Global linearization and fiber bundle structure of invariant manifolds. (2nd August 2018)

• Record Type: Journal Article
• Authors: Eldering, Jaap; Kvalheim, Matthew; Revzen, Shai
• Is Part Of: Nonlinearity. Volume 31:Number 9(2018:Sep.)
• Page Start: 4202
• Page End: 4245
• Publication Date: 2018-08-02
• Subjects: invariant manifolds -- normal hyperbolicity -- linearization -- singular perturbation theory -- slow-fast systems; 37D10 -- 34E15 -- 70K70; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aaca8d
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: 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.