Sufficiently dense Kuramoto networks are globally synchronizing

Abstract

Consider any network of $n$ identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least $\mu (n-1)$ other oscillators. There is a critical value of the connectivity, ${\mu }_c$, such that whenever $\mu >{\mu }_c$, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when $\mu <{\mu }_c$, there are networks with other stable states. The precise value of the critical connectivity remains unknown, but it has been conjectured to be ${\mu }_c=0.75$. In 2020, Lu and Steinerberger proved that ${\mu }_c\le 0.7889$, and Yoneda, Tatsukawa, and Teramae proved in 2021 that ${\mu }_c>0.6838$. In this paper, we prove that ${\mu }_c\le 0.75$ and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.

arXiv comment6 pages, 1 figure

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