Phase and frequency shifts in a population of phase oscillators

We investigate the synchronization behavior of a population of globally coupled phase oscillators with randomly distributed eigenfrequencies. A variety of synchronized states is rigorously analyzed by means of the center manifold theorem combined with a nonlinear renormalization procedure. This way, the different synchronized states are determined explicitly. We observe $n$-cluster states, where $n=1,\dots ,4\mathrm{}$. In a synchronized state all oscillators have the same frequency. Synchronization frequency and mutual phase differences depend on model parameters as well as on the configuration of the synchronized state, i.e., the number of oscillators within each cluster. The synchronization frequency may decisively differ from the mean of the eigenfrequencies, e.g., giving rise to frozen states, which are synchronized states with vanishing synchronization frequency. Unstable cluster states are associated with transitions between different synchronization frequencies. Our approach can easily be extended to a population of oscillators with randomly distributed coupling strengths. The different synchronized states are discussed in the context of neural coding.