$C^{1}$ Smoothness of the Conjugacy in a Delayed Nonautonomous Hartman--Grobman Theorem via $\mu$-Dichotomies

TL;DR

This paper proves that under certain conditions, the conjugacy in a nonautonomous delayed system with a linear part satisfying a $-dichotomy is a $C^{1}$ diffeomorphism, extending the Hartman--Grobman Theorem.

Contribution

It establishes the $C^{1}$ smoothness of the conjugacy in delayed nonautonomous systems with $-dichotomies, a significant extension of classical results.

Findings

Conjugacy is $C^{1}$ under specific conditions.

Extension of Hartman--Grobman Theorem to delayed systems.

Conditions on nonlinear perturbations ensure smooth conjugacy.

Abstract

We investigate the differentiability of the conjugacy in a nonautonomous version of the Hartman--Grobman Theorem for systems with finite delay, where the linear part satisfies a $\mu$-dichotomy. Under suitable conditions on the nonlinear perturbation, the conjugacy is shown to be a $C^{1}$ diffeomorphism.