Complexity reduction in the 3D Kuramoto model

Abstract

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.