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As has been shown by Watanabe and Strogatz (WS) [Phys. Rev. Lett., 70, 2391(1993)], a population of identical phase oscillators, sine-coupled to a commonfield, is a partially integrable system for any size: its dynamics reduces toequations for several collective variables. Here we develop a perturbationapproach for weakly nonidentical ensembles. We calculate corrections to the WSdynamics for two types of perturbations: due to a distribution of naturalfrequencies and of forcing terms, and due to small white noise. We demonstrate,that in both cases the complex mean field for which the dynamical equations arewritten, is close up to the leading order in the perturbation to the Kuramotoorder parameter. This supports validity of the dynamical reduction suggested byOtt and Antonsen [Chaos, 18, 037113 (2008)] for weakly inhomogeneouspopulations.

Dynamics of weakly inhomogeneous oscillator populations: Perturbation theory on top of Watanabe-Strogatz integrability