Smoothness of Topological Equivalence on the Half Line for Nonautonomous Systems

TL;DR

This paper investigates the smoothness of topological equivalence between stable linear nonautonomous systems and their nonlinear perturbations on the half line, establishing conditions for differentiability and stability preservation.

Contribution

It introduces a $C^{r}$--smooth homeomorphism on the positive half line and analyzes its properties in relation to system stability and topological equivalence.

Findings

Constructed a $C^{r}$--smooth homeomorphism inspired by Palmer's work.

Provided sufficient conditions for the homeomorphism's smoothness.

Studied the preservation of uniform stability properties under the homeomorphism.

Abstract

We study the differentiability properties of the topological equivalence between a uniformly asymptotically stable linear nonautonomous system and a perturbed system with suitable nonlinearities. For this purpose, we construct a uniformly continuous homeomorphism inspired in the Palmer's one restricted to the positive half line, providing sufficient conditions ensuring its $C^{r}$--smoothness. Additionally, we study the preservation of the uniform stability properties by this homeomorphism.