Low dimensional behavior of large systems of globally coupled oscillators

Because synchronous behavior in large groups consisting of many coupled oscillators has been widely observed in many situations, the behavior of such systems has long been of interest. Since the problem is difficult to solve in general, much work has been done on the simple paradigmatic case of globally coupled phase oscillators. Even in this simple context, however, much remains unclear, particularly when considering situations in which a large oscillator population interacts with external dynamical systems, or when there are communities of interacting oscillators with different community and connection characteristics, etc. In this paper we consider an approach that allows the study of the time evolving dynamical behavior of these types of systems by an exact reduction to a small number of ordinary differential equations. This reduction is achieved by considering a restricted class of states. In spite of this restriction, for at least one significant example (see Ref. 10), consideration of our derived ordinary differential equations appears to yield dynamics in precise agreement with results obtained from considerations not imposing this restriction. Thus we believe that our results may be useful in many other contexts.