  The dynamics of large systems of coupled oscillators is a subject of
increasing importance with prominent applications in several areas such as
physics and biology. The Kuramoto model, where a set of oscillators move around
a circle representing their phases, is a paradigm in this field, exhibiting a
continuous transition between disordered and synchronous motion. Reinterpreting
the oscillators as rotating unit vectors, the model was extended to allow
vectors to move on the surface of D-dimensional spheres, with $D=2$
corresponding to the original model. It was shown that the transition to
synchronous dynamics was discontinuous for odd D, raising a lot of interest.
Inspired by results in 2D, Ott et al proposed an ansatz for density function
describing the oscillators and derived equations for the ansatz parameters,
effectively reducing the dimensionality of the system. Here we take a different
approach for the 3D system and construct an ansatz based on spherical harmonics
decomposition of the distribution function. Our result differs significantly
from that proposed in Ott's work and leads to similar but simpler equations
determining the dynamics of the order parameter. We derive the phase diagram of
equilibrium solutions for several distributions of natural frequencies and find
excellent agreement with simulations. We also compare the dynamics of the order
parameter with numerical simulations and with the previously derived equations,
finding good agreement in all cases. We believe our approach can be generalized
to higher dimensions and help to achieve complexity reduction in other systems
of equations.
