Dangerous Bifurcations Revisited

A dangerous border collision bifurcation has been defined as the dynamical instability that occurs when the basins of attraction of stable fixed points shrink to a set of zero measure as the parameter approaches the bifurcation value from either side. This results in almost all trajectories diverging off to infinity at the bifurcation point, despite the eigenvalues of the fixed points before and after the bifurcation being within the unit circle. In this paper, we show that similar bifurcation phenomena also occur when the stable orbit in question is of a higher periodicity or is chaotic. Accordingly, we propose a generalized definition of dangerous bifurcation suitable for any kind of attracting sets. We report two types of dangerous border collision bifurcations and show that, in addition to the originally reported mechanism typically involving singleton saddle cycles, there exists one more situation where the basin boundary is formed by a repelling closed invariant curve.