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 $M$ from zero to a finite value as the parameter $\mu$ increases is found. Near the transition point ${\mu }_0$ a scaling $M\sim (\mu -{\mu }_0{)}^{\alpha }$ with $\alpha =0.5\mathrm{}$ 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.