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It is shown that, in the infinite size limit, certain systems of globallycoupled phase oscillators display low dimensional dynamics. In particular, wederive an explicit finite set of nonlinear ordinary differential equations forthe macroscopic evolution of the systems considered. For example, an exact,closed form solution for the nonlinear time evolution of the Kuramoto problemwith a Lorentzian oscillator frequency distribution function is obtained. Lowdimensional behavior is also demonstrated for several prototypical extensionsof the Kuramoto model, and time-delayed coupling is also considered.

Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators