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Phase resetting effects for robust cycles between chaotic sets

Ashwin, P., Field, M., Rucklidge, A.M. and Sturman, R. (2003) Phase resetting effects for robust cycles between chaotic sets. Chaos: An Interdisciplinary Journal of Nonlinear Science, 13 (3). pp. 973-981. ISSN 1054-1500

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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. This paper introduces and discusses an instructive example of an ODE where one can observe and analyse robust cycling behaviour. 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 Rossler system), and/or saddle equilibria.

For this model, we distinguish between cycling that include 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 non-phase resetting 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.

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 2003 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.
Institution: The University of Leeds
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: 19 Jun 2014 22:11
Published Version: http://dx.doi.org/10.1063/1.1586531
Status: Published
Refereed: Yes
Identification Number: 10.1063/1.1586531
URI: http://eprints.whiterose.ac.uk/id/eprint/996

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