宽带噪声激励下含分数阶导数的Duffing-van del Pol振子的稳态响应

Stationary response of Duffing-van del Pol oscillator with fractional derivative under wide-band noise excitations

应用广义谐波平衡技术,将分数阶导数表示的回复力分解为幅值依赖的等效拟线性阻尼力和拟线性回复力.然后,应用基于广义谐和函数的随机平均法得到关于系统幅值的平均伊藤随机微分方程,系统幅值的稳态概率密度通过求解与之相应的平稳FPK方程得到.最后,用原方程的Monte Carlo模拟结果验证了近似解析解的正确性.

First, the restoring force modeled by a fractional derivative is separated into the equiavalent quasi-linear dissipative force and quasi-linear restoring force by using the generalized harmonic balance technique. Then, an averaged It6 stochastic differential equation for the amplitude is derived by using the stochastic averaging method based on the generalized harmonic function, and the Fokker-Planck-Kolmogorov （FPK） equation associated with the averaged It6 equation is solved to obtain the stationary probability densities of amplitude. Finally, the approximate analytical solutions are validated by those from the Monte Carlo simulation of original system.