Asymptotic analysis and linearization of the randomly perturbed two-wells Duffing oscillator

The purpose of this paper is to approximate the dynamics of a non-linear oscillator with multiple static equilibrium excited by a white noise. An approach based on asymptotic expansions of dynamical systems driven by weak random perturbations is introduced. To illustrate the method, the Duffing oscillator with two symmetric potential wells is considered. An equivalent locally linear oscillator, which provides a good approximation of the dynamics of the studied oscillator, is obtained. This last approximation can be considered as a local linearization.