Boundary crisis bifurcation in two parameters

Abstract

The boundary crisis bifurcation is well known as a mechanism for destroying (or creating) a strange attractor by variation of one parameter: at the moment of the boundary crisis bifurcation the strange attractor touches the boundary of its own basin of attraction. Here we follow this codimension-one bifurcation in two parameters. One might expect that this leads to a differentiable curve in the two-parameter plane. Mathematically, a boundary crisis is effectively a homoclinic or heteroclinic tangency, the locus of which is a well-defined differentiable curve in a two-parameter system. However, instead of a boundary crisis, the transition through this tangency curve may lead to a basin boundary metamorphosis or an interior crisis bifurcation, in which the attractor persists. This phenomenon is again well known: at the point where the type of transition changes, the boundary crisis switches to another branch of homoclinic or heteroclinic tangencies, associated with manifolds of a periodic point with a different period than before. The curve of boundary crisis bifurcations is not differentiable at such points. In this paper, we show that the curve of boundary crisis bifurcations is, in fact, not even well defined as a piecewise-smooth curve. We illustrate that there are infinitely many gaps in much the same way as the one-parameter bifurcation diagram of the attractor contains infinitely many windows where the attractor is periodic and not strange or chaotic. Throughout, we use the Hénon map to illustrate our findings.