Dangerous bifurcation at border collision: when does it occur?

Abstract

It has been shown recently that border collision bifurcation in a piecewise smooth map can lead to a situation where a fixed point remains stable at both sides of the bifurcation point, and yet the orbit becomes unbounded at the point of bifurcation because the basin of attraction of the stable fixed point shrinks to zero size. Such bifurcations have been named "dangerous bifurcations". In this paper we provide explanation of this phenomenon, and develop the analytical conditions on the parameters under which such dangerous bifurcations will occur.