Global linearization and fiber bundle structure of invariant manifolds
| Autor: | Eldering, Jaap, Kvalheim, Matthew, Revzen, Shai |
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| Rok vydání: | 2017 |
| Předmět: | |
| Zdroj: | Nonlinearity 31.9 (2018) pp. 4202--4245 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1361-6544/aaca8d |
| Popis: | We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We restrict our attention to continuous-time dynamical systems, or flows. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a $C^k$ disk bundle structure if the local stable foliation is assumed $C^k$. We then show that the dynamics restricted to the stable manifold of a compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM. Moreover, we give conditions ensuring the existence of a global $C^k$ linearizing conjugacy. We also prove a $C^k$ global linearization result for inflowing NAIMs; we believe that even the local version of this result is new, and may be useful in applications to slow-fast systems. We illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel Normal Form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form. Comment: 40 pages, 4 figures. Version as accepted for publication with only minor changes |
| Databáze: | arXiv |
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