Two-parameter Locus of Boundary Crisis: Mind the Gaps!

Boundary crisis is a mechanism for destroying a chaotic attractor when one parameter is varied. In a two-parameter setting the locus of boundary crisis is associated with curves of homoor heteroclinic tangency bifurcations of saddle periodic orbits. It is known that the locus of boundary crisis contains many gaps, corresponding to channels (regions of positive measure) where a non-chaotic attractor persists. One side of such a subduction channel is a saddle-node bifurcation of a periodic orbit that marks the start of a periodic window in the chaotic regime; the other side of the channel is formed by a homoor heteroclinic tangency bifurcation associated with the saddle periodic orbit involved in the saddle-node bifurcation. We present a two-parameter study of boundary crisis in the Ikeda map, which models the dynamics of energy levels in a laser ring cavity. We confirm the existence of many gaps on the boundarycrisis locus. However, the gaps correspond to subduction channels that can have a rather different structure compared to what is known in the literature.