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Vibrational resonance and bifurcation in a fractional order quintic system with distributed time delay

In this paper, vibrational resonance and bifurcation phenomena are investigated in the fractional order quintic system with distributed time delay. The approximate theoretical expression for the response amplitude at low frequency is derived by using the method of fast and slow variable separation. In the presence of fractional order and distributed time delay, the vibrational resonance and bifurcation are explored in the single-well, double-well and triple-well potentials, respectively. It is found that the decay rate of the kernel function σ can cause bifurcation, which is closely related to the high frequency excitation amplitude F. It is worth noting that the range of ωr2 in which resonance occurs twice, three and four times decreases with the increase of σ. Furthermore, appropriate values of the decay rate σ, fractional order α and high frequency excitation amplitude F will enable to improve the response amplitude Q. The high match of the numerical simulation and analytical results confirms the validity of the theoretical analysis.