Robust bursting to the Origin: heteroclinic Cycles with Maximal Symmetry Equilibria

Robust attracting heteroclinic cycles have been found in many models of dynamics with symmetries. In all previous examples, robust heteroclinic cycles appear between a number of symmetry broken equilibria. In this paper we examine the first example where there are robust attracting heteroclinic cycles that include the origin, ie a point with maximal symmetry. The example we study is for vector fields on R3 with (Z2) 3 symmetry. We list all possible generic (codimension one) local and global bifurcations by which this cycle can appear as an attractor; these include a resonance bifurcation from a limit cycle, direct bifurcation from a stable origin and direct bifurcation from other and more familiar robust heteroclinic cycles. AMS classification scheme numbers: 34C37, 37C80.