Eldering, JaapKvalheim, Matthew D.Revzen, Shai2024-12-112024-12-112018-08-02Eldering, Jaap, Matthew Kvalheim, and Shai Revzen. “Global Linearization and Fiber Bundle Structure of Invariant Manifolds.” Nonlinearity 31, no. 9 (August 2018): 4202. https://doi.org/10.1088/1361-6544/aaca8d.https://iopscience.iop.org/article/10.1088/1361-6544/aaca8dhttp://hdl.handle.net/11603/37051We 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 Ck disk bundle structure if the local stable foliation is assumed Ck. 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 Ck linearizing conjugacy. We also prove a Ck 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 pagesen-USThis item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author.Text