Exploring the phase diagrams of multidimensional Kuramoto models

The multidimensional Kuramoto model describes the synchronization dynamics of particles moving on the surface of D-dimensional spheres, generalizing the original model where particles were characterized by a single phase. In this setup, particles are more easily represented by $D$-dimensional unit vectors than by $D-1$ spherical angles, allowing for the coupling constant to be extended to a coupling matrix acting on the vectors. As in the original Kuramoto model, each particle has a set of $D(D-1)/2$ natural frequencies, drawn from a distribution. The system has a large number of independent parameters, given by the average natural frequencies, the characteristic widths of their distributions plus $D^2$ constants of the coupling matrix. General phase diagrams, indicating regions in parameter space where the system exhibits different behaviors, are hard to derive analytically. Here we obtain the complete phase diagram for $D=2$ and Lorentzian distributions of natural frequencies using the Ott-Antonsen ansatz. We also explore the diagrams numerically for different distributions and some specific choices of parameters for $D=2$, $D=3$ and $D=4$. In all cases the system exhibits at most four different phases: disordered, static synchrony, rotation and active synchrony. Existence of specific phases and boundaries between them depend strongly on the dimension $D$, the coupling matrix and the distribution of natural frequencies.