Bifurcations in stochastic dynamical systems with simple singularities

The generalized Langevin stochastic dynamical system is introduced and the stationary probability density for its solution is investigated. The stochastic field is assumed to be singular with a simple singularity, and noise in the control parameters is modelled as dychotomous Markov noises. A classification of bifurcation diagrams for the stationary density probability is obtained. Two examples encountered from physics, the dye laser model and the Verhulst model, are investigated.