Science and Technology Production
Article
Authorship
De Vita, Cecilia
;
GROISMAN, PABLO JOSE
;
Huang, Ruojun
Date
2026
Publishing House and Editing Place
SIAM PUBLICATIONS
Magazine
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS,
vol. 25
(pp. 765-788)
SIAM PUBLICATIONS
Summary
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The Kuramoto model is a classical nonlinear ODE system designed to study synchronization phenomena. Each equation represents the phase of an oscillator, and the coupling between them is determined by a graph. There is an increasing interest in understanding the relation between the graph topology and the spontaneous synchronization of the oscillators. Abdalla, Bandeira, and Invernizzi [SIAM J. Appl. Dyn. Syst., 23 (2024), pp. 779?790] considered random geometric graphs on the 𝑑-dimensio...
The Kuramoto model is a classical nonlinear ODE system designed to study synchronization phenomena. Each equation represents the phase of an oscillator, and the coupling between them is determined by a graph. There is an increasing interest in understanding the relation between the graph topology and the spontaneous synchronization of the oscillators. Abdalla, Bandeira, and Invernizzi [SIAM J. Appl. Dyn. Syst., 23 (2024), pp. 779?790] considered random geometric graphs on the 𝑑-dimensional sphere and proved that the system synchronizes with high probability as long as the mean number of neighbors and the dimension 𝑑 go to infinity. They posed the question about the behavior when 𝑑 is small. In this paper, we prove that synchronization holds for random geometric graphs on the two-dimensional sphere, with high probability as the number of nodes goes to infinity, as long as the initial conditions converge to a smooth function. We conjecture a similar behavior for more general simply connected closed Riemannian manifolds, but we expect global synchronization to fail if the manifold is not simply connected, as was shown in De Vita, Bonder, and Groisman [SIAM J. Appl. Dyn. Syst., 24 (2025), pp. 1?15] and suggested in Cirelli et al. [SIAM J. Appl. Math., 85 (2025), pp. 1719?1748].
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Key Words
Kuramoto modelinteracting dynamical systemsrandom geometric graphssynchronization