Emergent behaviors of the singular continuum Kuramoto model and its graph limit

TL;DR

This paper analyzes the emergent synchronization behaviors of the singular continuum Kuramoto model and its graph limit, highlighting the role of singularities and natural frequency functions in phase synchronization phenomena.

Contribution

It introduces a rigorous derivation of the graph limit from finite models and characterizes synchronization behaviors under various conditions.

Findings

Complete phase synchronization occurs in finite time for identical natural frequencies.

Practical synchronization emerges with nonidentical natural frequencies, with phase diameter inversely proportional to coupling strength.

Numerical simulations support the theoretical analysis.

Abstract

We study the emergent dynamics of the singular continuum Kuramoto model (in short, SCKM) and its graph limit. The SCKM takes the form of an integro-differential equation exhibiting two types of nonlocal singularities: a nonlocal singular interaction weight and a nonlocal singular alignment force. The natural frequency function determines the emergent dynamics of the SCKM, and we emphasize that singularity plays a crucial role in the occurrence of sticking phenomena. For the identical natural frequency function, we derive the complete phase synchronization in finite time under a suitable set of conditions for system parameters and initial data. In contrast, for a nonidentical natural frequency function, we show the emergence of practical synchronization, meaning that the phase diameter is proportional to the inverse of coupling strength asymptotically. Furthermore, we rigorously establish a graph limit from the singular Kuramoto model with a finite system size to the SCKM. We also provide several numerical simulations to illustrate our theoretical results.