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

摘要

The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinicudcycle connecting two equilibria and a periodic orbit is investigated. This type of system is knownudto exhibit complicated, possibly chaotic dynamics including irregular switching of sign of variousudphase space variables, but details of the mechanisms underlying the complicated dynamics haveudnot previously been investigated. We identify global bifurcations that induce the onset of chaoticuddynamics and switching near a heteroclinic cycle of this type, and by construction and analysisudof approximate return maps, locate the global bifurcations in parameter space. We find there is audthreshold in the size of certain symmetry-breaking terms below which there can be no persistentudswitching. Our results are illustrated by a numerical example.