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Populations of globally coupled phase oscillators are described in thethermodynamic limit by kinetic equations for the distribution densities, orequivalently, by infinite hierarchies of equations for the order parameters.Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariantfinite-dimensional subspace on which the dynamics is described by one complexvariable per population. For oscillators with Cauchy distributed frequencies orfor those driven by Cauchy white noise, this subspace is weakly stable and thusdescribes the asymptotic dynamics. Here we report on an exactfinite-dimensional reduction of the dynamics outside of the Ott-Antonsensubspace. We show, that the evolution from generic initial states can bereduced to that of three complex variables, plus a constant function. Foridentical noise-free oscillators, this reduction corresponds to theWatanabe-Strogatz system of equations [Phys. Rev. Lett. 70, 2391 (1993)]. Wediscuss how the reduced system can be used to explore the transient dynamics ofperturbed ensembles.

Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories