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A dangerous bifurcation has been defined as a situation where a stable period-1 orbit occurs before and after the bifurcation, and yet the basin of attraction shrinks to zero size at the bifurcation point. It is known that this phenomenon can occur is non-smooth systems. In this paper we generalize the definition to one in which any attracting orbit may exist before and after the bifurcation, and their basins of attraction shrink to zero size at the bifurcation point, resulting in divergence of orbits starting from all initial conditions. Using the normal form of a 2D piecewise smooth map, we develop the conditions and show the parameter space regions where this phenomenon occurs. |