Primary and subharmonic simultaneous resonance of fractional-order Duffing oscillator

The primary and subharmonic simultaneous resonance of Duffing oscillator with fractional-order derivative is studied. Firstly, the approximately analytical solution of the resonance is obtained by the method of multiple scales, and the correctness and satisfactory precision of the analytical solution are verified by numerical simulation. Then, the amplitude-frequency curve equation and phase-frequency curve equation are derived from the analytical solution. The stability condition of the steady-state response is obtained by Lyapunov's first method, and the state switching between two stable periodic orbits is demonstrated. Finally, the effects of nonlinear factor on the system response are analyzed, and the difference between stiffness softening and stiffness hardening system is demonstrated. The influence of fractional-order term on the system is analyzed in depth, and the effect mechanism of fractional-order term is revealed, i.e., the focus and intensity of effect are determined by the order and coefficient of the fractional-order derivative, respectively.