中文摘要：

基于广义谐和函数与随机平均原理，研究了具有强非线性的Duffing—Rayleigh—Mathieu系统在色噪声激励下的稳态响应。通过vanderPol坐标变换，将系统运动方程转化为关于幅值与初始相位角的随机微分方程。应用Stratonovich—Khasminskii极限定理，作随机平均，得到近似的二维扩散过程。在此基础上，考虑共振情形，引入相位差变量，做确定性的平均，得到关于幅值与相位差的It6随机微分方程。建立对应的Fokker-Planck—Kolmogorov（FPK）方程，结合边界条件与归一化条件，用Crank—Nicolson型有限差分法求解稳态的FPK方程，得到平稳状态下系统的联合概率分布。用MonteCarlo数值模拟法验证了理论方法的有效性。

英文摘要：

Based on generalized harmonic functions, a new stochastic averaging procedure for strongly nonlinear oscillators under combined periodical and wide-band colored noise excitations is used to investigate the stationary response of DuffingRayleigh-Mathieu system under colored noise excitation. The damping is linear plus nonlinear. The periodical excitation is parametric and the random excitations are external plus parametric. The motion equation of the original system is transformed to a set of stochastic differential equations by van der Pol transformation. According to Stratonovich-Khasminskii limit theorem and by introducing phase difference, the stochastic averaging and deterministic averaging procedures are completed and the original system is approximated by two diffusion processes in the case of resonance. The reduced FPK equation is established and solved numerically with boundary conditions and normalization condition, All theoretical results are verified by Monte Carlo digital simulation.