Multiscale linearization of nonautonomous systems

TL;DR

This paper establishes broad conditions under which linear nonautonomous systems and their nonlinear perturbations are topologically conjugate, without requiring asymptotic behavior assumptions, applicable to both discrete and continuous cases.

Contribution

It introduces general criteria for topological conjugacy of nonautonomous systems with nonlinear perturbations, allowing different control along various directions without asymptotic constraints.

Findings

Provided sufficient conditions for topological conjugacy

Applicable to both discrete and continuous systems

No asymptotic behavior assumptions needed

Abstract

We present sufficient conditions under which a given linear nonautonomous system and its nonlinear perturbation are topologically conjugated. Our conditions are of a very general form and provided that the nonlinear perturbations are well-behaved, we do not assume any asymptotic behaviour of the linear system. Moreover, the control on the nonlinear perturbations may differ along finitely many mutually complementary directions. We consider both the cases of one-sided discrete and continuous dynamics.