Phase synchronizations: Transitions from high- to low-dimensional tori through chaos

Phase synchronized entrainment of coupled nonidentical limit cycles and chaotic oscillators is a generic phenomenon in complicated and chaotic dynamics. Recent developments in phase synchronization are reviewed. Two approaches: The statistical approach and the dynamical approach are proposed. From a statistical viewpoint, phase entrainment exhibits a nonequilibrium phase transition from the disordered state to the ordered state. Dynamically, phase synchronization among oscillators shows a tree-like bifurcation and a number of clustered states are experienced. The route from partial to complete phase synchronization for coupled limit cycles is identified as a cascade of transitions from high- to low-dimensional tori (quasiperiodicity) interrupt ed by intermittent chaos. For coupled per iodic oscillators, desynchronization-induced chaos originates from the mixing of intermittent ON-OFF duration time scales. For coupled chaotic cases, the route to phase entrainment is identified as transitions from high- to low-dimensional chaos.