DIFFUSION ACROSS CHARACTERISTIC BOUNDARIES WITH CRITICAL POINTS.

The authors consider the problems of the effect of small white noise perturbations on a deterministic dynamical system in the plane with (i) an asymptotically stable equilibrium point or limit cycle and (ii) an equilibrium point surrounded by closed trajectories. The mean exit time and the distribution of exit points for each problem is determined by solving singularly perturbed elliptic boundary value problems in domains with closed characteristic boundaries with critical points. Uniformly valid asymptotic solutions are constructed for each of the problems. For the asymptotically stable equilibrium point, the method of matched asymptotic expansions with the integral condition of Matkowsky and Schuss is employed. A method of averaging combined with boundary layer analysis is used for the problem of an equilibrium point surrounded by closed trajectories. The influence on the solutions, of the critical points on the boundary, is exhibited and explained. An application to the physical pendulum is given. Finally the results are shown to be in close agreement with simulations.