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Propagation delay arises in a coupling channel due to the finite propagation speed of signals and the dispersive nature of the channel. In this paper, we study the effects of propagation delay that appears in the indirect coupling path of direct (diffusive)–indirect (environmental) coupled oscillators. In sharp contrast to the direct coupled oscillators where propagation delay induces amplitude death, we show that in the case of direct–indirect coupling, even a small propagation delay is conducive to an oscillatory behavior. It is well known that simultaneous application of direct and indirect coupling is the general mechanism for amplitude death. However, here we show that the presence of propagation delay hinders the death state and helps the revival of oscillation. We demonstrate our results by considering chaotic time-delayed oscillators and FitzHugh–Nagumo oscillators. We use linear stability analysis to derive the explicit conditions for the onset of oscillation from the death state. We also verify the robustness of our results in an electronic hardware level experiment. Our study reveals that the effect of time delay on the dynamics of coupled oscillators is coupling function dependent and, therefore, highly non-trivial.
Time delay plays a crucial role in determining the dynamics of coupled systems. Understanding of the effects of delay in coupled oscillators is, therefore, important in natural and man-made systems. It is well established that propagation delay in diffusive (or direct) coupling paths of coupled oscillators induces suppression of oscillations. However, in sharp contrast, here we show that propagation delay in the indirect coupling path of a system of direct–indirect coupled oscillators supports oscillations. The simultaneous introduction of diffusive (direct) and environmental (indirect) coupling in coupled oscillators is the general mechanism for obtaining amplitude death. Here, we find that the presence of propagation delay hinders the death state; rather, it is conducive for a revival of oscillations. We demonstrate our results by considering chaotic time-delayed oscillators and the FitzHugh–Nagumo (FHN) oscillator. We construct an electronic hardware level experiment to verify that the results are robust under practical conditions, such as parameter fluctuation and experimental noise. Our study reveals that time delay indeed affects the dynamics of coupled oscillators in a manifold non-trivial manner.
