Effects of Time Delay on Stability of an Unstable State in a Bistable System with Correlated Noises

Effects of time delay on stability of an unstable state in a time-delayed bistable system are investigated. The analytic expression of the transition rate W(x_u,τ) from unstable state x_u to stable state x₊ is derived. The numerical calculation results of W(x_u,τ) indicate that W(x_u,τ) decreases with the increasing multiplicative noise intensity, the additive noise intensity and the strength of correlations between the multiplicative and the additive noise increase, but W(x_u,τ) increases with increasing delay time. Namely, the multiplicative noise, the additive noise and the correlations between the multiplicative and the additive noises enhance the stability of the unstable state in the time-delayed bistable system but the stability is weakened by time delay.