Stochastic resonance in a bistable system with time-delayed feedback loops

The phenomenon of stochastic resonance of a bistable system subjected to linear time-delayed feedback loops driven by multiplicative Gaussian coloured noise and additive Gaussian white noise is investigated. Firstly, the analytic expression of the quasi-steady distribution function P_s(x,t) is derived by applying the unified coloured noise approximation and the Novikov Theorem; Secondly, the expression of the signal-to-noise ratio (SNR) is obtained in the adiabatic limit to quantify the stochastic resonance. Finally, the effects of the linear coefficient a, the nonlinear coefficient b, the linear time-delayed feedback coefficient c and the delay time τ on P _s(x,t) and SNR^± are discussed. It is found that the effects of the linear coefficient and the nonlinear coefficient, the positive linear time-delayed feedback coefficient and the negative linear time-delayed feedback coefficient, the positive delayed time and the negative delayed time on P_s(x,t) and SNR^± are different, respectively. This discussion would be helpful to the study of the system reliability and controlling stochastic resonance.