Global linearization and fiber bundle structure of invariant manifolds
| Autor: | Matthew D. Kvalheim, Jaap Eldering, Shai Revzen |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
Dynamical systems theory 37D10 34E15 70K70 Applied Mathematics 010102 general mathematics Invariant manifold General Physics and Astronomy Statistical and Nonlinear Physics Dynamical Systems (math.DS) 01 natural sciences Stable manifold 010305 fluids & plasmas Bundle 0103 physical sciences FOS: Mathematics Fiber bundle Normally hyperbolic invariant manifold 0101 mathematics Invariant (mathematics) Mathematics - Dynamical Systems Topological conjugacy Mathematics::Symplectic Geometry Mathematical Physics Mathematics |
| 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. 40 pages, 4 figures. Version as accepted for publication with only minor changes |
| Databáze: | OpenAIRE |
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