Sufficiently dense Kuramoto networks are globally synchronizing.

Consider any network of n identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least μ ( n − 1 ) other oscillators. There is a critical value of the connectivity, μ c, such that whenever μ > μ c, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when μ 0.6838. This paper proves that μ c ≤ 0.75 and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.