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Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection

Ashwin, P. and Rucklidge, A.M. (1998) Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection. Physica D: Nonlinear Phenomena, 122 (1-4). pp. 134-154. ISSN 0167-2789

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Abstract

We examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets; this ‘cycling chaos’ manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets. Such behaviour is robust to perturbations that preserve the symmetry of the system; we examine bifurcations of this state.

We discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace. This differs from the standard scenario for the blowout bifurcation in that in our case, the blowout is neither subcritical nor supercritical. The robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors.

The model we consider is a ninth-order truncated ordinary differential equation (ODE) model of three-dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient. Symmetries of the model lead to the existence of invariant subspaces for the dynamics; in particular there are invariant subspaces that correspond to regimes of two-dimensional flows, with variation in the vertical but only one of the two horizontal directions. Stable two-dimensional chaotic flow can go unstable to three-dimensional flow via the cross-roll instability. We show how the bifurcations mentioned above can be located by examination of various transverse Liapunov exponents. We also consider a reduction of the ODE to a map and demonstrate that the same behaviour can be found in the corresponding map. This allows us to describe and predict a number of observed transitions in these models. The dynamics we describe is new but nonetheless robust, and so should occur in other applications.

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 1998 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: heteroclinic cycle, symmetry, chaotic dynamics, magnetoconvection
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: 06 Feb 2006
Last Modified: 16 Jun 2014 17:03
Published Version: http://dx.doi.org/10.1016/S0167-2789(98)00174-2
Status: Published
Refereed: Yes
Identification Number: 10.1016/S0167-2789(98)00174-2
URI: http://eprints.whiterose.ac.uk/id/eprint/985

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