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.

This paper investigated primary resonance suppression in nonlinear spur gear systems using a hybrid proportional and fractional-order derivative displacement feedback (P-FDDF) controller. The dynamic model of the system was established through a second-order non-autonomous differential equation incorporating time-varying meshing stiffness, backlash, and external excitations. The amplitude-frequency response equation of primary resonance was derived via the multiple scale method, while Lyapunov stability theory was employed to analyze the stability of steady-state solutions. Numerical analyses examined the effects of meshing damping, load fluctuations, meshing stiffness variations, and control parameters on resonance characteristics. Time history responses and phase diagrams demonstrated that the P-FDDF strategy achieves simultaneous resonant amplitude suppression and frequency tuning. The fractional-order component's frequency-weighting and memory properties enhance adaptability to complex nonlinear dynamics induced by time-varying meshing stiffness and backlash, establishing the P-FDDF as a reliable solution for gear system vibration control.

This paper analyzed the primary resonance of a novel fractional-order coronary artery model, exploring cardiovascular diseases related to vascular behavior from the nonlinear dynamics perspective. By applying the averaging method, approximate analytical solutions and the amplitude-frequency equation were derived, whereas Lyapunov stability theory was utilized to analyze the steady-state behavior. Numerical simulations validated the accuracy of the analytical approach, demonstrating close agreement between theoretical predictions and computational results. Key findings include the identification of parameter-driven bifurcations that modulate resonance amplitude and stability. Specifically, a lower fractional order $q$ and reduced damping $\mu$ amplify resonant responses, while increased pulse pressure $F$ correlates with elevated spasm risk. These results provide a theoretical framework for understanding how blood vessel viscoelasticity and external stimuli influence cardiovascular dynamics, offering insights for developing diagnostic tools and therapeutic strategies to mitigate coronary artery spasm and related cardiovascular diseases.