分数阶van der Pol振子的超谐与亚谐联合共振

Super-harmonic and sub-harmonic simultaneous resonance of fractional-order van der Pol oscillator

基于平均法研究了分数阶van der Pol振子3次超谐与1/3次亚谐联合共振时的动力学特性。得到了系统的一阶近似解析解,提出了超、亚谐联合共振时等效线性阻尼和等效线性刚度的概念。建立了联合共振定常解幅频曲线的解析表达式,又结合变分方程进行线性化处理,推导出分数阶van der Pol振子在联合共振时的周期解稳定性判断准则。通过与单一谐波下超谐共振、亚谐共振的对比,发现在不同基本参数下该系统可分别表现出单谐波超谐共振、单谐波亚谐共振以及两者共存时的特征现象。研究表明,分数阶微分项参数通过等效线性阻尼和等效线性刚度的形式对系统的响应幅值、共振频率、定常解稳定性、周期解数量、共振区域、曲线拓扑结构及跳跃现象等复杂动力学特性均产生重要影响。

Based on the averaging method,the dynamical characteristics of third-order super-harmonic and one-third order sub-harmonic simultaneous resonance of a van der Pol oscillator with a fractional-order differential term are analytically studied.The first-order approximate analytical solution is obtained,and the definitions of the equivalent linear damping coefficient and the equivalent linear stiffness efficient for super-harmonic and sub-harmonic simultaneous resonance are presented.The analytical amplitude-frequency equation for steady-state solution of the simultaneous resonance is established.Combined with the variational equation for linearization,the criteria for the periodic solution stability of the van der Pol oscillator under simultaneous resonance are derived.Through the comparison with super-harmonic resonance and sub-harmonic resonance under single harmonic excitation,it is found that the system can exhibit characteristic phenomenon of single harmonic super-harmonic resonance,single harmonic sub-harmonic resonance and both existence of these two resonances under different system parameters.The results show that the system parameters in fractional-order differential term have important influence on the response amplitude,the resonance frequency,the stability of stationary solution,the number of periodic solutions,the resonance region,topological structure of the amplitude-frequency curve,jumping phenomena and other complex dynamic characteristics through the equivalent linear damping and equivalent linear stiffness.