In this paper, some resonance behaviors and stability for the solutions of the Duffing-van der Pol system under multi-frequency excitation are studied. First, the first-order asymptotic solution of the system under resonance circumstance is obtained by the method of multiple scales, the approximation sketch is compared with the numerical solution of the system, and the result indicates the validity of the asymptotic approximate solution. Then, the amplitude-frequency equation of steady-state response is derived. The amplitude-frequency equation of the system shows phenomenon of multiple solutions. Through numerical simulation, the influences of the Duffing parameter, van der Pol parameter and external force amplitude on the dynamic behavior and stability of the system were studied. Compared to ones of the Duffing system or van der Pol system, the amplitude-frequency relationship of the Duffing-van der Pol system is more complex. Finally, based on Lyapunov stability theory, the stability condition for a steady-state solution involving system parameters in case of the combination resonance is obtained. The research achievements provide a theoretical basis for the analysis and control design of such systems in engineering.

本文研究了多频激励下 Duffing-van der Pol 系统解的一些共振行为和稳定性。首先，采用多尺度方法得到了系统在共振情况下的一阶渐近解，并将近似图与系统的数值解进行了比较，结果表明了渐近近似解的有效性。然后，推导了稳态响应的幅频方程。系统幅频方程呈现多解现象。通过数值模拟，研究了Duffing参数、van der Pol参数和外力幅值对系统动态行为和稳定性的影响。与Duffing系统或van der Pol系统相比，Duffing-van der Pol系统的幅频关系更为复杂。最后，基于Lyapunov稳定性理论，得到了组合谐振情况下涉及系统参数的稳态解的稳定性条件。研究成果为工程中此类系统的分析和控制设计提供了理论依据。