中文摘要：

研究了乘性色噪声作用下三稳态vanderPol-Duffing振子的随机P-分岔问题．首先应用随机平均法得到系统振动幅值稳态概率密度函数的表达式，进而应用奇异性理论，得到刻画随机P-分岔发生的临界参数条件的转迁集以及系统存在的典型稳态概率密度曲线，并通过Monte-Carlo数值模拟进行了验证．以此为基础讨论了噪声强度、相关时间、系统线性阻尼系数对随机P-分岔和系统稳态响应行为的影响．

英文摘要：

This article aims at studying the stochastic P-bifurcation of tri-stable van der Pol-Duffing oscillator subjected to multiplicative colored noise. First, the stationary probability density of amplitude is derived by using the stochastic averaging method. Then the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. And the different types of stationary probability densities of amplitude are also obtained, which are in good agreement with the results from Monte-Carlo numerical simulation. Based on these results, the effects of the noise correlation time, noise intensity and linear damping coefficient on the P-bifurcation and the stable response behavior of the system are studied.