含分数阶微分项的Van Del Pol振子的动力学分析

Dynamical analysis of a Van Del Pol oscillator with fractional-order derivative

研究了含分数阶微分项的Van Del Pol振子。利用平均法得出系统的近似解析解,发现在近似解中分数阶微分项的系数和阶次以等效线性阻尼和线性刚度的形式影响着系统的振幅、共振频率以及稳定性,这与现有文献中直接把分数阶微分项单纯看做阻尼处理的方式不同。随后把解析解和数值解进行了比较,发现二者吻合较好,证明了该解析解的正确性。通过数值仿真进一步验证了理论推导的正确性。

A Van Del Pol oscillator with fractional-order derivative was studied by the averaging method,and the approximately analytical solution was obtained.The effects of the system parameters on response amplitude,frequency and stability,including the fractional coefficient and the fractional order,were characterized by the equivalent linear damping and the equivalent stiffness.The conclusion is entirely different from the published results which directly treat the fractional-order derivative as simple damping.The comparison of the analytical solution with the numerical results verifies the correctness of the approximately analytical results.The following analysis about the effects of the fractional parameters on the amplitude frequency was fulfilled.The correctness of the theoretical derivation was verified by numerical simulation.