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The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry

Max Lipton, Renato Mirollo, Steven H. Strogatz

Abstract

We study a system of $N$ interacting particles moving on the unit sphere in $d$-dimensional space. The particles are self-propelled and coupled all to all, and their motion is heavily overdamped. For $d=2$, the system reduces to the classic Kuramoto model of coupled oscillators; for $d=3$, it has been proposed to describe the orientation dynamics of swarms of drones or other entities moving about in three-dimensional space. Here we use group theory to explain the recent discovery that the model shows low-dimensional dynamics for all $N \ge 3$, and to clarify why it admits the analog of the Ott-Antonsen ansatz in the continuum limit $N \rightarrow \infty$. The underlying reason is that the system is intimately connected to the natural hyperbolic geometry on the unit ball $B^d$. In this geometry, the isometries form a Lie group consisting of higher-dimensional generalizations of the Möbius transformations used in complex analysis. Once these connections are realized, the reduced dynamics and the generalized Ott-Antonsen ansatz follow immediately. This framework also reveals the seamless connection between the finite and infinite-$N$ cases. Finally, we show that special forms of coupling yield gradient dynamics with respect to the hyperbolic metric, and use that fact to obtain global stability results about convergence to the synchronized state.

The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry

Abstract

We study a system of interacting particles moving on the unit sphere in -dimensional space. The particles are self-propelled and coupled all to all, and their motion is heavily overdamped. For , the system reduces to the classic Kuramoto model of coupled oscillators; for , it has been proposed to describe the orientation dynamics of swarms of drones or other entities moving about in three-dimensional space. Here we use group theory to explain the recent discovery that the model shows low-dimensional dynamics for all , and to clarify why it admits the analog of the Ott-Antonsen ansatz in the continuum limit . The underlying reason is that the system is intimately connected to the natural hyperbolic geometry on the unit ball . In this geometry, the isometries form a Lie group consisting of higher-dimensional generalizations of the Möbius transformations used in complex analysis. Once these connections are realized, the reduced dynamics and the generalized Ott-Antonsen ansatz follow immediately. This framework also reveals the seamless connection between the finite and infinite- cases. Finally, we show that special forms of coupling yield gradient dynamics with respect to the hyperbolic metric, and use that fact to obtain global stability results about convergence to the synchronized state.

Paper Structure

This paper contains 17 sections, 2 theorems, 99 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Kuramoto Model on a Sphere
  4. General philosophy: Lie groups and reducible systems
  5. Hyperbolic geometry and Möbius transformations
  6. Möbius transformations in higher dimensions
  7. Infinitesimal generators
  8. Reduced Equations
  9. Comparison of Z Versus W Coordinates
  10. Continuum Limit
  11. Complex Case
  12. Relation to Previous Research
  13. An Example: First-Order Linear Order Parameter Gives Gradient System
  14. A Computer Visualization
  15. Existence of Hyperbolic Gradient
  16. ...and 2 more sections

Key Result

Lemma \oldthetheorem

Any fixed point for the flow flow in $B^d$ is repelling.

Figures (3)

  • Figure 1: A first-order linear Kuramoto system on the two-dimensional sphere $S^2$ with equal weights $a_i=1/N$, and randomly chosen initial conditions. The states shown are at $t = 0, t = 10,$ and $t = 40$ respectively. This simulation was written in Python and visualized with Plotly.
  • Figure 2: A first-order linear Kuramoto system on $S^2$ with weights distributed according to a Riemann sum which approximates the integral of a normal distribution, and randomly chosen initial conditions. Pink particles contribute to the order parameter with greater weights than the blue particles do. The states shown are at $t = 0, t = 10,$ and $t = 40$ respectively.
  • Figure 3: A first-order linear Kuramoto system on $S^2$ with a majority cluster, where one particle is chosen to have a weight which exceeds the combined weights of all other particles (or equivalently, where all the particles have equal weight but a majority of them cluster into a single point and therefore act if they were a single giant particle; hence the name "majority cluster"). The states shown are at $t = 0, t = -10,$ and $t = -40$ respectively; we have chosen to depict time running backward to highlight that the backwards-time limit tends toward an antipodal configuration. In this simulation, one particle, depicted in pink, was chosen to have a weight of $0.6$, and the remaining $99$ particles, depicted as blue, were chosen to have equal weights of $0.4/99$.

Theorems & Definitions (5)

  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • proof