Three forms of chaos and their associated attractors
type: Item
instance of: scholarly article
title: Three forms of chaos and their associated attractors
zbMATH Open document ID: 0752.26004
published in: Real Analysis Exchange
publication date: 1992-09-26T00:00:00Z
review text: There are different ways how to define chaos if we consider a continuous map of a real compact interval into itself. Namely, a map can be regarded as chaotic if its topological entropy is positive or if it possesses a scrambled set in the sense of Li and Yorke. A further possibility can be obtained by measuring the separation of trajectories via the map \(\omega_ f: x\to\omega(x,f)\), where \(\omega(x,f)\) is the set of all cluster points of the trajectory \(\{f^ n(x)\}^ \infty_{n=0}\). The property \(\omega_ f\not\in B_ 1\), where \(B_ 1\) is the first Baire class, can serve as a third definition of chaos which is strictly intermediate to the first two mentioned. The author investigates the existence of certain kinds of \(\omega\)-limit sets for functions chaotic in all discussed senses. Among others, a necessary and sufficient condition for \(\omega_ f\not\in B_ 1\) is given.
Mathematics Subject Classification ID: 26A18
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 26A21
zbMATH DE Number: 54526
zbMATH Keywords: \(\omega\)-limit set
zbMATH Keywords: chaos
zbMATH Keywords: topological entropy
zbMATH Keywords: first Baire class
author: Andrew M. Bruckner
reviewed by: Katarina Janková
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 54526