高斯白噪声激励下分数阶Duffing-Van der Pol系统的稳态响应

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.

Response of duffing-van der pol oscillator with fractional derivative under gaussian white noise

This paper aimed to investigate 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 can be deduced by utilizing the stochastic averaging method and Stratonovich-Khasminskii theorem in the first place. And then we can solve the FPK equation to obtain the stationary probability densities of amplitude, which in fact can be used to describe the response of system. Then one found that reducing fractional derivative order is able to enhance the response of system and that increasing fractional coefficient can weaken the response of system. So the fractional derivative damping term has a great effect on the response of Duffing-Van der Pol oscillator. In addition,the response can also be influenced by other system parameters. Finally, the above analytical results are confirmed to be pretty fit with that of the Monte Carlo simulation.