From synchronous to one-time delayed dynamics in coupled maps

We study the completely synchronized states (CSSs) of a system of coupled logistic maps as a function of three parameters: interaction strength (ɛ), range of the interaction (α), that can vary from first neighbors to global coupling, and a parameter (β) that allows one to scan continuously from nondelayed to one-time delayed dynamics. In the α-ɛ plane we identify periodic orbits, limit cycles, and chaotic trajectories, and describe how these structures change with delay. These features can be explained by studying the bifurcation diagrams of a two-dimensional nondelayed map. This allows us to understand the effects of one-time delays on CSSs, e.g., regularization of chaotic orbits and synchronization of short-range coupled maps, observed when the dynamics is moderately delayed. Finally, we substitute the logistic map with cubic and logarithmic maps, in order to test the robustness of our findings.