On the emergent dynamics of the infinite set of Kuramoto oscillators

TL;DR

This paper introduces an infinite Kuramoto model to analyze emergent synchronization phenomena across different network topologies, revealing unique behaviors such as constant phase diameter and distinct asymptotic states.

Contribution

It extends the Kuramoto model to an infinite set of oscillators, providing new insights into synchronization dynamics and phase behaviors in complex network topologies.

Findings

Homogeneous ensemble exhibits complete synchronization and gradient flow structure.

Weak synchronization (practical) observed in heterogeneous ensembles.

Constant phase diameter possible in some infinite network topologies.

Abstract

We propose an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row(or column)-summable network topology, we show that a homogeneous ensemble exhibits complete synchronization, and the infinite Kuramoto model can cast as a gradient flow, whereas we obtain a weak synchronization estimate, namely practical synchronization for a heterogeneous ensemble. Unlike with the finite Kuramoto model, phase diameter can be constant for some class of network topologies which is a novel feature of the infinite model. We also consider a second class of network topology (so-called a sender network) in which coupling strengths are proportional to a constant that depends only on sender's index number. For this network topology, we have a better control on emergent dynamics. For a homogeneous ensemble, there are only two possible asymptotic states, complete phase synchrony or bi-cluster configuration in any positive coupling strengths. In contrast, for a heterogeneous ensemble, complete synchronization occurs exponentially fast for a class of initial configuration confined in a quarter arc.