Dynamical Systems: An International Journal
23 (2008) 43--74.
The effect of symmetry breaking on the dynamics near a structurally
stable heteroclinic cycle between equilibria and a periodic orbit
(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
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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