Phase resetting effects for robust cycles between chaotic sets

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Phase resetting effects for robust cycles between chaotic sets

Show simple item record Ashwin, Peter en_GB Field, Michael en_GB Rucklidge, Alastair M. en_GB Sturman, Rob en_GB
dc.contributor.department University of Exeter; University of Houston, Texas; University of Leeds en_GB 2008-02-27T14:17:32Z en_GB 2011-01-25T10:33:42Z en_US 2013-03-20T12:32:23Z 2003 en_GB
dc.description.abstract In 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.citation 13 (3), pp. 973-981 en_GB
dc.identifier.doi 10.1063/1.1586531 en_GB
dc.identifier.uri en_GB
dc.language.iso en en_GB
dc.publisher American Institute of Physics en_GB
dc.relation.url en_GB
dc.subject chaos en_GB
dc.subject set theory en_GB
dc.subject Lyapunov methods en_GB
dc.title Phase resetting effects for robust cycles between chaotic sets en_GB
dc.type Article en_GB 2008-02-27T14:17:32Z en_GB 2011-01-25T10:33:42Z en_US 2013-03-20T12:32:23Z
dc.identifier.issn 1054-1500 en_GB
dc.description Copyright © 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 en_GB
dc.identifier.journal Chaos en_GB

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