  The Kuramoto model, describing the synchronization dynamics of coupled
oscillators, has been generalized in many ways over the past years. One recent
extension of the model replaces the oscillators, originally characterized by a
single phase, by particles with D-1 internal phases, represented by a point on
the surface of the unit D-sphere. Particles are then more easily represented by
D-dimensional unit vectors than by D-1 spherical angles. However, numerical
integration of the state equations should ensure that the propagated vectors
remain unit and that particles rotate on the sphere as predicted by the
dynamical equations. As discussed in [1] integration of the three-dimensional
Kuramoto model using Euler's method with time step $\Delta t$ not only changes
the norm of the vectors but produces a small rotation of the particles around
the wrong axis. Importantly, the error in the axis' direction does not vanish
in the limit $\Delta t \rightarrow 0$. Therefore, instead of displacing the
unit vectors in the direction of the velocity one should performed a sequence
of direct small rotations, as dictated by the equations of motion. This keeps
the particles on the sphere at all times, ensuring exact norm preservation, and
rotates the particles around the proper axis for small $\Delta t$ [1]. Here I
propose an alternative way to do such integration by rotations in 3D that can
be generalized to more dimensions using Cayley-Hamilton's theorem. Explicit
formulas are provided for 2, 3 and 4 dimensions. I also compare the results
with the forth order Runge-Kutta method, which seems to provide accurate
results even requiring renormalization of the vectors after each integration
step.
