Vibrational resonance in a fractional order system with asymmetric bistable potential and time delay feedback

Abstract

In this paper, the bifurcation and vibrational resonance phenomena of fractional order system with time delay feedback and asymmetric potential are investigated, and approximate expression for the response amplitude at low frequency is obtained by the fast and slow variables separation method. We found that the bifurcation and resonance phenomena can be induced by asymmetric parameter, fractional order, time delay and so on, respectively. Importantly, we demonstrate that the introduction of asymmetric parameter can increase the number of resonance. In addition, the equilibrium points of the system and resonance phenomena change periodically with time delay. It is peculiar that the maximum response amplitude from both the shallower potential and the deeper potential are decreasing at first and then increasing under the influence of the fractional order α. But with the impact of the asymmetric parameter a, the maximum response amplitude from the shallower potential first shows an opposite trend to the deeper, then both of them are exhibiting the same trend. Ultimately, we use the numerical solution as a baseline and introduce relative error to confirm the validity of the theoretical analysis method.