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arXiv · 2605.22983

Global analysis of the Kuramoto flow

Daniel Burns·Gregorio Malajovich·Charles Pugh·Indika Rajapakse·Steve Smale
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Abstract

Kuramoto's differential equation describes a synchronization process between several harmonic oscillators. It has been used to model biological phenomena such as the synchronization of heart cells, the circadian rhythm, or brain waves. It is also used in power system control. The simplest possible model assumes that all oscillators are identical and connected to each other with equal pairwise attraction. In this paper, we give a full geometric description of its global dynamics in terms of Morse theory and dynamical systems. Most of this description is stable in the sense that it is topologically preserved under small perturbations of the parameters.

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Daniel Burns, Gregorio Malajovich, Charles Pugh, Indika Rajapakse, Steve Smale. 2026-05-21. Global analysis of the Kuramoto flow. https://arxiv.org/abs/2605.22983

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