Pseudo-riddling in chaotic systems

Abstract

Previous investigations of riddling have focused in the case where the dynamical invariant set in the symmetric invariant subspace of the system is a chaotic attractor. A situation expected to arise commonly in dynamical systems, however, is that the dynamics in the invariant subspace is in a periodic window. In such a case, there are both a stable periodic attractor and a coexisting non-attracting chaotic saddle in the invariant subspace. We show that riddling can still occur in a generalized sense. In particular, we argue that the basin of the periodic attractor in the invariant subspace contains both an open set and a set of measure zero with riddled holes that belong to the basin of another attractor. We call such basin pseudo-riddled and we argue that pseudo-riddling can be more pervasive than riddling of chaotic attractors because it can occur regardless of whether the chaotic saddle in the invariant subspace is transversely stable or unstable. We construct an analyzable model, derive a scaling law for pseudo-riddling, and provide numerical support. We also show that the chaotic saddle in the invariant subspace can undergo an enlargement into the full phase space when it becomes transversely unstable.