Nonuniform Almost Reducibility of Nonautonomous Linear Differential Equations

TL;DR

This paper demonstrates that nonautonomous linear differential systems with nonuniform hyperbolicity can be approximately diagonalized with small perturbations, extending the concepts of reducibility to nonuniform settings.

Contribution

It introduces the notions of nonuniform almost reducibility and nonuniform contractibility, generalizing existing uniform concepts to nonuniform hyperbolic systems.

Findings

Systems can be expressed as diagonal plus small perturbation

Diagonal terms lie within the nonuniform exponential dichotomy spectrum

Extends reducibility concepts to nonuniform hyperbolic systems

Abstract

We prove that a linear nonautonomous differential system with nonuniform hyperbolicity on the half line can be expressed as diagonal system with a perturbation which is small enough. Moreover we show that the diagonal terms are contained in the nonuniform exponential dichotomy spectrum. For this purpose we introduce the concepts of \textit{nonuniform almost reducibility} and \textit{nonuniform contractibility} which are generalization of this notions originally defined in a uniform context.