Stochastic bifurcation: concepts and examples

Deterministic bifurcation theory deals with qualitative changes in parametrized families of vector fields or dynamical systems. "Qualitative change" has been successfully formalized by means of topological equivalence and structural stability. We review the generic elementary bifurcation scenarios.

Stochastic bifurcation theory should deal with qualitative changes in parametrized families of random dynamical systems (systems perturbed by noise), and should reduce to the deterministic theory if noise is absent. While the concept of a random dynamical system can be unanimously formalized, this is not the case with "qualitative change". I offer a "phenomenological approach" introduced by physicists already in the eighties, and a "dynamical approach" based on Lyapunov exponents, invariant measures and random attractors.

We then try to find the stochastic versions of the elementary bifurcation scenarios, in particular of the Hopf bifurcation, mainly by way of prototypical examples like the Duffing-van der Pol oscillator with multiplicative noise and the Duffing (Kramers) oscillator with additive noise.