Return map analysis of chaotic phase synchronization

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We investigate chaotic phase synchronization (CPS) observed in a mutually coupled map system and a forced map oscillator by using return maps (RM) of phase difference. RM visualizes the presence or absence of the channel for phase slip which characterizes the onset of CPS. It is found that CPS occurs when the channel starts to be completely closed, and that the saddle node point is approximately equal to the point at which the channel is closed on the average. Furthermore, the proposed RM describes the critical characteristics of the phase difference and reveals the existence of the quasi-synchronized state below the synchronization point.