Pugh's global linearization for the nonautonomous unbounded system with $μ$-dichotomy via Lyapunov theory
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909692341518336 |
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| author | Lu, Weijie Xia, Yonghui |
| author_facet | Lu, Weijie Xia, Yonghui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02855 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pugh's global linearization for the nonautonomous unbounded system with $μ$-dichotomy via Lyapunov theory Lu, Weijie Xia, Yonghui Dynamical Systems Classical Analysis and ODEs 37D25, 37C86, 37C60 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. |
| title | Pugh's global linearization for the nonautonomous unbounded system with $μ$-dichotomy via Lyapunov theory |
| topic | Dynamical Systems Classical Analysis and ODEs 37D25, 37C86, 37C60 |
| url | https://arxiv.org/abs/2506.02855 |