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

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.