高斯白噪声激励下分数阶Duffing-Van der Pol系统的稳态响应

摘要

This paper investigates the stationary response of Duffing-Van der Pol oscillator with fractional derivative damping term under Gaussian white noise excitation.The corresponding Fokker-Planck-Kolmogorov (FPK) equation is firstly deduced by utilizing the stochastic averaging method and Stratonovich-Khasminskii theorem.And we then solve the FPK equation to obtain the stationary probability densities of system amplitude,which in fact can be used to describe the system response.It is then found that the reducing fractional derivative order enhances the system response,while the increasing fractional coefficient weakens the system response.Therefore,the fractional derivative damping term shows a great effect on the response of Duffing-Van der Pol oscillator.In addition,the response is also be influenced by other system parameters.Finally,the above analytical results reasonably fit with that of the Monte Carlo simulation.%应用随机平均法研究了高斯白噪声激励下含有分数阶阻尼项的Duffing-Van der Pol系统的稳态响应.首先应用基于广义谐和函数的随机平均法得到系统关于幅值的平均伊藤微分方程并建立相应的平稳FPK方程,求解该平稳FPK方程的近似理论解得到系统幅值的稳态概率密度.分析幅值、位移和速度的稳态概率密度探究分数阶阻尼项以及其它参数对系统稳态响应的影响.发现降低分数阶的阶数可以增强系统的响应而增大分数阶的系数可以减弱系统响应.最后对原系统进行Monte Carlo数值模拟验证近似理论解的有效性.