Effects of cross correlation on the relaxation time of a bistable system driven by cross-correlated noise

We study the effects of correlations between additive and multiplicative noise on relaxation time in a bistable system driven by cross-correlated noise. Using the projection-operator method, we derived an analytic expression for the relaxation time $T_c$ of the system, which is the function of additive (α) and multiplicative $\mathrm{}\left(D\right)\mathrm{}$ noise intensities, correlation intensity λ of noise, and correlation time τ of noise. After introducing a noise intensity ratio and a dimensionless parameter $R=D/\alpha ,$ and then performing numerical computations, we find the following: (i) For the case of $R<\mathrm{}1,$ the relaxation time $T_c$ increases as R increases. (ii) For the cases of $R>~1,$ there is a one-peak structure on the $T_c-R$ plot and the effects of cross-correlated noise on the relaxation time are very notable. (iii) For the case of $R<\mathrm{}1,$ $T_c$ almost does not change with both λ and τ, and for the cases of $R>~1,$ $T_c$ decreases as λ increases, however $T_c$ increases as τ increases. λ and τ play opposite roles in $T_c,$ i.e., λ enhances the fluctuation decay of dynamical variable and τ slows down the fluctuation decay of dynamical variable.