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

TL;DR

This paper investigates how gamma distributed and uniformly distributed time-delayed feedback influence vibrational resonance in the Duffing oscillator, providing theoretical expressions and confirming results with numerical simulations.

Contribution

It introduces a theoretical framework for analyzing vibrational resonance with distributed time delays in the Duffing oscillator, extending previous models.

Findings

Identification of parameter regions with different resonance behaviors

Good agreement between theoretical predictions and numerical simulations

Demonstration of strong influence of time-delayed feedback on resonance

Abstract

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.