Existence of partial entrainment and stability of phase locking behavior of coupled oscillators

We study a network of all-to-all interconnected phase oscillators as modeled 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. As the coupling strength increases, the distance between each pair of phases decreases. Stability of each phase locking solution is proven for general frequency distributions. There exist one unique asymptotically stable phase locking solution. Furthermore a description is given of partial entrainment, which can be regarded as the finite number analogon of partial synchronization in the infinite number case. When the network is partially entraining some phase differences possess an upper and lower bound. Partial entrainment of the three-cell network is analyzed: an estimate of the onset of partial entrainment is given and the existence of partial entrainment is proven. Furthermore, local stability of partial entrainment is proven for the three-cell network with two identical oscillators.