Properties of zero-lag long-range synchronization via dynamical relaying

In 2006 Fischer et al. reported the phenomenon of zero-lag long-range isochronous synchronization via dynamical relaying in systems with delay [Phys. Rev. Lett. 97, 123902 (2006)]. They reported that when one has two coupled systems A and C, with delay between them, then the introduction of a third coupled element B between A and C will allow them to synchronize even in regions of the parameter space where this was not possible without the presence of B. Here we studied this phenomenon in detail in coupled Rössler oscillators and found that this happens in three different ways, that is, we verified that when there is feedback between A and B this kind of synchronization can happen via antiphase or in-phase synchronization depending on the delay value. The third case occurs via anticipating synchronization and is found when B drives A and C without getting feedback from them. We found similar phenomena in coupled Kuramoto oscillators, which present periodic orbits only and this allowed us to perform analytical calculations. Finally, we studied coupled logistic maps and identify the region where zero-lag long-range synchronization is possible as the region characterized by a negative transversal Lyapunov exponent.