{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $f \\colon \\R^n \\to \\R^n$ be a locally Lipschitz vector field whose [[autonomous system|0673D85QKZ]] $x' = f(x)$ has [[chaotic|DBKXCGDKTD]] solutions.\nLet $\\varepsilon \u003e 0$ and let $\\Gamma$ be an $n \\times n$ matrix.\nConsider the coupled system\n$$x' = f(x) + \\varepsilon \\Gamma (y - x), \\qquad y' = f(y) + \\varepsilon \\Gamma (x - y).$$\nThe diagonal $\\set{(x, y) : x = y}$ is invariant under this system, and on it each component follows the chaotic system $x' = f(x)$.\nThe coupled system shows *complete synchronization* from a set $U \\subseteq \\R^n \\times \\R^n$ of initial states if every solution with $(x(0), y(0)) \\in U$ satisfies\n$$\\lim_{t \\to \\infty} \\norm{x(t) - y(t)} = 0.$$\nSynchronization of chaos refers to such behaviour, with $U$ a neighbourhood of a chaotic attractor on the diagonal, and to its weaker forms.\n\n## Remarks\n\nThe difference $e = x - y$ satisfies, to first order, $e' = \\big(Df(x) - 2 \\varepsilon \\Gamma\\big) e$.\nSynchronization is stable when the largest Lyapunov exponent of this linear equation along the synchronized orbit, the transverse Lyapunov exponent, is negative.\n\nExample: let $f(x) = a x$ be scalar with $a \u003e 0$, which is not chaotic but shows the mechanism, and $\\Gamma = 1$.\nThen $e' = (a - 2\\varepsilon) e$ exactly, so $e(t) \\to 0$ for every initial state exactly when $\\varepsilon \u003e a/2$.\n\nNon-example: for $\\varepsilon = 0$ the two copies are uncoupled, and for a chaotic $f$ nearby initial states separate, so $\\norm{x(t) - y(t)}$ does not tend to $0$ for most initial states.\n\nWeaker forms are phase synchronization, where only the phases of oscillations lock, and generalized synchronization, where $y(t) - \\psi(x(t)) \\to 0$ for some function $\\psi$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "YP1PMW0HE3",
    "kind": "definition",
    "msc": [
      "34D06",
      "37D45"
    ],
    "name": "Synchronization of chaos",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Synchronization_of_chaos"
      }
    ],
    "tag": "NZBKMKB8CV",
    "uses": [
      "0673D85QKZ",
      "DBKXCGDKTD"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/NZBKMKB8CV/",
  "links": [
    {
      "field": "defines",
      "target": "YP1PMW0HE3"
    },
    {
      "field": "uses",
      "target": "0673D85QKZ"
    },
    {
      "field": "uses",
      "target": "DBKXCGDKTD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/NZBKMKB8CV.md",
  "msc": [
    "34D06",
    "37D45"
  ],
  "name": "Synchronization of chaos",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NZB/NZBKMKB8CV.md",
  "status": null,
  "tag": "NZBKMKB8CV",
  "uri": "m2://NZBKMKB8CV",
  "url": "https://tamnd.github.io/math2/wiki/synchronization-of-chaos-definition/",
  "writing": {
    "tag": "NZBKMKB8CV",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}