Duffing系统的主-超谐联合共振

Simultaneous primary and super-harmonic resonance of Duffing oscillator

非线性动力系统极易发生共振,在多频激励下可能发生联合共振或组合共振,目前关于非线性系统的主-超谐联合共振的研究少见报道.本文以Duffing系统为对象,研究系统在主-超谐联合共振时的周期运动和通往混沌的道路.应用多尺度法得到系统的近似解析解,并利用数值方法对解析解进行验证,结果吻合良好.基于Lyapunov第一方法得到稳态周期解的稳定性条件,并分析了非线性刚度对稳态周期解的幅值和稳定性的影响.此外,由于近似解只能描述周期运动,不足以描述系统的全局特性,因而应用Melnikov方法对系统进行全局分析,得到系统进入Smale马蹄意义下混沌的条件,依据该条件以及主-超谐联合共振的条件选取一组参数进行数值仿真.分岔图和最大Lyapunov指数显示出两个临界值:当激励幅值通过第一个临界值时,异宿轨道破裂,混沌吸引子突然出现,系统以激变方式进入混沌;激励幅值通过第二个临界值时,系统在混沌态下再次发生激变,进入另一种混沌态.利用Melnikov方法考察了第一个临界值在多种频率组合下的变化趋势,并用数值仿真验证了解析结果的正确性.

There are many resonance phenomena in a nonlinear dynamical system subjected to forced excitation,especially the excitation with multiple frequencies.Duffing oscillator subjected to the excitation with multiple frequencies may exhibit some complex resonance phenomena,such as simultaneous resonance and combination resonance.In this paper,the simultaneous primary and super-harmonic resonance of Duffing oscillator is studied,and it is analyzed in periodic motion and chaotic motion.Firstly,the approximate analytical solution is obtained by the method of multiple scales,and the correctness and accuracy of the analytical solution are verified through numerical simulation.Furthermore,the amplitude-frequency equation and phase-frequency equation of the steady-state response are derived from the approximate solution,and the stability of the steadystate response is analyzed based on Lyapunov’s first method.It is found that there are at most two stable periodic solutions and one unstable periodic solution.The effects of nonlinear stiffness on steady-state response is also analyzed through numerical simulation.However,the approximate solution obtained by the singular perturbation method is not sufficient to describe the global characteristics of the system,therefore,the necessary condition for the chaos in the sense of Smale horseshoes is derived based on the Melnikov method.Finally,one-demonstrational system that meets the condition of simultaneous resonance is analyzed through numerical simulation,and the bifurcation diagram shows the two thresholds of the demonstration system.At the first threshold,the heteroclinic orbit of the system breaks,and the system goes to chaos in crisis way.At the second threshold,the crisis reappears and the new strange attractor appears.The variation of the first critical value under various frequency combinations is investigated based on the Melnikov method,and the results are compared with the results of numerical simulation.The analytical and numerical results are qualitatively the same although there is a quantitative difference between them.