Topological chaos, braiding and bifurcation of almost-cyclic sets

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dc.contributorVirginia Techen
dc.contributor.authorGrover, Piyushen
dc.contributor.authorRoss, Shane D.en
dc.contributor.authorStremler, Mark A.en
dc.contributor.authorKumar, Pankajen
dc.contributor.departmentBiomedical Engineering and Mechanicsen
dc.date.accessed2013-11-20en
dc.date.accessioned2013-12-04T14:59:21Zen
dc.date.available2013-12-04T14:59:21Zen
dc.date.issued2012-12-01en
dc.description.abstractIn certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or " ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.en
dc.description.sponsorshipNational Science Foundation 1150456en
dc.format.mimetypeapplication/pdfen
dc.identifier.citationGrover, Piyush and Ross, Shane D. and Stremler, Mark A. and Kumar, Pankaj, “Topological chaos, braiding and bifurcation of almost-cyclic sets,” Chaos 22, 043135 (2012), DOI:http://dx.doi.org/10.1063/1.4768666en
dc.identifier.doihttps://doi.org/10.1063/1.4768666en
dc.identifier.issn1054-1500en
dc.identifier.urihttp://hdl.handle.net/10919/24395en
dc.identifier.urlhttp://scitation.aip.org/content/aip/journal/chaos/22/4/10.1063/1.4768666en
dc.language.isoenen
dc.publisherAmerican Institute of Physicsen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectDynamical-systemsen
dc.subjectInvarient setsen
dc.subjectCoherent structuresen
dc.subject2-dimensional mapsen
dc.subjectFluid mechanicsen
dc.subjectEntropyen
dc.subjectTransporten
dc.subjectAdvectionen
dc.subjectApproximationen
dc.subjectManifoldsen
dc.titleTopological chaos, braiding and bifurcation of almost-cyclic setsen
dc.title.serialChaosen
dc.typeArticle - Refereeden
dc.type.dcmitypeTexten

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