2025/06/03 by Weijie Lu, Yonghui Xia, Lu, Weijie +1
Engineering · Mathematics · Physics and Astronomy · #37C60 #37C86 #37D25 #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2506.02855
openalex publication_date 2025/06/03 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
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 μ-dichotomy (more general than classical exponential dichotomy). To this end, we first develop a comprehensive Lyapunov function framework for systems exhibiting nonuniform μ-dichotomy. Subsequently, we establish a characterization of nonuniform μ-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 μ-dichotomy assumption. In the proof, we give a splitting lemma for nonuniform μ-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.