Department of Mathematics, Cornell University, Ithaca, New York 14853, USA.
Chaos. 2021 Jul;31(7):073135. doi: 10.1063/5.0057659.
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, such that whenever μ>μ, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when μ<μ, there are networks with other stable states. The precise value of the critical connectivity remains unknown, but it has been conjectured to be μ=0.75. In 2020, Lu and Steinerberger proved that μ≤0.7889, and Yoneda, Tatsukawa, and Teramae proved in 2021 that μ>0.6838. This paper proves that μ≤0.75 and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.
考虑一个由 n 个相同的 Kuramoto 振荡器组成的网络，其中每个振荡器以单位强度双向耦合到至少 μ(n-1)个其他振荡器。存在一个连接的临界值 μ，使得只要 μ>μ，系统几乎可以保证对于所有初始条件都收敛到全同相同步状态，但当 μ<μ 时，网络会存在其他稳定状态。临界连接的确切值仍然未知，但有人推测 μ=0.75。2020 年，Lu 和 Steinerberger 证明了 μ≤0.7889，而 Yoneda、Tatsukawa 和 Teramae 在 2021 年证明了 μ>0.6838。本文证明了 μ≤0.75，并解释了为什么这是通过纯线性稳定性分析可以获得的最佳上界。
