Global linearization and fiber bundle structure of invariant manifolds

Author(s)

Eldering, Jaap ; Kvalheim, Matthew ^[1] ; Revzen, Shai ^{[1, 2]}

Affiliation

^[1] Univ Michigan, Dept EECS, Ann Arbor, MI 48109 - USA

^[2] Univ Michigan, Dept EEB, Ann Arbor, MI 48109 - USA

• Document type: Journal article
• Source: Nonlinearity; v. 31, n. 9, p. 4202-4245, SEP 2018.
• Web of Science Citations: 1

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. (AU)

• FAPESP's process: 15/25947-6 - Stability and synchronization in power networks
• Grantee: Jacob Eldering
• Support Opportunities: Scholarships in Brazil - Post-Doctoral