Sharpness of $C^0$ conjugacy for the non-autonomous differential equations with Lipschitzian perturbation

TL;DR

This paper investigates the regularity of conjugacies in non-autonomous differential equations with Lipschitz perturbations, demonstrating that the conjugacy can be Lipschitz but its inverse may only be Hölder continuous, and conjecturing this regularity is optimal.

Contribution

The paper constructs a counterexample showing the conjugacy's regularity is sharp and proves the conjecture for systems with linear contraction, advancing understanding of conjugacy regularity.

Findings

Counterexample shows conjugacy is Lipschitz, inverse is Hölder continuous

Conjecture that regularity is sharp is confirmed for systems with linear contraction

Special cases related to spectrum are analyzed

Abstract

The classical $C^0$ linearization theorem for the non-autonomous differential equations states the existence of a $C^0$ topological conjugacy between the nonlinear system and its linear part. That is, there exists a homeomorphism (equivalent function) $H$ sending the solutions of the nonlinear system onto those of its linear part. It is proved in the previous literature that the equivalent function $H$ and its inverse $G=H^{-1}$ are both H\"{o}lder continuous if the nonlinear perturbation is Lipschitzian. Questions: is it possible to improve the regularity? Is the regularity sharp? To answer this question, we construct a counterexample to show that the equivalent function $H$ is exactly Lipschitzian, but the inverse $G=H^{-1}$ is merely H\"{o}lder continuous. Furthermore, we propose a conjecture that such regularity of the homeomorphisms is sharp (it could not be improved anymore). We prove that the conjecture is true for the systems with linear contraction. Furthermore, we present the special cases of linear perturbation, which are closely related to the spectrum.