Kuramoto model  exactly satisfies a *two*-dimensional dynamical system, for the special case where the frequency distribution $g(\omega)$ is Lorentzian and the initial state satisfies certain strong analyticity properties with respect to $\omega$.

In this paper we analyze the two-dimensional system derived from the analysis of Ott and Antonsen \[2008\]. Our results give the first complete picture of the bifurcation structure for the forced Kuramoto model. We obtain explicit formulas for the system’s saddle-node and Hopf bifurcation curves, as well as the codimension-2 Takens-Bogdanov point from which they emanate. Bifurcation theory predicts that a curve of homoclinic bifurcations should also emerge from the Takens-Bogdanov point; we compute this homoclinic curve numerically.

The rest of the paper is organized as follows. Section 2 reviews the approach of Ott and Antonsen \[2008\], leading up to their derivation of the reduced equations for the order parameter dynamics. Section 3 presents new results about the bifurcations in this system and resolves the issue of how all the transition curves fit together. The final section discusses the implications of the results, their relation to prior work, the limitations of the approach used here, and some of the questions that remain.

Derivation of the reduced equations
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