Synchronization of regular and chaotic systems

We investigate the synchronization between two systems consisting of coupled circle maps that have a common drive, which may be chaotic or regular. We observe several new aspects of chaotic and regular synchronization. In the chaotic regime the transition from synchronization to nonsynchronization corresponds to the transition from one to two positive Liapunov exponents. We find regions in the parameter space with periodic motion where synchronization is always achieved, never achieved or, depending on the initial conditions, sometimes achieved. The nonsynchronization or synchronization are stable in the presence of a weak chaotic (or noisy) signal.