Differentiability of Palmer's linearization Theorem and converse result for density functions

TL;DR

This paper investigates the differentiability of the homeomorphism in Palmer's linearization theorem for stable linear systems and extends a converse result for density functions in nonautonomous nonlinear systems.

Contribution

It provides sufficient conditions for the homeomorphism to be a $C^{2}$ diffeomorphism and generalizes a converse density function result to nonautonomous systems.

Findings

Conditions for $H$ to be a $C^{2}$ diffeomorphism are established.

A generalized converse result for density functions in nonautonomous systems is presented.

Theoretical insights into the differentiability properties of Palmer's linearization are provided.

Abstract

We study differentiability properties in a particular case of the Palmer's linearization Theorem, which states the existence of an homeomorphism $H$ between the solutions of a linear ODE system having exponential dichotomy and a quasilinear system. Indeed, if the linear system is uniformly asymptotically stable, sufficient conditions ensuring that $H$ is a $C^{2}$ preserving orientation diffeomorphism are given. As an application, we generalize a converse result of density functions for a nonlinear system in the nonautonomous case.