Phase synchronization in coupled chaotic oscillators with time delay

The phase synchronization (PS) of two Rössler oscillators with time-delayed signal coupling is studied. We find that time delay can always lead to PS even when the delay is very long. Moreover, with the increase of time delay, the coupling strength at the transition to PS undergoes a nearly periodic wave distribution. At some fixed time-delayed signal coupling, a PS region is followed by a non-PS region when the coupling strength increases. However, an increase of the coupling leads to the PS state again. This phenomenon occurs in systems with a relatively large PS transition point.