Pugh's global linearization for the nonautonomous unbounded system with $\mu$-dichotomy via Lyapunov theory

TL;DR

This paper extends the classical global linearization theorem to nonautonomous systems with unbounded perturbations, using Lyapunov functions and the concept of nonuniform ichotomy, broadening the scope of linearization results.

Contribution

It introduces a new unbounded global linearization theorem for nonautonomous systems with ichotomy, utilizing Lyapunov functions and a splitting lemma for the first time.

Findings

Established a Lyapunov function framework for ichotomy systems

Characterized ichotomy via quadratic Lyapunov functions

Derived a linearization transformation for nonuniform ichotomy systems

Abstract

The classical global linearization theorem for autonomous system given in [C. Pugh, Amer. J. Math., 91 (1969) 363-367] requires that nonlinear system with hyperbolicity satisfies boundedness and Lipschitz continuity.In this paper, we establish an {\em unbounded} global linearization theorem for nonautonomous systems subject to unbounded Lipschitz perturbations, under the assumption that the linear system admits a nonuniform $\mu$-dichotomy (more general than classical exponential dichotomy). To this end, we first develop a comprehensive Lyapunov function framework for systems exhibiting nonuniform $\mu$-dichotomy. Subsequently, we establish a characterization of nonuniform $\mu$-dichotomy in terms of strict quadratic Lyapunov functions. Building upon these theoretical foundations, we then employ these Lyapunov functions to derive a linearization result under the nonuniform $\mu$-dichotomy assumption. In the proof, we give a splitting lemma for nonuniform $\mu$-dichotomy to decouple hyperbolic system into a contractive system and an expansive system. Then we construct a transformation to linearize contractive/expansive system, which is defined by the crossing time with respect to the unit sphere.