It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.
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This follows by noting that the Cauchy–Riemann conditions imply that the complex function satisfies Laplace’s equation in , , where . Setting , this gives . Since in , cannot assume a maximum anywhere interior to the lower-half complex -plane. Hence the maximum value of in must occur on , i.e., on the real -axis.
While the time-asymptotic behavior of the order-parameter obtained from the dynamics of the full system is seen to correspond to the attractors of the reduced systems (10) and (13) [where Eq. (13) is treated in Ref. 10], we emphasize that does not need to be attracting for the microscopic state . This can be simply seen in the case of the Kuramoto problem (4), for which and for all is a solution for and any choice of perturbations , .