer}, and \textit{J. A. Yorke} [Chaos. An introduction to dynamical systems. New York, NY: Springer (1996; Zbl 0867.58043)].
reviewed by: Peter Giesl
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 34C28
Mathematics Subject Classification ID: 34D45
zbMATH DE Number: 5542789
zbMATH Keywords: chaos
zbMATH Keywords: attractor
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5542789