Linearization of a nonautonomous unbounded system with nonuniform contraction: A Spectral Approach

TL;DR

This paper develops a spectral approach to establish topological equivalence between a nonautonomous linear system with nonuniform contraction and its unbounded nonlinear perturbation, using homeomorphisms based on spectral properties and Lyapunov functions.

Contribution

It introduces a novel spectral method to construct topological equivalences for nonautonomous systems with nonuniform contraction, extending classical linearization techniques.

Findings

Constructed a topological equivalence via homeomorphisms.

Reduced the linear system to a diagonal form using spectral properties.

Linked the nonlinear perturbation to the linear system through crossing times.

Abstract

For a nonautonomous linear system with nonuniform contraction, we construct a topological equivalence between this system and an unbounded nonlinear perturbation. This topological equivalence is constructed as a composition of homeomorphisms. The first one is set up by considering the fact that linear system is almost reducible to diagonal system with a small enough perturbation where the diagonal entries belong to spectrum of the nonuniform exponential dichotomy; and the second one is constructed in terms of the crossing times with respect to unit sphere of an adequate Lyapunov function associated to the linear system.