Regular and chaotic phase synchronization of coupled circle maps

We study the effects of regular and chaotic phase synchronization in ensembles of coupled nonidentical circle maps (CMs) and find phase-locking regions for both types of synchronization. We show that synchronization of chaotic CMs is crucially influenced by the three quantities: (i) rotation number difference, (ii) variance of the phase evolution, and (iii)relative duration of intervals of phase increase respect decrease. In the case of regular CMs, only variance and rotation number difference are important. It is demonstrated that with increase of noncoherence of phase evolutionsin the regular and chaotic regime, the regions of the main (1:1) synchronization are usually decreased. We present a chaotic synchronization in the systems of coupled nonidentical circle maps where phase entrainment occurs and it is not accompanied by bifurcations of the chaotic set. For ensembles (chains) of coupled CMs with linear and random distributions of the individual frequencies soft and hard transitions to global synchronization are found.