随机激励van der Pol-Duffing方程三峰P-分岔

Three-peak P-bifurcations in stochastically excited van der Pol-Duffing oscillator

本文讨论三稳态van der Pol-Duffing振子的随机P-分岔问题及参数影响.首先由随机平均法导出振动幅值的稳态概率密度函数,再应用突变理论得到系统发生随机P-分岔的临界参数条件.结果表明:参数变化时,系统经两次随机P-分岔,幅值稳态概率密度分布曲线峰的个数从1增加到3.随机激励的强度、系统阻尼系数对概率密度分布有重要影响,概率密度曲线峰的最大数目与确定性系统吸引子的数目相等.

This paper investigates the P-bifurcations and the parameters＇ effects in the tri-stable van der Pol-Duffing oscillator. By using the stochastic averaging method, the stationary probability density function of amplitude is obtained. Then the explicit condition for P-bifurcation is deduced by employing the catastrophe theory. It is shown that the number of peaks on the stationary probability density curve increases from 1 to 3 after two stochastic P-bifurcations induced by the variation of the two parameters, random excitation strength and linear damping coefficient. It is also shown that the maximal number of peaks of the stationary probability density equals to that of coexisted attractors of the corresponding deterministic system.