Stochastically Perturbed Resonance

Abstract

We analyze the motion of a periodically forced oscillator whose natural frequency is the superposition of a slowly varying component and a small stochastic perturbation. When the slowly varying component of natural frequency passes through the value corresponding to the forcing, there is resonance. The purpose of this study is to understand the effect of the resonance on the stochastic processes of the oscillator’s amplitude and phase. J. B. Kelley’s smoothing method is employed to derive a diffusion equation that governs the joint distribution of amplitude and phase. We study this equation in the limit where the time scale of resonance is much shorter than the time scale of diffusion. Far away from resonance, the distribution function satisfies to leading order the same diffusion equation as in the unforced case. The effect of the resonance is to create an internal layer in the solution at the time of resonance in which the distribution undergoes a small but abrupt change. A simple calculation gives the magnitude of the jump.