中文摘要：

分析了乘性和加性噪声作用下三稳态VanderPol-Duffing振子的随机P分岔．首先用随机平均法得到系统的随机微分方程，求得系统响应幅值的稳态概率密度函数．然后应用分岔分析的奇异性理论，求得随机P分岔发生的临界参数条件，得到多种定性不同的稳态概率密度曲线．讨论了2种激励噪声强度和系统阻尼对响应稳态概率密度曲线峰的个数、各峰值相对大小的影响．通过Monte-Carlo数值模拟对理论计算结果进行了验证．该方法可用于其他系统的随机P分岔分析．

英文摘要：

This paper aims to investigate the stochastic P-blturcatlons m me tn-staole Vau der Pol-Duffing oscillator with additive and multiplicative Gauss noise. By using the stochastic averaging method, the stationary probability density function of amplitude is obtained. Then based on the singularity theory of the deterministic system, the explicit parameter conditions for P-bifurcation are deduced, and eleven types of qualitatively different probability density curves are founded. Finally the effects of three coefficients, one for linear damping and two for random excitation strength, are discussed. The results are verified by Monte-Carlo numerical simulations. The method used here is also suitable for other systems＇ P-bifurcation analysis.