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

Abstract

In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold in a finite-dimensional smooth manifold of an autonomous system is homeomorphic to the submanifold’s tubular neighborhood. The compactness of the submanifold is crucial, without which this result is false; two counterexamples are provided to demonstrate this. • We show a fundamental result about the domain of attraction of a compact asymptotically stable submanifold. • We provide counterexamples to emphasize the key role of the compactness of the submanifold. • The aforementioned result refines the existing one and facilitates the topological analysis of autonomous systems.