Global linearization and fiber bundle structure of invariant manifolds

dc.contributor.authorEldering, Jaap
dc.contributor.authorKvalheim, Matthew D.
dc.contributor.authorRevzen, Shai
dc.date.accessioned2024-12-11T17:02:19Z
dc.date.available2024-12-11T17:02:19Z
dc.date.issued2018-08-02
dc.description.abstractWe 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.
dc.description.sponsorshipJaap Eldering performed the major share of his contribution to this work while holding a postdoctoral position at ICMC, University of São Paulo, São Carlos, Brazil, supported by FAPESP grant 2015/25947-6. Matthew Kvalheim and Shai Revzen were supported by ARO grants W911NF-14-1-0573 and W911NF-17-1-0306 to Revzen. Kvalheim would like to thank Ralf Spatzier for many helpful conversations related to Theorem 1. We also thank the two anonymous referees and the editors for finding a mistake in a lemma, for bringing the reference [JT09] to our attention, and for their many useful suggestions.
dc.description.urihttps://iopscience.iop.org/article/10.1088/1361-6544/aaca8d
dc.format.extent40 pages
dc.genrejournal articles
dc.genrepreprints
dc.identifierdoi:10.13016/m2dqud-u47z
dc.identifier.citationEldering, 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.
dc.identifier.urihttps://iopscience.iop.org/article/10.1088/1361-6544/aaca8d
dc.identifier.urihttp://hdl.handle.net/11603/37051
dc.language.isoen_US
dc.publisherIOP Science
dc.relation.isAvailableAtThe University of Maryland, Baltimore County (UMBC)
dc.relation.ispartofUMBC Mathematics and Statistics Department
dc.rightsThis 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.
dc.titleGlobal linearization and fiber bundle structure of invariant manifolds
dc.typeText
dcterms.creatorhttps://orcid.org/0000-0002-2662-6760

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