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 they are characterized by a single phase. Particles are represented by D-dimensional unit vectors and the coupling constant can be extended to a coupling matrix acting on the vectors. The system has a large number of independent parameters, given by the characteristic widths of the distributions of natural frequencies and the D2 entries of the coupling matrix. Moreover, as the coupling matrix breaks the rotational symmetry, the average values of the natural frequencies also play a key role in the dynamics. 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, for arbitrary coupling matrices and Lorentzian distributions of natural frequencies. We show that the system exhibits four different phases: disordered and static synchrony (as in the original Kuramoto model), rotation of the synchronized cluster (similar to the Kuramoto-Sakaguchi model with frustration) and active synchrony, a new phase where the module of the order parameter oscillates as it rotates on the sphere. We also explore the diagrams numerically for higher dimensions, D=3 and D=4, for particular choices of coupling matrices and frequency distributions. We find that the system always exhibits the same four phases, but their location in the space of parameters depends strongly on the dimension D being even or odd, on the coupling matrix and on the shape of the distribution of natural frequencies.