Minimal attractors and bifurcations of random dynamical systems

Abstract

We consider attractors for certain types of random dynamical systems. These are skew–product systems whose base transformations preserve an ergodic invariant measure. We discuss definitions of invariant sets, attractors and invariant measures for deterministic and random dynamical systems. Under assumptions that include, for example, iterated function systems, but that exclude stochastic differential equations, we demonstrate how random attractors can be seen as examples of Milnor attractors for a skew–product system. We discuss the minimality of these attractors and invariant measures supported by them. As a further connection between random dynamical systems and deterministic dynamical systems, we show how dynamical or D–bifurcations of random attractors with multiplicative noise can be seen as blowout bifurcations, and we relate the issue of branching at such D–bifurcations to branching at blowout bifurcations.