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Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory

Ashwin, P., Rucklidge, A.M. and Sturman, R. (2004) Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory. Physica D: Nonlinear Phenomena, 194 (1-2). pp. 30-48. ISSN 0167-2789

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Abstract

We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limit cycle, motivated by a low-order model for magnetic activity in a stellar dynamo. This model consists of coupled interactions between a saddle-node and two Hopf bifurcations, where the saddle-node bifurcation is assumed to have global reinjection of trajectories. The model can produce chaotic behaviour within each of a pair of invariant subspaces, and also it can show attractors that are stuck-on to both of the invariant subspaces. We investigate the detailed intermittent dynamics for such an attractor, investigating the effect of breaking the symmetry between the two Hopf bifurcations, and observing that it can appear via blowout bifurcations from the invariant subspaces. We give a simple Markov chain model for the two-state intermittent dynamics that reproduces the time spent close to the invariant subspaces and the switching between the different possible invariant subspaces; this clarifies the observation that the proportion of time spent near the different subspaces depends on the average residence time and also on the probabilities of switching between the possible subspaces.

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 2004 Elsevier Science B.V. This is an author produced version of an article published in Physica D: Nonlinear Phenomena. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.
Keywords: dynamo theory, bifurcation with symmetry, intermittency, cycling chaos
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: 14 Jun 2014 18:41
Published Version: http://dx.doi.org/10.1016/j.physd.2004.02.002
Status: Published
Refereed: Yes
Identification Number: 10.1016/j.physd.2004.02.002
URI: http://eprints.whiterose.ac.uk/id/eprint/998

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