中文摘要：

为了深入研究窄带噪声作用下随机动力系统的特性，将复规范形法用于窄带随机动力系统研究了Duffing、Rayleigh和Vanderpol方程在谐和与窄带随机参数激励联合作用下的主共振响应和稳定性．由复规范形法得到了此系统响应振幅和相住所满足的方程，再由摄动法分析了系统的主共振响应和稳定性，并用随机增维精细积分法验证了方程理论分析结果的正确性，用数值法计算了平凡解的Lyaputov指数曲面．结果表明，随着窄带随机扰动强度的增加，系统稳态解的相图从极限环变为扩散的极限环．研究证实了复规范形法用于窄带随机动力系统是有效的．

英文摘要：

In order to study the property of random dynamic systems excited by narrow-band noise, the complex normal form method was applied to narrow-band random dynamic systems. The principal resonance and stability of Duffing, Rayleigh and Van der pol oscillator under combined harmonic and narrow-band random parametric excitation were investigated. Equations of the amplitude and phase were obtained by using the complex normal form method. Then the perturbation method was used to analyze principal resonance and stability. The theoretical results were verified by stochastic precise integration method. The Lyapunov exponent three-dimensional surface was also obtained by numerical method. Theoretical analyses and numerical simulation showed that when the intensity of the random excitation increases, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. The results proved the applicability of the complex normal form method for narrow-band random dynamics systems.