Abstract: An effective approach to calculate the transient probability densities of the multi-delayed nonlinear system driven by colored noise excitations is developed. The system with time delay is simplified to an equivalent system without time delay. The linearization technique and the stochastic averaging method are adopted to obtain the averaged It? stochastic differential equation and the corresponding FPK（Fokker-Planck-Kolmogorov） equation for the amplitude process. A set of orthogonal base functions is obtained from the degenerated linear system. The Galerkin method is applied in the orthogonal base space to obtain the approximate probability densities. The proposed procedure is applied to the Duffing-Van Der Pol oscillator with two time delays and an external colored noise. The reliability of the theoretical results is verified by MCS（Monte Carlo simulation）. Effects of the colored noise and the time delay are also discussed. The results show that the proposed method is effective at studying the transient probability densities of the time-delayed nonlinear system driven by colored noises; the theoretical calculation efficiency is higher than that of MCS; both of the colored noises and time delay affect the transient responses.

Key words: time delay, colored noise, FPK（Fokker-Planck-Kolmogorov） equation, transient probability density, Galerkin method

摘要： 建立了色噪声与时滞联合作用的非线性系统模型，提出求解其瞬态概率密度的高效近似算法。利用等价变换将时滞系统简化为非时滞系统；通过线性化方法和随机平均原理得到原系统振幅过程的平均It?随机微分方程和相应的Fokker-Planck-Kolmogorov（FPK）方程。基于退化线性系统导出一组正交基，在该基空间内进行Galerkin变分得到近似瞬态概率密度。将该方法应用到受色噪声激励的双时滞Duffing-Van Der Pol振子得到理论解，采用蒙特卡罗模拟（MCS）验证理论解的正确性。分析了色噪声参数和时滞参数对瞬态响应的影响。研究结果表明：所提理论方法可有效求解受色噪声激励的时滞非线性系统的瞬态概率密度；算法求解效率高于MCS;色噪声和时滞均明显影响了系统瞬态响应。

关键词: 时滞, 色噪声, FPK方程, 瞬态概率密度, Galerkin算法