THE RELAXATION TIME OF A BISTABLE SYSTEM WITH TWO DIFFERENT KINDS OF TIME DELAYS

In this paper, we study the relaxation time of a bistable system driven by correlated noises when there are two different kinds of time delays in the deterministic (τ) and random (θ) forces, respectively. The expression for the relaxation time Tc is derived by the projection operator method, in which the effects of the memory kernels are taken into account. After introducing a dimensionless parameter R (R = D/α, where D is the strength of multiplicative noise and α is the strength of additive noise), and then performing numerical computations, we find the following: (1) for the case of R ≤ 1, the relaxation time Tc increases as τ increases, i.e. τ slows down the fluctuation decay of the dynamical variable for the case of R ≤ 1; (2) however, for the case of R > 1, Tc decreases as τ increases, i.e. τ enhances the fluctuation decay of the dynamical variable for the case of R > 1; (3) Tc decreases as θ increases, i.e. θ enhances the fluctuation decay of the dynamical variable for the three cases of R (R > 1, R=1, and R < 1).

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.