Resonant dynamics of a periodically driven noisy oscillator

A unified approach to study the dynamics of single degree of freedom systems excited by both weakly periodic and random additive perturbations is developed. The near-resonant motion of such systems is not well understood. This problem is studied with the aim of discovering a common geometric structure in the phase space, and determining the effects of noisy perturbations on the passage of trajectories through the resonance zone. We consider the noisy, periodically driven Duffing equation as a prototypical single degree of freedom system and achieve a model-reduction through stochastic averaging. The solution of the reduced model can be approximated by a Markov process. Depending on the strength of the noise, the reduced Markov process takes its values on a line or on a graph with certain gluing conditions at the vertices of the graph. The reduced model provides a framework for computing standard statistical measures of dynamics and stability such as probability density functions. This work also explains a counter-intuitive phenomenon in stochastic resonance, in which a weak periodic forcing in a nonlinear system can be enhanced by the addition of external noise.