A generalized global Hartman-Grobman theorem for asymptotically stable semiflows

TL;DR

This paper extends a generalized global Hartman-Grobman theorem to include possibly discontinuous vector fields generating asymptotically stable semiflows, broadening its applicability beyond continuous fields.

Contribution

It generalizes the theorem to discontinuous vector fields and semiflows, removing the hyperbolicity assumption and expanding the scope of stability analysis.

Findings

The theorem applies to a broader class of vector fields, including discontinuous ones.

It leverages topological properties of Lyapunov functions for stability analysis.

The extension covers semiflows, not just flows, under asymptotic stability.

Abstract

Recently, Kvalheim and Sontag provided a generalized global Hartman-Grobman theorem for equilibria under asymptotically stable continuous vector fields. By leveraging topological properties of Lyapunov functions, their theorem works without assuming hyperbolicity. We extend their theorem to a class of possibly discontinuous vector fields, in particular, to vector fields generating asymptotically stable semiflows.