Exact solution at a transition to frequency synchronization of three coupled phase oscillators

Abstract

A model of three bidirectionally coupled phase oscillators in a ring is studied at the transition to a complete frequency synchronization. Analytic expressions for the critical coupling strengths, at which oscillators synchronize to a common frequency, are obtained. These expressions are determined for cases when the initial oscillators’ frequencies are arranged arbitrarily or they are assigned according to fixed separations. Three unidirectionally coupled phase oscillators are synchronized in an analogous manner to the bidirectional system. This finding allows us to find an analytic equation for the critical coupling strength in the case of the model of the unidirectionally coupled phase oscillators. The bifurcation diagrams show excellent agreement between the analytic formulas and the numerical solutions of the differential equations that describe the models.