Subharmonic Resonance of Fractional‐Quintic van der Pol–Duffing Oscillator

In this paper, the dynamical characteristics of nonlinear fractional van der Pol–Duffing systems are studied. The harmonic balance method is utilized to determine the main resonance of the system, and the approximate analytical solution of 1/3 subharmonic resonance of the system is calculated by means of the averaging method. The analytical expressions of amplitude‐frequency and phase‐frequency characteristic curves based on the harmonic balance method and the averaging method are established. The analytical solutions of the system are analyzed by Routh–Hurwitz and the stability conditions of the system are obtained. The numerical simulation is used to compare the analytical solution with the numerical solution. The results show that the numerical solution and the analytical solution are consistent in trend. The paper focuses on the research and analysis of the fork bifurcation and saddle‐node bifurcation of the system and explores the influence of cubic stiffness, quintic stiffness, linear damping, nonlinear damping, and external excitation amplitude on the bifurcation law of the system. Finally, we analyze the chaos threshold under external excitation, as well as the time history diagram, phase trajectory and Poincaré section of the system with and without chaos, so as to verify the correctness of the derivation of chaos theory in this paper. The results indicate that as the excitation frequency and amplitude change, the system enters chaotic motion through a dual period bifurcation.