Vibrational Resonance in Time-Delayed Nonlinear Systems

Time-delay is ubiquitous in many dynamical systems. The role of single and multiple time-delay on vibrational resonance in a single Duffing oscillator and in a system of n Duffing oscillators coupled unidirectionally and driven by both a lowand a high-frequency periodic force is presented. The investigation is performed through both theoretical approach and numerical simulation. Theoretically determined values of the amplitude of the high frequency force and the delay-time at which resonance occurs are in very good agreement with the numerical simulation. A major consequence of time-delay feedback is that it gives rise to a periodic or quasiperiodic pattern of vibrational resonance profile with respect to the timedelay parameter. For the system of n-coupled oscillators with a single time-delay coupling, the response amplitudes of the oscillators are shown to be independent of the time-delay. In the case of a multi time-delayed coupling, undamped signal propagation occurs for coupling strength (ı) above a certain critical value (denoted as ıu). Further, the response amplitude approaches a limiting value QL with the oscillator number i . Analytical expressions for both ıu and QL are determined.