Stability of phase locking in a ring of unidirectionally coupled oscillators

We discuss the dynamic behaviour of a finite group of phase oscillators unidirectionally coupled in a ring. The dynamics are based on the Kuramoto model. In the case of identical oscillators, all phase locking solutions and their stability properties are obtained. For nonidentical oscillators it is proven that there exist phase locking solutions for sufficiently strong coupling. An algorithm to obtain all phase locking solutions is proposed. These solutions can be classified into classes, each with its own stability properties. The stability properties are obtained by means of a novel extension of Gershgorin's theorem. One class of stable solutions has the property that all phase differences between neighbouring cells are contained in (-pi/2, pi/2). Contrary to intuition, a second class of stable solutions is established with exactly one of the phase differences contained in (pi/2, 3pi/2). The stability results are extended from sinusoidal interconnections to a class of odd functions. To conclude, a connection with the field of active antenna arrays is made, generalizing some results earlier obtained in this field.