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Cycling chaotic attractors in two models for dynamics with invariant subspaces

Ashwin, P., Rucklidge, A.M. and Sturman, R. (2004) Cycling chaotic attractors in two models for dynamics with invariant subspaces. Chaos: An Interdisciplinary Journal of Nonlinear Science, 14 (3). pp. 571-582. ISSN 1054-1500

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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. (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 (2001).

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 2004 American Institute of Physics.This is an author produced version of an article published in Chaos. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Depositing User: A. M. Rucklidge
Date Deposited: 10 Feb 2006
Last Modified: 08 Feb 2013 17:02
Published Version: http://dx.doi.org/10.1063/1.1769111
Status: Published
Refereed: Yes
Identification Number: 10.1063/1.1769111
URI: http://eprints.whiterose.ac.uk/id/eprint/1001

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