Smoothness of linearization by mixing parameters of dichotomy, bounded growth and perturbation

TL;DR

This paper investigates the smoothness of conjugacies between linear systems with nonuniform dichotomies and their quasilinear perturbations, showing how parameter adjustments can enhance smoothness intervals.

Contribution

It introduces a parameter-based framework to analyze and extend the smoothness of conjugacies in nonuniform linear systems with perturbations.

Findings

Broader smoothness intervals achieved through parameter modifications

Conditions for nonuniform exponential dichotomy and bounded growth detailed

Method applicable to a class of nonautonomous systems

Abstract

We study the smoothness properties of a global and nonautonomous topological conjugacy between a linear system and a quasilinear perturbation. The linear system exhibits a nonuniform exponential dichotomy with a nontrivial projector and nonuniform bounded growth property. Additionally, the quasilinear perturbation is dominated by an increasing exponential function. Emphasis is placed on employing a set of parameters to describe the conditions of dichotomy, bounded growth and quasilinear perturbations. Finally, we prove that modifying these conditions enables us to achieve a broader smoothness interval.