Asymptotic stability equals exponential stability -- while you twist your eyes

TL;DR

This paper proves that if two vector fields make a point globally asymptotically stable, then a continuous transformation exists between these fields that preserves this stability.

Contribution

It establishes a homotopy between vector fields maintaining global asymptotic stability, linking stability properties through continuous deformation.

Findings

Existence of a homotopy preserving GAS

Stability equivalence under continuous deformation

Theoretical connection between asymptotic and exponential stability

Abstract

Suppose that two vector fields on a smooth manifold render some equilibrium point globally asymptotically stable (GAS). We show that there exists a homotopy between the corresponding semiflows such that this point remains GAS along this homotopy.