Sánchez-Gabites, J. J. (2010) Unstable attractors in manifolds. Transactions of the American Mathematical Society, Vol.362 (No.7). pp. 3563-3589. ISSN 0002-9947

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Abstract

Assume that K is a compact attractor with basin of attraction A(K) for some continuous flow phi in a space M. Stable attractors are very well known, but otherwise (without the stability assumption) the situation can be extremely wild. In this paper we consider the class of attractors with no external explosions, where a mild form of instability is allowed. After obtaining a simple description of the trajectories in A(K) - K we study how K sits in A(K) by performing an analysis of the Poincare polynomial of the pair (A(K), K). In case M is a surface we obtain a nice geometric characterization of attractors with no external explosions, as well as a converse to the well known fact that the inclusion of a stable attractor in its basin of attraction is a shape equivalence. Finally, we explore the strong relations which exist between the shape (in the sense of Borsuk) of K and the shape (in the intuitive sense) of the whole phase space M, much in the spirit of the Morse-Conley theory.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Attractors (Mathematics), Manifolds (Mathematics)
Journal or Publication Title:	Transactions of the American Mathematical Society
Publisher:	American Mathematical Society
ISSN:	0002-9947
Date:	July 2010
Volume:	Vol.362
Number:	No.7
Number of Pages:	27
Page Range:	pp. 3563-3589
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	Spain. Dirección General de Investigación Científica y Técnica (DGICYT)
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URI:	http://wrap.warwick.ac.uk/id/eprint/5699

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