Global linearization and fiber bundle structure of invariant manifolds

TL;DR

This paper investigates the global structure of invariant manifolds in dynamical systems, establishing their fiber bundle properties and conditions for global linearization, with applications to geometric singular perturbation theory.

Contribution

It proves that the global stable foliation of a NAIM forms a topological disk bundle and provides new conditions for global $C^k$ linearization, extending local results to the entire stable manifold.

Findings

Global stable foliation has a topological disk bundle structure.

Conditions for $C^k$ global linearization are established.

Applications include extending Fenichel Normal Form to entire stable manifolds.

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.