Asymmetric bistable systems subject to periodic and stochastic forcing in the strongly nonlinear regime: Switching time distributions

A detailed theoretical analysis of the dynamics of a sinusoidally driven noisy asymmetric bistable system is presented. The results are valid for any two-state system, however, the specific case of the Duffing potential is considered in detail. The dynamics are considered in the weak noise limit, i.e., when the response of the system to the external periodic field is strongly nonlinear. The system asymmetry is created by a nonzero dc component of the external force, and manifests itself as an asymmetry between the mean switching times between the potential wells. We obtain explicit analytic expressions for the whole hierarchy of switching time distributions (including the residence time and return time distributions). We also obtain expressions for the average residence times and describe how they depend on asymmetry, together with an explicit expression for the difference between the residence times in the weak noise limit; the results are presented in the context of using the switching dynamics to detect weak dc target signals