Noise Perturbations of Nonlinear Dynamical

We investigate analytical methods to predict the long-time cumulative eects of noise perturbations on nonlinear dynamical systems. The study of such perturbations addresses the problem of the global stability of nonlinear systems with multiple coexisting attractors. Indeed small random perturbations can cause large excursions of the system from a nominal steady state, and hence induce transitions from one attractor to another one. Noise-induced diusion in state space and stochastic escape of trajectories can be studied by asymptotic analysis of the response probability density function and the mean exit-time from an attractor's domain in the limit of weak noise. Two types of methods are reviewed based on either functional integral representations or on singular perturbation techniques of appropriate boundary value problems. The theory is exemplied on the problem of escape in a two-state Dung oscillator excited by harmonic forces.