University of Ghent,Belgium

**Stability of Phase Locking Behavior of Coupled Oscillators**

We study a network of all-to-all interconnected phase oscillators as given by the Kuramoto model. For coupling strengths larger than a critical value, we show the existence of a collective behavior called phase locking: the phase differences between all oscillators are constant in time.

Necessary and sufficient conditions for the existence of phase locking behavior are given. Moreover, the local stability properties of all phase locking solutoins is examined and it is proved that for a sufficiently large coupling strength there exists just one phase locking solution which is locally asymptotically stable.

This type of behavior is different from the partial synchronization behavior described in the literature for a continuum of oscillators, where only a subset of the oscillators is phase locked.

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