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The Ott-Antonsen ansatz shows that, for certain classes of distribution ofthe natural frequencies in systems of $N$ globally coupled Kuramotooscillators, the dynamics of the order parameter, in the limit $N\to \infty$,evolves, under suitable initial conditions, in a manifold of low dimension.This is not possible when the frequency distribution, continued in the complexplane, has an essential singularity at infinite; this is the case for example,of a Gaussian distribution. In this work we propose a simple approximationscheme that allows to extend also to this case the representation of thedynamics of the order parameter in a low dimensional manifold. Using as aworking example the Gaussian frequency distribution, we compare the dynamicalevolution of the order parameter of the system of oscillators, obtained by thenumerical integration of the $N$ equations of motion, with the analogousdynamics in the low dimensional manifold obtained with the application of theapproximation scheme. The results confirm the validity of the approximation.The method could be employed for general frequency distributions, allowing thedetermination of the corresponding phase diagram of the oscillator system.

The study of the dynamics of the order parameter of coupled oscillators in the Ott-Antonsen scheme for generic frequency distributions