Chain recurrence, semiflows, and gradients
type: Item
instance of: scholarly article
title: Chain recurrence, semiflows, and gradients
zbMATH Open document ID: 0832.34041
DOI: 10.1007/BF02219371
published in: Journal of Dynamics and Differential Equations
publication date: 1996-02-20T00:00:00Z
review text: Let \((X, d)\) be a metric space. A semiflow is a continuous map \(\phi: [0, \infty)\times X\to X\) satisfying (i) \(\forall x\in X\); \(\phi(0, x)= x\), and (ii) \(\forall s, t\geq 0 \forall x\in X\); \(\phi(s, \phi(t, x))= \phi(s+ t, x)\). The time-one map of \(\phi\) is denoted by \(h: h(x)= \phi(1, x)\). In the case of \(X\) compact, \textit{C. Conley} [Isolated invariant sets and the Morse index, Am. Math. Soc., Providence, Rhode Island (1978; Zbl 0397.34056)] gave a characterization of the chain recurrent set \({\mathcal C}{\mathcal R}(\phi)\) of \(\phi\) in terms of attractors of \(\phi\). Namely, he proved that \(X\backslash {\mathcal C}{\mathcal R}(\phi)= \bigcup[B(A)\backslash A]\) as \(A\) varies over all attractors of \(\phi\) and where \(B(A)\) denotes the basin of attraction of \(A\). In the present paper, the author extends that result to the case of an arbitrary metric space. Moreover, the author shows that the chain recurrent set of \(\phi\) is the same as the chain recurrent set of its time-one map \(h\). Applying these results, the author establishes some conditions on a real- valued function which ensure that the semiflow generated by its gradient has only equilibria in its chain recurrent set.
reviewed by: J.Rogowski
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 800417
zbMATH Keywords: semiflow
zbMATH Keywords: attractors
zbMATH Keywords: chain recurrent set
author: Mike Hurley
MaRDI profile type: Publication
cites work: Q5558469
cites work: Q4180600
cites work: Attractors: Persistence, and Density of Their Basins
cites work: Chain recurrence and attraction in non-compact spaces
cites work: Noncompact Chain Recurrence and Attraction
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cites work: Analytic Extensions of Differentiable Functions Defined in Closed Sets
cites work: A function not constant on a connected set of critical points
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description: scientific article; zbMATH DE number 800417