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Effect of multiple time-delay on vibrational resonance

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10.1063/1.4793542

### Abstract

We report our investigation on the effect of 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 low- and a high-frequency periodic force. For the single oscillator, we obtain analytical expressions for the response amplitude *Q* and the amplitude *g* of the high-frequency force at which resonance occurs. The regions in parameter space of enhanced *Q* at resonance, as compared to the case in absence of time-delay, show a bands-like structure. For the two-coupled oscillators, we explain all the features of variation of *Q* with the control parameter *g*. 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 takes place for coupling strength (δ) above a certain critical value (denoted as ). Moreover, the response amplitude approaches a limiting value with the oscillator number *i*. We obtain analytical expressions for both and .

© 2013 American Institute of Physics

Received 28 September 2012
Accepted 08 February 2013
Published online 06 March 2013

Lead Paragraph:

Time-delay is ubiquitous in many dynamical systems and the investigation of various nonlinear phenomena in time-delay systems has received a great deal of interest in recent years. In certain systems, time-delay is found to induce various phenomena which do not occur in its absence. The presence of a single time-delay is shown to alter the stability of an equilibrium point, gives birth to a limit cycle, leads to bifurcation, chaos and different types of synchronization.^{1–3} More than one time-delay is applied in certain practical systems. For example, in a laser system multi time-delays arise when it is subjected to more than one optical reflection. Therefore, it is important to analyse the effect of multi time-delay on various nonlinear phenomena. In the present work, we explore the influence of multi time-delay on vibrational resonance in the single Duffing oscillator and signal transduction in a system of *n*-coupled Duffing oscillators.

Acknowledgments: C.J. expresses his gratitude to University Grants Commission (UGC), India for the financial support in the form of UGC meritorious fellowship. M.A.F.S. acknowledges the financial support from the Spanish Ministry of Science and Innovation under Project No. FIS2009-09898.

Article outline:

I. INTRODUCTION

II. SINGLE DUFFING OSCILLATOR

A. Theoretical expression for the response amplitude *Q*

B. Resonanceanalysis

C. Determination of the value of *g* at which resonance occurs

III. RESONANCE IN TWO-COUPLED DUFFING OSCILLATORS WITH MULTIPLE TIME-DELAYED COUPLING

A. Theoretical approach

B. Resonanceanalysis

IV. SIGNAL PROPAGATION IN A SYSTEM OF *N*-COUPLED OSCILLATORS

V. CONCLUSIONS

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2013-03-06

2014-04-25

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