Determination of the critical coupling for oscillators in a ring.

We study a model of coupled oscillators with bidirectional first nearest neighbors coupling with periodic boundary conditions. We show that a stable phase-locked solution is decided by the oscillators at the borders between the major clusters, which merge to form a larger one of all oscillators at the stage of complete synchronization. We are able to locate these four oscillators depending only on the set of the initial frequencies. Using these results plus an educated guess (supported by numerical findings) of the functional dependence of the corrections due to periodic boundary conditions, we are able to obtain a formula for the critical coupling, at which the complete synchronization state occurs. Such formula fits well in very good accuracy with the results that come from numerical simulations. This also helps to determine the sizes of the major clusters in the vicinity of the stage of full synchronization.