{
  "id": "1911.12336",
  "version": "v1",
  "published": "2019-11-27T18:33:33.000Z",
  "updated": "2019-11-27T18:33:33.000Z",
  "title": "Synchronization of Kuramoto Oscillators in Dense Networks",
  "authors": [
    "Jianfeng Lu",
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.OC",
    "math.DS",
    "nlin.AO"
  ],
  "abstract": "We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let $G=(V,E)$ be a connected graph and $(a_{ij})_{i,j=1}^{n}$ denotes its adjacency matrix. Let the function $f:\\mathbb{T}^n \\rightarrow \\mathbb{R}$ be given by $$ f(\\theta_1, \\dots, \\theta_n) = \\sum_{i,j=1}^{n}{ a_{ij} \\cos{(\\theta_i - \\theta_j)}}.$$ This function has a global maximum when $\\theta_i = \\theta$ for all $1\\leq i \\leq n$. It is known that if every vertex is connected to at least $\\mu(n-1)$ other vertices for $\\mu$ sufficiently large, then every local maximum is global. Taylor proved this for $\\mu \\geq 0.9395$ and Ling, Xu \\& Bandeira improved this to $\\mu \\geq 0.7929$. We give a slight improvement to $\\mu \\geq 0.7889$. Townsend, Stillman \\& Strogatz suggested that the critical value might be $\\mu_c = 0.75$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-27T18:33:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kuramoto oscillators",
      "dense networks",
      "study synchronization properties",
      "slight improvement",
      "adjacency matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}