Unfolding a chaotic bifurcation

The differential equation of the sinusoidally forced pendulum is studied by digital simulation in a régime where two simple, symmetrically related chaotic attractors grow and merge continuously as the forcing amplitude is increased. By introducing a small constant bias in the forcing to break the symmetry, two discontinuous bifurcations unfold from the single merging event. Considering both forcing amplitude and bias together as controls, a codimension two bifurcation of chaotic attractors is defined, whose geometric structure in control-phase space is closely related to the elementary cusp catastrophe. The chaotic bifurcations are explained in terms of homoclinic structures (Smale cycles) in the Poincaré map.