Unstable attractors in manifolds

TL;DR

This paper investigates a specific class of attractors in dynamical systems called attractors with no external explosions, focusing on their cohomological properties and the relationship between their shape and the topology of the phase space.

Contribution

It introduces and studies the class of attractors with no external explosions, bridging the gap between stable and unstable attractors, and explores their cohomological and topological properties.

Findings

Characterization of cohomological properties of attractors with no external explosions.

Establishment of strong relations between shape and phase space topology.

Insights into the intermediate nature of these attractors between stable and unstable types.

Abstract

Let $M$ be a locally compact metric space endowed with a continuous flow $\phi : M \times \mathbb{R} \longrightarrow M$. Frequently an attractor $K$ for $\phi$ exists which is of interest, not only in itself but also the dynamics in its basin of attraction $\mathcal{A}(K)$. In this paper the class of {\sl attractors with no external explosions}, which is intermediate between the well known {\sl stable attractors} and the extremely wild {\sl unstable attractors}, is studied. We are mainly interested in their cohomological properties, as well as in the strong relations which exist between their shape (in the sense of Borsuk) and the topology of the phase space.