Averaging of globally coupled oscillators

Abstract

We study a specific system of symmetrically coupled oscillators using the method of averaging. The equations describe a series array of Josephson junctions. We concentrate on the dynamics near the splay-phase state (also known as the antiphase state, ponies on a merry-go-round, or rotating wave). We calculate the Floquet exponents of the splay-phase periodic orbit in the weak-coupling limit, and find that all of the Floquet exponents are purely imaginary; in fact, all the Floquet exponents are zero except for a single complex conjugate pair. Thus, nested two-tori of doubly periodic solutions surround the splay-phase state in the linearized averaged equations. We numerically integrate the original system, and find startling agreement with the averaging results on two counts: The observed ratio of frequencies is very close to the prediction, and the solutions of the full equations appear to be either periodic or doubly periodic, as they are in the averaged equations. Such behavior is quite surprising from the point of view of generic dynamical systems theory-one expects higher-dimensional tori and chaotic solutions. We show that the functional form of the equations, and not just their symmetry, is responsible for this nongeneric behavior.