Two-parameter locus of boundary crisis: Mind the gaps!

Pages: 1148 - 1157, Issue Special, September 2011

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Hinke M. Osinga - Bristol Center for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR, United Kingdom (email)
James Rankin - NeuroMathComp Project Team, INRIA Sophia-Antipolis, 2004 Route des Lucioles-BP 93, Sophia Antipolis, 06902, France (email)

Abstract: 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 homo- or 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 homo- or 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 boundary- crisis locus. However, the gaps correspond to subduction channels that can have a rather different structure compared to what is known in the literature.

Keywords:  Ikeda map, homoclinic tangency, saddle-node bifurcation, subduction, double-crisis vertex
Mathematics Subject Classification:  Primary: 37G25, 37M20; Secondary: 65P30

Received: July 2010;      Revised: August 2010;      Published: October 2011.