The effect of the fractional derivative order on vibrational resonance in a special fractional quintic oscillator

In this work, we make the general study of the vibratory dynamics of a special fractional quintic oscillator discovered in complex media. First of all, we consider that the system is excited by a combination of both low-frequency force and high-frequency force. Then, we analyze the occurrence of vibrational resonance, where the response consists of a slow motion and a fast motion respectively with low and high frequencies. Through this, we obtain an approximate analytical expression of the response amplitude and we determine the values of the low frequency and the amplitude of the high-frequency force at which vibrational resonance occurs. The theoretical predictions are found to be in good agreement with numerical results. Moreover, for fixed values of the system parameters, varying the order of the fractional derivatives can introduce new vibrational resonance phenomena. We found that low value of the fractional derivative order favor the occurrence of the first vibrational resonance with cross-well motions.