Effects of time delay on the statistical fluctuations for a bistable system driven by cross-correlated noises

We study the effects of time delay on the normalized correlation function C(s) and the associated relaxation time T _c for a bistable system with correlations between multiplicative and additive white noises under the condition of small time delay. Using the projection operator method, the expressions of T _c and C(s) are obtained. Based on numerical computations, it is found that the delay time τ slows down the rate of fluctuation decay of dynamical variable for the presence of positive feedback intensity (∈ > 0), while speeds up the rate of fluctuation decay of dynamical variable for the presence of negative feedback intensity (∈ < 0). The effects of the delay time τ on the T _c and C(s) are entirely opposite for ∈ 〉 0 and ∈ < 0.