On topological properties of closed attractors

TL;DR

This paper investigates the topological structure of closed, non-compact attractors in dynamical systems, establishing conditions under which they are homotopy equivalent to their domains of attraction to enhance understanding of feedback stabilization.

Contribution

It characterizes when non-compact, closed attractors are homotopy equivalent to their attraction domains, advancing the structural analysis of such systems.

Findings

Homotopy equivalence conditions identified for non-compact attractors

Enhanced understanding of feedback stabilization in non-compact settings

Structural insights into attractor domains in locally compact spaces

Abstract

The notion of an attractor has various definitions in the theory of dynamical systems. Under compactness assumptions, several of those definitions coincide and the theory is rather complete. However, without compactness, the picture becomes blurry. To improve our understanding, we characterize in this work when a closed, not necessarily compact, asymptotically stable attractor on a locally compact metric space is homotopy equivalent to its domain of attraction. This enables a further structural study of the corresponding feedback stabilization problem.