Noise perturbations of a non-linear system with multiple steady states

Abstract

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 times of first-passage to the separatrix starting from a point in the domain of attraction of each stable state. This analysis is done in the limit of small damping and small noise intensity by use of an averaging scheme which reduces the dimensionality of the problem from two to one; this yields valuable information about the relative stability of the stable states. The obtained theoretical results are supported by digital simulation data. The analytical theory gives good agreement even for large noise.