Complex Resonance Behaviors of Weak Nonlinear Duffing-van der Pol Systems Under Multi-frequency Excitation

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.