Unstable attractors in manifolds

Abstract

Assume that K is a compact attractor with basin of attraction A(K) for some continuous flow Φ 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 Poincaré 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. Assume that K is a compact attractor with basin of attraction A(K) for some continuous flow Φ 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 Poincaré 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. © 2010 American Mathematical Society.