Synchronization in an array of globally coupled maps with delayed interactions

We study synchronization of a one-dimensional array of coupled logistic maps in the regime where the individual maps, without coupling, evolve in a periodic orbit. We investigate the effect of a delay in the coupling that takes into account the finite velocity of propagation of interactions. Two qualitatively different synchronization regimes are found, depending on the value of the coupling strength. For weak coupling the array divides into clusters, and the behavior of the individual elements within each cluster depends on the delay times. For strong enough coupling, the array synchronizes into a single cluster. The evolution of the elements is periodic and their relative phases depend on the delay times.