Catastrophic bifurcation from riddled to fractal basins - art. no. 056228

Most existing works on riddling assume that the underlying dynamical system possesses an invariant subspace that usually results from a symmetry. In r ealistic applications of chaotic systems, however, there exists no perfect symmetry. The aim of this paper is to examine the consequences of symmetry- breaking on riddling. In particular, we consider smooth deterministic pertu rbations that destroy the existence of invariant subspace, and identify, as a symmetry-breaking parameter is increased from zero, two distinct bifurca tions. In the first case, the chaotic attractor in the invariant subspace i s transversely stable so that the basin is riddled. We find that a bifurcat ion from riddled to fractal basins can occur in the sense that an arbitrari ly small amount of symmetry breaking can replace the riddled basin by fract al basins. We call this a catastrophe of riddling. In the second case, wher e the chaotic attractor in the invariant subspace is transversely unstable so that there is no riddling in the unperturbed system, the presence of a s ymmetry breaking, no matter how small, can immediately create fractal basin s in the vicinity of the original invariant subspace. This is a smooth-frac tal basin boundary metamorphosis. We analyze the dynamical mechanisms for b oth catastrophes of riddling and basin boundary metamorphoses, derive scali ng laws to characterize the fractal basins induced by symmetry breaking, an d provide numerical confirmations. The main implication of our results is t hat while riddling is robust against perturbations that preserve the system symmetry, riddled basins of chaotic attractors in the invariant subspace, on which most existing works are focused, are structurally unstable against symmetry-breaking perturbations.