Sufficiently dense Kuramoto networks are globally synchronizing

TL;DR

This paper investigates the critical connectivity threshold in dense Kuramoto oscillator networks, establishing an improved upper bound for the transition to global synchronization using linear stability analysis.

Contribution

It proves that the critical connectivity  is at most 0.75, refining previous bounds and explaining the limitations of linear stability methods.

Findings

Established  .75 as an upper bound for synchronization

Connected the bound to linear stability analysis limitations

Compared with previous bounds of 0.7889 and 0.6838

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 $\mu_c\leq 0.7889$, and Yoneda, Tatsukawa, and Teramae proved in 2021 that $\mu_c > 0.6838$. In this paper, we prove that $\mu_c\leq 0.75$ and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.