Bifurcation and vibration resonance in the time delay Duffing system with fractional internal and external damping

This paper is focused on investigating the bifurcation and vibration resonance problems of fractional double-damping Duffing time delay system driven by external excitation signal with two wildly different frequencies omega and Omega Firstly, the approximate expressions of the critical bifurcation point and response amplitude Q at low-frequency omega are obtained by means of the direct separation of the slow and fast motions. And then corresponding numerical simulation is made to show that it is a good agreement with the theoretical analysis. Next, the influence of system parameters, including internal damping order alpha, external damping order lambda, high-frequency amplitude F, and time delay size tau, on the vibration resonance is discussed. Some significant results are obtained. If the fractional orders alpha and lambda are treated as a control parameter, then alpha and lambda can induce vibration resonance of the system in three different types when the response amplitude Q changes with the high-frequency amplitude F. If the high-frequency amplitude F is treated as a control parameter, then F can induce vibration resonance of the system as well at some particular points. If the time delay tau is treated as a control parameter, not only can tau induce three types of vibration resonance, but the response amplitude Q views periodically with tau. In addition, the resonance behaviors of the considered system are more abundant than those in other similar systems since the internal damping order alpha, external damping order lambda, time delay tau and cubic term coefficient beta are introduced into the system which changes the shapes of the effective potential function.