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dc.contributor.authorAshwin, Peteren_GB
dc.contributor.authorField, Michaelen_GB
dc.contributor.authorRucklidge, Alastair M.en_GB
dc.contributor.authorSturman, Roben_GB
dc.contributor.departmentUniversity of Exeter; University of Houston, Texas; University of Leedsen_GB
dc.date.accessioned2008-02-27T14:17:32Zen_GB
dc.date.accessioned2011-01-25T10:33:42Zen_GB
dc.date.accessioned2013-03-20T12:32:23Z
dc.date.issued2003en_GB
dc.description.abstractIn the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated, owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena, including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. In this paper we introduce and discuss an instructive example of an ordinary differential equation where one can observe and analyze robust cycling behavior. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a Rössler system), and/or saddle equilibria. For this model, we distinguish between cycling that includes phase resetting connections (where there is only one connecting trajectory) and more general non(phase) resetting cases, where there may be an infinite number (even a continuum) of connections. In the nonresetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability, whereas more general cases may give rise to "stuck on" cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase resetting and connection selection.en_GB
dc.identifier.citation13 (3), pp. 973-981en_GB
dc.identifier.doi10.1063/1.1586531en_GB
dc.identifier.urihttp://hdl.handle.net/10036/19256en_GB
dc.language.isoenen_GB
dc.publisherAmerican Institute of Physicsen_GB
dc.relation.urlhttp://link.aip.org/link/?cha/13/973en_GB
dc.subjectchaosen_GB
dc.subjectset theoryen_GB
dc.subjectLyapunov methodsen_GB
dc.titlePhase resetting effects for robust cycles between chaotic setsen_GB
dc.typeArticleen_GB
dc.date.available2008-02-27T14:17:32Zen_GB
dc.date.available2011-01-25T10:33:42Zen_GB
dc.date.available2013-03-20T12:32:23Z
dc.identifier.issn1054-1500en_GB
dc.descriptionCopyright © 2003 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Chaos 13 (2003) and may be found at http://link.aip.org/link/?cha/13/973en_GB
dc.identifier.journalChaosen_GB


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