Unstable attractors in manifolds
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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Official URL: http://www.ams.org/journals/tran/2010-362-07/S0002...
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|
|Official Date:||July 2010|
|Number of Pages:||27|
|Page Range:||pp. 3563-3589|
|Access rights to Published version:||Open Access|
|Funder:||Spain. Dirección General de Investigación Científica y Técnica (DGICYT)|
1. K. Athanassopoulos, Explosions near isolated unstable attractors, Pacific J. Math. 210 (2003),
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