  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 common
field, is a partially integrable system for any size: its dynamics reduces to
equations for several collective variables. Here we develop a perturbation
approach for weakly nonidentical ensembles. We calculate corrections to the WS
dynamics for two types of perturbations: due to a distribution of natural
frequencies 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 are
written, is close up to the leading order in the perturbation to the Kuramoto
order parameter. This supports validity of the dynamical reduction suggested by
Ott and Antonsen [Chaos, 18, 037113 (2008)] for weakly inhomogeneous
populations.
