Complexity reduction in the 3D Kuramoto model

• Published in: Chaos, solitons and fractals Vol. 149; p. 111090
• Main Authors: Barioni, Ana Elisa D., de Aguiar, Marcus A.M.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.08.2021
• Subjects: Dynamics; Equilibrium; Kuramoto; Oscillators; Syncronization; Kuramoto; Dynamics; Equilibrium; Oscillators; Syncronization
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2021.111090

•We propose a new type of complexity reduction formalism for the 3D Kuramoto model.•We accurately reproduce the first order transition for different distributions.•Our order parameter dynamics is equally accurate and simpler than previous works.•This novel derivation brings out computational advantages.•The approach might be extended to larger dimensions or other systems of coupled equations. 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 higher dimensions by allowing 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. Inspired by results in 2D, Ott et al proposed an ansatz for the density function describing the oscillators and derived equations for the ansatz parameters, effectively reducing the dynamics complexity. 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 from 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 numerical solutions for the full system dynamics. We believe our approach can be generalized to higher dimensions, leading to complexity reduction in other systems of coupled equations.