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Unstable attractors in manifolds
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SánchezGabites, J. J.. (2010) Unstable attractors in manifolds. Transactions of the American Mathematical Society, Vol.362 (No.7). pp. 35633589. ISSN 00029947

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Official URL: http://www.ams.org/journals/tran/201036207/S0002...
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 MorseConley 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:  00029947 
Official Date:  July 2010 
Volume:  Vol.362 
Number:  No.7 
Number of Pages:  27 
Page Range:  pp. 35633589 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Open Access 
Funder:  Spain. Dirección General de Investigación Científica y Técnica (DGICYT) 
References:  1. K. Athanassopoulos, Explosions near isolated unstable attractors, Pacific J. Math. 210 (2003), 
URI:  http://wrap.warwick.ac.uk/id/eprint/5699 
Data sourced from Thomson Reuters' Web of Knowledge
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