A mechanism for dangerous border collision bifurcations

In this paper we investigate a mechanism causing dangerous border collision bifurcations which is numerically characterized by exhibiting a stable fixed point before and after the critical bifurcation point, but the unbounded behavior of orbits at the critical bifurcation point. In particular, we provide that at the critical bifurcation value μ₀, the qualitative type of the fixed point without having Jacobian information is saddle, which can be induced by invariant manifolds of the periodic saddle orbit on the boundary of the basin of attractor at infinity at the parameter μ ≠ μ₀. In addition, we show that invariant sets of such fixed point are nonsmooth curve, and also point out a dangerous situation of numerical simulation.