Delay-Induced Resonance in the Time-Delayed Duffing Oscillator

Abstract

The phenomenon of delay-induced resonance implies that in a nonlinear system a time-delay term may be used as an effective enhancer of the oscillations caused by an external forcing maintaining the same frequency. This is possible for the parameters for which the time-delay induces sustained oscillations. Here, we study this type of resonance in the overdamped and underdamped time-delayed Duffing oscillators, and we explore some new features. One of them is the conjugate phenomenon: the oscillations caused by the time-delay may be enhanced by means of the forcing without modifying their frequency. The resonance takes place when the frequency of the oscillations induced by the time-delay matches the ones caused by the forcing and vice versa. This is an interesting result as the nature of both perturbations is different. Even for the parameters for which the time-delay does not induce sustained oscillations, we show that a resonance may appear following a different mechanism.

Extremely Broadband Stochastic Resonance of Light and Enhanced Energy Harvesting Enabled by Memory Effects in the Nonlinear Response

“…The increase in x rms =x 0 rms as τ → γ −1 from above indicates that the limit cycles emerge from the interplay of different timescales (τ and γ −1 ). Since somewhat similar limit cycles have been observed in nonlinear oscillators with constant time delay in their linear response [59], we analyze the relevance of our noninstantaneous nonlinearity for different V. the fixed energy harvesting time. Therefore, effective nonlinear energy harvesting is restricted to a narrow range of potentials in the Markovian regime.…”

Delay-induced resonance suppresses damping-induced unpredictability

“…As previously mentioned, the time delay induces sustained oscillations in the system for certain values of the parameters ( γ , τ ). The dependence of the amplitude of these oscillations on τ has been studied in the case of the absence of dissipation, μ = 0, in [16]. The results are summarized in figure 1 a , where we show the peak-to-peak values of the amplitude A versus the time-delay τ .…”

“…The delay term can play the role of one of the periodic forcings typically used in vibrational resonance, and the resonance can be triggered by the cooperation of the time delay and just one external periodic forcing as shown in [15]. The latter phenomenon has been called delay-induced resonance and has been studied among other fields in the context of meteorology, as for example the ENSO model in [16]. Other branches of science where time delay is relevant (neural networks, population dynamics or electronics, as cited before) will be an important focus of study in the following years.…”

Effect of Different Forms of Periodic Piecewise Linear Forces on Duffing Oscillator

“…Duffing oscillator equation represents a second-order ordinary differential equation with cubic nonlinearity and periodic excitation. From the last century, the Duffing-type nonlinear dynamical systems have been investigated uninterruptedly in various fields conducted by plenty of researches, such as physics, engineering, chemistry, economics and biological and social sciences [12][13][14][15][16][17][18][19][20]. The effect of smooth single or multiple forces has been studied extensively in Duffing oscillator [12,13,[21][22][23][24][25].…”