Random attractors--general properties, existence and applications to stochastic bifurcation theory

This paper is concerned with attractors of randomly perturbed dynamical systems, called random attractors. The framework used is provided by the theory of random dynamical systems. We first define, analyze, and prove existence of random attractors. The main result is a technique, similar to Lyapunov's direct method, to ensure existence of random attractors for random differential equations. This method is formulated as a generally applicable procedure. As an illustration we shall apply it to the random Duffing-van der Pol equation. We then show, by the same example, that random attractors provide an important tool to analyze the bifurcation behavior of stochastically perturbed dynamical systems. We introduce new methods and techniques, and we investigate the Hopf bifurcation behavior of the random Duffing-van der Pol equation in detail. In addition, the relationship of random attractors to invariant measures and unstable sets is studied.