Hard bifurcations in dynamical systems with bounded random perturbations

We study bifurcations in dynamical systems with bounded random perturbations. Such systems, which arise quite naturally, have been nearly ignored in the literature, despite a rich body of work on systems with unbounded, usually normally distributed, noise. In systems with bounded random perturbations, new kinds of bifurcations that we call 'hard' may happen and in fact do occur in many situations when the unperturbed deterministic systems experience elementary, codimension-one bifurcations such as saddle-node and homoclinic bifurcations. A hard bifurcation is defined as discontinuous change in the density function or support of a stationary measure of the system.