Stability of phase locking in a ring of unidirectionally coupled oscillators

In this paper we discuss the behavior of a finite group of phase oscillators unidirectionally coupled in a ring. The dynamics is described by the Kuramoto model. In the case of identical oscillators, all phase locking solutions are obtained explicitly, together with their stability properties. For nonidentical oscillators it is proven that there exist phase locking solutions for sufficiently strong coupling. An algorithm for obtaining all phase locking solutions is given. These solutions can be subdivided into classes, the stability properties of each of which are established separately. The stability results are extended to interconnections belonging to a class of odd functions. Finally, a connection with the field of active antenna arrays is made, generalizing some results earlier obtained in this field.