Amplitude death in intrinsic time-delayed chaotic oscillators with direct–indirect coupling: the existence of death islands

We investigate the transition between oscillatory and amplitude death (AD) states and the existence of death islands in intrinsic time-delayed chaotic oscillators under the simultaneous presence of diffusive (direct) and environmental (indirect) coupling. Studies in two-parameter space reveal that depending upon parameters and intrinsic time delays the coupling can bring the oscillators to the AD state and again can revive the system to oscillatory states, thus creating death islands in parameter space; this observation is in sharp contrast to the death scenario of non-delayed oscillators under the same coupling scheme where no death islands are formed. Using a linear stability analysis, we derive the explicit conditions for different transition scenarios. We use a continuation package for the time-delay systems to precisely identify the zone of AD and its islands and their origin. We also extend our study to the network of oscillators and show that the observed results are general for a large number of oscillators, too. Finally, we demonstrate our results experimentally to verify the analytical and numerical findings.