Complexity reduction in the 3D Kuramoto model

The dynamics of large systems coupled oscillators is a subject increasing importance with prominent applications in several areas such as physics and biology. Kuramoto model, where set move around circle representing their phases, paradigm this field, exhibiting continuous transition between disordered synchronous motion. Reinterpreting the rotating unit vectors, model was extended to higher dimensions by allowing vectors on surface D-dimensional spheres, D=2 corresponding original model. It shown that discontinuous for odd D. Inspired results 2D, Ott et al proposed an ansatz density function describing derived equations parameters, effectively reducing complexity. Here we take different approach 3D system construct based spherical harmonics decomposition distribution function. Our result differs from Ott’s work leads similar but simpler determining order parameter. We derive phase diagram equilibrium solutions distributions natural frequencies find excellent agreement numerical full dynamics. believe our can be generalized dimensions, leading complexity reduction other equations.