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arXiv · 2309.16538

On the emergent dynamics of the infinite set of Kuramoto oscillators

Seung-Yeal Ha·Euntaek Lee·Woojoo Shim
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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.

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Seung-Yeal Ha, Euntaek Lee, Woojoo Shim. 2023-09-28. On the emergent dynamics of the infinite set of Kuramoto oscillators. https://arxiv.org/abs/2309.16538

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