Synergistic effect of time delay, diffusion, and environment in coupled oscillators

We explore the emergent dynamics of coupled oscillators which arise through the interplay of diffusive and environmental coupling in the presence of time delays. We show that unlike environmental coupling delay that induces revival of oscillations, the propagation delay in the diffusive path gives rise to death islands, i.e., both amplitude death and its revival. Moreover, revival from amplitude death shows an interesting phenomenon: if it is achieved through the diffusive coupling delay, the system shows an out-of-phase synchronized oscillations, whereas if the revival is achieved through the environmental coupling delay, an in-phase synchronized solution sets in. We carry out linear stability analysis and numerical analysis to establish our results. Finally, we perform a hardware-level electronic circuit experiment to support our theoretical observations. The present study deepens our understanding of the dynamics and their control in coupled oscillators in the presence of time delay.