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.