  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.
