Phase order in chaotic maps and in coupled map lattices

By defining a direction phase as the direction of two sequential iterations of the logistic map, a transition of a net direction phase ill from zero to a finite value as the parameter mu increases is found. Near the transition point mu(0) a scaling M similar to (mu - mu(0))(alpha) with alpha = 0.5 is obtained. The order state of the direction phases in a coupled map lattice is also studied. A phase synchronization of the direction phases is found although the lattices still remain chaotic.