Universality in the periodicity manifestations in the non-locally coupled map lattices in the turbulent regime

In the turbulent regime of coupled map lattice with non-local interaction the maps systematically form periodic cluster attractors and their remnants by synchronization due to the foliation of periodic windows of the element map. We examine these periodicity manifestations in three types of coupled map lattices in $D=1, 2, 3$. In the first two, the interaction is all-to-all but the coupling decreases with distance in a power and an exponential law. In the third, the interaction is uniform but cut off sharply. We find that in all three models and in all dimensions periodicity manifests universally from turbulence when the same suppression of the local mean field fluctuation is achieved by the non-local averaging.