Cycling chaotic attractors in two models for dynamics with invariant subspaces

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Cycling chaotic attractors in two models for dynamics with invariant subspaces

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dc.contributor.author Ashwin, Peter en_GB
dc.contributor.author Rucklidge, Alastair M. en_GB
dc.contributor.author Sturman, Rob en_GB
dc.contributor.department University of Exeter; University of Leeds en_GB
dc.date.accessioned 2008-03-13T12:05:01Z en_GB
dc.date.accessioned 2011-01-25T10:33:48Z en_US
dc.date.accessioned 2013-03-20T12:31:12Z
dc.date.issued 2004 en_GB
dc.description.abstract Nonergodic attractors can robustly appear in symmetric systems as structurally stable cycles between saddle-type invariant sets. These saddles may be chaotic giving rise to "cycling chaos." The robustness of such attractors appears by virtue of the fact that the connections are robust within some invariant subspace. We consider two previously studied examples and examine these in detail for a number of effects: (i) presence of internal symmetries within the chaotic saddles, (ii) phase-resetting, where only a limited set of connecting trajectories between saddles are possible, and (iii) multistability of periodic orbits near bifurcation to cycling attractors. The first model consists of three cyclically coupled Lorenz equations and was investigated first by Dellnitz et al. [Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 1243–1247 (1995)]. We show that one can find a "false phase-resetting" effect here due to the presence of a skew product structure for the dynamics in an invariant subspace; we verify this by considering a more general bi-directional coupling. The presence of internal symmetries of the chaotic saddles means that the set of connections can never be clean in this system, that is, there will always be transversely repelling orbits within the saddles that are transversely attracting on average. Nonetheless we argue that "anomalous connections" are rare. The second model we consider is an approximate return mapping near the stable manifold of a saddle in a cycling attractor from a magnetoconvection problem previously investigated by two of the authors. Near resonance, we show that the model genuinely is phase-resetting, and there are indeed stable periodic orbits of arbitrarily long period close to resonance, as previously conjectured. We examine the set of nearby periodic orbits in both parameter and phase space and show that their structure appears to be much more complicated than previously suspected. In particular, the basins of attraction of the periodic orbits appear to be pseudo-riddled in the terminology of Lai [Physica D 150, 1–13 (2001)]. en_GB
dc.identifier.citation 14 (3), pp. 571-582 en_GB
dc.identifier.doi 10.1063/1.1769111 en_GB
dc.identifier.uri http://hdl.handle.net/10036/20632 en_GB
dc.language.iso en en_GB
dc.publisher American Institute of Physics en_GB
dc.relation.url http://link.aip.org/link/?cha/14/571 en_GB
dc.subject chaos en_GB
dc.subject bifurcation en_GB
dc.subject stability en_GB
dc.subject set theory en_GB
dc.subject nonlinear dynamical systems en_GB
dc.subject convection en_GB
dc.title Cycling chaotic attractors in two models for dynamics with invariant subspaces en_GB
dc.type Article en_GB
dc.date.available 2008-03-13T12:05:01Z en_GB
dc.date.available 2011-01-25T10:33:48Z en_US
dc.date.available 2013-03-20T12:31:12Z
dc.identifier.issn 1054-1500 en_GB
dc.description Copyright © 2004 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 14 (2004) and may be found at http://link.aip.org/link/?cha/14/571 en_GB
dc.identifier.journal Chaos en_GB


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