On Wilson's theorem about domains of attraction and tubular neighborhoods

TL;DR

This paper proves that the domain of attraction of a compact asymptotically stable submanifold in a smooth manifold is homeomorphic to its tubular neighborhood, emphasizing the importance of compactness with counterexamples.

Contribution

It establishes a topological equivalence between the domain of attraction and tubular neighborhoods for compact stable submanifolds, extending Wilson's theorem.

Findings

Domain of attraction is homeomorphic to tubular neighborhood for compact attractors.

Counterexamples show the necessity of compactness for this homeomorphism.

Highlights the role of compactness in stability analysis.

Abstract

In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold of a finite-dimensional smooth manifold of an autonomous system is homeomorphic to its tubular neighborhood. The compactness of the attractor is crucial, without which this result is false; two counterexamples are provided to demonstrate this.