Can a border collision bifurcation of a chaotic attractor lead to its expansion?

Abstract

Recently, we reported that a chaotic attractor in a discontinuous one-dimensional map may undergo a so-called exterior border collision bifurcation, which causes additional bands of the attractor to appear. In the present paper, we suppose that the chaotic attractor’s basin boundary contains a chaotic repeller, and discuss a bifurcation pattern consisting of an exterior border collision bifurcation and an expansion bifurcation (interior crisis). In the generic case, where neither the border point the chaotic attractor collides with, nor any of its images belong to the chaotic repeller, the exterior border collision bifurcation is followed by the expansion bifurcation, and the distance between both bifurcations may be arbitrarily small but positive. In the non-generic (codimension-2) case, these bifurcations occur simultaneously, so that a border collision bifurcation of a chaotic attractor leads directly to its expansion.