Analysis of a quintic system with fractional damping in the presence of vibrational resonance

In the present paper, the phenomenon of the vibrational resonance in a quantic oscillator that possesses a fractional order damping and is driven by both the low and the high frequency periodic signals is investigated, and the approximate theoretical expression of the response amplitude at the low-frequency is obtained by utilizing the method of direct partition of motions. Based on the definition of the Caputo fractional derivative, an algorithm for simulating the system is introduced, and the new method is shown to have higher precision and better feasibility than the method based on the Grünwald -Letnikov expansion. Due to the order of the fractional derivative, various new resonance phenomena are found for the system with single-well, double-well, and triple-well potential, respectively. Moreover, the value of fractional order can be treated as a bifurcation parameter, through which, it is found that the slowly-varying system can be transmitted from a bistability system to a monostabillity one, or from tristability to bistability, and finally to monostabillity. Unlike the cases of the integer-order system, the critical resonance amplitude of the high-frequency signal in the fractional system does depend on the damping strength and can be significantly affected by the fractional-order damping. The numerical results given by the new method is found to be in good agreement with the analytical predictions.