Vibrational Resonance in the Duffing Oscillator with Distributed Time-Delayed Feedback

We analyze the vibrational resonance in the Duffing oscillator system in the presence of (i) a gamma distributed time-delayed feedback and (ii) integrative time-delayed (uniformly distributed time delays over a finite interval) feedback. Particularly, applying a theoretical procedure we obtain an expression for the response amplitude $Q$ at the low-frequency of the driving biharmonic force. For both double-well potential and single-well potential cases we are able to identify the regions in parameter space where either (i) two resonances, (ii) a single resonance or (iii) no resonance occur. Theoretically predicted values of $Q$ and the values of a control parameter at which resonance occurs are in good agreement with our numerical simulation. The analysis shows a strong influence of both types of time-delayed feedback on vibrational resonance.