Compactification for asymptotically autonomous dynamical systems: Theory, applications and invariant manifolds

dc.check.date2022-05-13
dc.check.infoAccess to this article is restricted until 12 months after publication by request of the publisher.en
dc.contributor.authorWieczorek, Sebastian
dc.contributor.authorXie, Chun
dc.contributor.authorJones, Chris K.R.T.
dc.contributor.funderOffice of Naval Researchen
dc.contributor.funderNational Science Foundationen
dc.contributor.funderHorizon 2020en
dc.date.accessioned2021-06-25T08:34:46Z
dc.date.available2021-06-25T08:34:46Z
dc.date.issued2021-05-13
dc.date.updated2021-06-25T08:22:30Z
dc.description.abstractWe develop a general compactification framework to facilitate analysis of nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyze the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework, and their analysis can be greatly simplified by the compactification.en
dc.description.sponsorshipOffice of Naval Research (Grant No. N00014-18-1-2204); National Science Foundation (Grant No. DMS-1722578)en
dc.description.statusPeer revieweden
dc.description.versionAccepted Versionen
dc.format.mimetypeapplication/pdfen
dc.identifier.citationWieczorek, S., Xie, C. and Jones, C. K.R.T. (2021) 'Compactification for asymptotically autonomous dynamical systems: Theory, applications and invariant manifolds', Nonlinearity, 34(5), pp. 2970-3000. doi: 10.1088/1361-6544/abe456en
dc.identifier.doi10.1088/1361-6544/abe456en
dc.identifier.eissn1361-6544
dc.identifier.endpage3000en
dc.identifier.issn0951-7715
dc.identifier.issued5en
dc.identifier.journaltitleNonlinearityen
dc.identifier.startpage2970en
dc.identifier.urihttps://hdl.handle.net/10468/11487
dc.identifier.volume34en
dc.language.isoenen
dc.publisherIOP Publishingen
dc.relation.projectinfo:eu-repo/grantAgreement/EC/H2020::MSCA-ITN-ETN/643073/EU/Critical Transitions in Complex Systems/CRITICSen
dc.rights© 2021, IOP Publishing Ltd & London Mathematical Society. This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6544/abe456en
dc.subjectAsymptotically autonomous dynamical systemsen
dc.subjectCompactificationen
dc.subjectNonautonomous instabilitiesen
dc.subjectNonlinear wavesen
dc.subjectRate-induced tippingen
dc.titleCompactification for asymptotically autonomous dynamical systems: Theory, applications and invariant manifoldsen
dc.typeArticle (peer-reviewed)en
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