The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit, by Kirk and Rucklidge

Dynamical Systems: An International Journal 23 (2008) 43--74. doi:10.1080/14689360701709088

The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit

V. Kirk(1) and A.M. Rucklidge(2)

(1) Department of Mathematics,
University of Auckland, Private Bag 92019,
Auckland, New Zealand
(2) Department of Applied Mathematics,
University of Leeds, Leeds, LS2 9JT, UK

Abstract. The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various phase space variables, but details of the mechanisms underlying the complicated dynamics have not previously been investigated. We identify global bifurcations that induce the onset of chaotic dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis of approximate return maps, locate the global bifurcations in parameter space. We find there is a threshold in the size of certain symmetry-breaking terms, below which there can be no persistent switching. Our results are illustrated by a numerical example.

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