# Exponential trichotomy and global linearization of non-autonomous differential equations

## TL;DR

This paper extends the Hartman-Grobman linearization theorem to non-autonomous differential equations with exponential trichotomy, establishing new conditions and properties for the linearization functions, including Hölder continuity and non-periodicity.

## Contribution

It introduces the first linearization results for systems with exponential trichotomy, generalizing Palmer's theorem beyond dichotomy and analyzing the properties of the linearization functions.

## Key findings

- Established linearization theorems for systems with exponential trichotomy.
- Proved Hölder continuity of the linearization functions and their inverses.
- Showed that for periodic systems, the linearization functions lack periodicity or asymptotic periodicity.

## Abstract

Hartman-Grobman theorem was initially extended to the non-autonomous cases by Palmer. Usually, dichotomy is an essential condition of Palmer's linearization theorem. Is Palmer's linearization theorem valid for the systems with trichotomy? In this paper, we obtain new versions of the linearization theorem if linear system admits exponential trichotomy on $\mathbb{R}$. { Furthermore, the equivalent function $\mathscr H(t,x)$ and its inverse $\mathscr L(t,y)$ of our linearization theorems are H\"{o}lder continuous}. In addition, if a system is periodic, we find the equivalent function $\mathscr H(t,x)$ and its inverse $\mathscr L(t,y)$ of our linearization theorems do not have periodicity or asymptotical periodicity. To the best of our knowledge, this is the first paper studying the linearization with exponential trichotomy.