The Kuramoto model describes how coupled oscillators synchronize their phases as the intensity of the coupling increases past a threshold. The model was recently extended by reinterpreting the oscillators as particles moving on the surface of unit spheres in a $D$-dimensional space. Each particle is then represented by a $D$-dimensional unit vector; for $D=2$ the particles move on the unit circle and the vectors can be described by a single phase, recovering the original Kuramoto model. This multidimensional description can be further extended by promoting the coupling constant between the particles to a matrix $\mathbf{K}$ that acts on the unit vectors. As the coupling matrix changes the direction of the vectors, it can be interpreted as a generalized frustration that tends to hinder synchronization. In a recent paper we studied in detail the role of the coupling matrix for $D=2$. Here we extend this analysis to arbitrary dimensions. We show that, for identical particles, when the natural frequencies are set to zero, the system converges either to a stationary synchronized state, given by one of the real eigenvectors of $\mathbf{K}$, or to an effective two-dimensional rotation, defined by one of the complex eigenvectors of $\mathbf{K}$. The stability of these states depends on the set eigenvalues and eigenvectors of the coupling matrix, which controls the asymptotic behavior of the system, and therefore, can be used to manipulate these states. When the natural frequencies are not zero, synchronization depends on whether $D$ is even or odd. In even dimensions the transition to synchronization is continuous and rotating states are replaced by active states, where the module of the order parameter oscillates while it rotates. If $D$ is odd the phase transition is discontinuous and active states can be suppressed for some distributions of natural frequencies.

• Received 11 January 2023
• Accepted 24 March 2023

DOI:https://doi.org/10.1103/PhysRevE.107.044205