Indeterminate jumps to resonance from a tangled saddle-node bifurcation

We identify in this paper a new type of bifurcation which yields an unpredictable outcome and therefore seems to be an important new ingredient of nonlinear dissipative dynamics. It arises when the unstable manifold of the saddle of a saddle-node bifurcation is heteroclinically tangled with the inset of a distant saddle which is itself homoclinically tangled so that it forms a fractal basin boundary between two remote attractors. At the saddle-node fold, a slowly evolving system will thus find itself sitting precisely on a fractal basin boundary, and in the presence of even infinitesimal noise we cannot predict to which of the two remote attractors the system will jump. We show here that such an indeterminate tangled saddle-node bifurcation is a common ingredient in the resonance of softening systems.