Primary resonance and feedback control of the fractional Duffing-van der Pol oscillator with quintic nonlinear-restoring force

In the present paper, the primary resonance and feedback control of the fractional Duffing-van der Pol oscillator with quintic nonlinear-restoring force is studied. The approximately analytical solution and the amplitude-frequency equation are obtained using the multiple scale method. Based on the Lyapunov theory, the stability conditions for the steady-state solution are obtained. The bifurcations of primary resonance for system parameters are analyzed, and the influence of parameters on fractional-order model is also studied. Numerical simulation shows that when the parameter values are fixed, the curve bends to the right or left, resulting in jumping phenomena and multi-valued amplitudes. As the excitation frequency changes, the typical hardening or softening characteristics of the oscillator are observed. In addition, the comparisons of approximate analytical solution and numerical solution are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution.