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arXiv · 2512.06399

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

Li Chen·Seung-Yeal Ha·Xinyu Wang·Valeriia Zhidkova
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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.

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Li Chen, Seung-Yeal Ha, Xinyu Wang, Valeriia Zhidkova. 2025-12-06. Emergent behaviors of the singular continuum Kuramoto model and its graph limit. https://arxiv.org/abs/2512.06399

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