Unstable attractors in manifolds

Abstract

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.

An approach to the shape Conley index without index pairs

“…Now, it was first proved by Hastings [13] that if an attractor K is stable then the inclusion i : K −→ A(K) is a shape equivalence (subsequently different versions have been given in, for example, [10,11] and [26]), and if the converse implication were true it would certainly be a nice criterion to detect stability, and an easy one to check too. It is known to be true for flows on surfaces (see [23]) and we shall see now that it is also true for attractors with no external explosions: …”

“…As an immediate consequence of the claim above and the fact that W u (K)/K is closed (because W u (K) = K is compact) we see that the mapping r : M/K −→ W u (K)/K given by We will not give any examples where the corollary above can be applied because the papers [18] and [23] contain plenty of them. However, the methods used in both articles are very manifold oriented, so to speak; they do not work in arbitrary phase spaces whereas the techniques presented here do.…”

“…Proposition 4.1 means that the simplest unstable attractors are those whose explosion points lie all in K. These receive a special name, first introduced by Athanassopoulos in [1]: an attractor with no external explosions is an attractor K such that J + (p) ⊆ K for all p ∈ A(K) − K. They have been studied in [1], [18] and [23] when M is a manifold, and we will see that many of the results contained in those papers still hold for more general phase spaces. The reasons why this generality is of interest are presented at the end of the section.…”

“…It was shown in [18] and [23] that there exists manifolds M which cannot contain unstable attractors without external explosions, or otherwise stated every isolated connected attractor in M is either stable or has external explosions. The next corollary goes in the same line but it is applicable to arbitrary phase spaces.…”

Unstable manifold, Conley index and fixed points of flows

“…For a complete treatment of shape theory we refer the reader to [7,12,13,27,26,38] . The use of shape in dynamics is illustrated by the papers [18,15,19,21,24,32,33,36]. For information about basic aspects of dynamical systems we recommend [5,34,44] and for algebraic topology the books written by Hatcher [22] and Spanier [42] are very useful.…”