On riddling and weak attractors

Abstract

We propose general definitions for riddling and partial riddling of a subset V of Rm with non-zero Lebesgue measure and show that these properties are invariant for a large class of dynamical systems. We introduce the concept of a weak attractor, a weaker notion than a Milnor attractor and use this to re-examine and classify riddled basins of attractors. We find that basins of attraction can be partially riddled but if this is the case then any partial riddling must be evident near the attractor. We use these concepts to aid classification of bifurcations of attractors from invariant subspaces. In particular, our weak attractor is a generalisation of the absorbing area investigated by other authors and we suggest that a transition of a basin to riddling is usually associated with loss of stability of a weak attractor.