Global linearization and fiber bundle structure of invariant manifolds

J Eldering, M Kvalheim, Shai Revzen

Nonlinearity, vol. 31, no. 9, pp. 4202, 2018

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.

BibTeX

@article{eldering-2018-glfb,
  author = {Eldering, J and Kvalheim, M and Revzen, Shai},
  title = {Global linearization and fiber bundle structure of invariant manifolds},
  journal = {Nonlinearity},
  year = {2018},
  volume = {31},
  number = {9},
  pages = {4202},
  doi = {10.1088/1361-6544/aaca8d},
  editor = {de la Llave, Rafael},
  cvtopic = {dsctrl}
}