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 $\ensuremath{\mu}$ increases is found. Near the transition point ${\ensuremath{\mu}}_{0}$ a scaling $M\ensuremath{\sim}(\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{0}{)}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.5$ 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.