Stochastic Nonlinear Dynamics Modulated by External Periodic Forces

Abstract

The stochastic theory is developed for overdamped, nonlinear stochastic systems with periodic forcing. By use of a generalized Floquet theory we show that such systems averaged over the random phase ϕ are not strongly mixing, but exhibit ever present undamped oscillations, e.g. the power spectrum contains δ-function peaks at multiples of the driving frequency Ω. For the archetypal periodically driven, bistable stochastic flow, = x−x3 + A cos(Ωt + ϕ) + ξ(t), we evaluate by means of matrix continued fraction techniques the stationary probability Wst(x, θ = Ωt + ϕ) and the ϕ averaged, complex-valued dynamical susceptibility. The stationary probability has a most interesting, rich topology in (x, θ)-phase space exhibiting several, competing modulation-induced escape paths.