Stability concepts in topological dynamics by nonstandard methods
type: Item
instance of: scholarly article
title: Stability concepts in topological dynamics by nonstandard methods
zbMATH Open document ID: 0891.54016
DOI: 10.1023/A:1005840309425
author: Karl-Georg Schlesinger
published in: Acta Applicandae Mathematicae
publication date: 1998-07-12T00:00:00Z
review text: Motivated by, and using, methods of nonstandard analysis, the paper introduces four topologically-defined concepts of stability of a continuous semiflow on a compact subset \(K\) of \(R^{n}\). Standard versions of the results are also presented. Unfortunately, the paper lacks examples showing that the stability definitions it presents are interesting. The set of semiflows is topologized by \(d(f,g) = \sup_{x\in K, 0\leq t\leq 1}|f(t,x) - g(t,x)|\). The first (and most restrictive) of the definitions states that a flow \(f\) is strongly s-stable if for any \(\varepsilon > 0\) one can choose a neighborhood \(G\) of \(f\) such that for any \(g\in G\), \(|f(t,x) - g(t,x)|< \varepsilon \) for all \(t\) and \(x\). It seems to the reviewer that if the omega limit set for \(f\) of some point consists of more than one point, then defining \(g(t,x) \equiv f(kt, x)\) for some \(k\) near 1 shows that \(f\) is not strongly s-stable. The second stability definition (`deformational stability') requires that the semiflow have the property that \(q\) lie on the closure of the forward orbit of \(p\) whenever it is possible to find an \(\varepsilon\)-chain from \(p\) to \(q\) for each \(\varepsilon >0\). It is shown that any minimal flow has this property; for flows it seems clear that if the phase space is connected then the converse also holds. (On a connected phase space, if there is any ordered pair of points \((p,q)\) and \(\varepsilon >0\) where it is not possible to find an \(\varepsilon\)-chain from \(p\) to \(q\), then there is a nontrivial attractor-repellor pair \((A,A*)\) with \(A\), \(A*\) nonempty disjoint, invariant, and there are \(\varepsilon\)-chains going from \(A*\) to \(A\).) The third stability concept is related to the author's version of the pseudo-orbit tracing property (POTP). Both of these definitions suffer from the failure to consider the necessity of reparameterization of the \(t\) coordinates. The final stability concept is approximate stability. Roughly…
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 54J05
zbMATH DE Number: 1097677
reviewed by: Mike Hurley
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1097677