Vibration resonance and fork bifurcation of under-damped Duffing system with fractional and linear delay terms

In this paper, we mainly study the bifurcation and resonance of under-damped Duffing systems with fractional order delay and fractional order under-damped Duffing systems with linear delay. Based on the separation method of fast and slow variables, the high-frequency excitation components in the system are eliminated, and the equivalent system of slow variables is obtained. For the under-damped Duffing system with fractional delay term, the harmonic balance method is used to solve the amplitude and phase analytic solution of the slow variable system, and for the under-damped fractional Duffing system with linear delay term, the average method is used to solve the amplitude and phase analytic solution of the slow variable system. Then the resonance and bifurcation of bistable and monostable systems with different parameters are analyzed. In the last section of this paper, numerical simulation is carried out to study the influence of fractional order, control parameters, delay quantity and other factors on the two systems, and the correctness of the analytical analysis is verified by comparing the numerical simulation results.