Vibrational Resonance in a Duffing System with a Generalized Delayed Feedback

We investigate the vibrational resonance in the Duffing system with different kinds of delayed feedback. Our approach is to consider the delayed feedback as a generalized delayed feedback in a fractional-order differential version. For three special cases, the generalized delayed feedback corresponds to displacement delayed feedback, velocity delayed feedback, and acceleration delayed feedback respectively. At first, based on the vibrational mechanism, the approximate solution of the system is obtained. Then, we give conditions for all resonance patterns. The theoretical predictions are verified by numerical simulations. Furthermore, the theoretical results are in good agreement with the numerical simulations. Since the delayed feedback is in a generalized form, our results can be regarded as universal for the vibrational resonance in nonlinear systems with different kinds of delayed feedback.