Theory of frequency and phase synchronization in a rocked bistable stochastic system

We investigate the role of noise in the phenomenon of stochastic synchronization of switching events in a rocked, overdamped bistable potential driven by white Gaussian noise, the archetype description of Stochastic Resonance. We present a new approach to the stochastic counting process of noise-induced switching events: starting from the Markovian dynamics of the nonstationary, continuous particle dynamics one finds upon contraction onto two states a non-Markovian renewal dynamics. The output frequency is determined as the velocity of the underlying discrete phase dynamics. The phenomenon of noise-assisted phase synchronization is investigated in terms of an effective, instantaneous phase diffusion. The theory is applied to rectangular-shaped rocking signals versus increasing input-noise strengths. Precise numerical simulations corroborate very favorably our analytical results. The novel theoretical findings are also compared with prior findings.