Noise Perturbations of a Nonlinear System with Multiple Steady States

We examine the effects of small white Gaussian noise perturbations on a harmonically forced Duffing oscillator. For specific values of the parameters, the noise-free system admits two coexisting steady-state attractors. The presence of noise induces transitions from one attractor to the other, however small the noise intensity may be. As a first step, the equation of motion is transformed into a system of stochastic differential equations for the slowly-varying van der Pol variables, by assuming that the predominant frequency of response is that of the forcing term. The condition for bistability, the stable fixed points, and the separatrix defining the domains of attraction are then examined. In the second step, the effects of noise perturbations on the system are analyzed by determining the steady-state probability density of the fluctuations of the response, as well as the probabilities of escape from one attractor to the other. The latter are found by determination of the expected..