{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "8TEV7CSX2H"
    },
    {
      "field": "uses",
      "target": "NZBKMKB8CV"
    }
  ],
  "body": "\n## Summary\n\nChaos theory studies deterministic [[dynamical systems|BBZP1R872N]] whose long-time behaviour looks random and cannot be predicted in practice.\nThe key feature is sensitive dependence on initial conditions, the [[butterfly effect|G0ZT1SDQGS]]: tiny differences in the starting state grow until they dominate.\nTypical tools are [[Lyapunov exponents|NJ1FJSNK6A]], [[strange attractors|9WKCZGG906]], symbolic dynamics and invariant measures, and a standard mathematical definition of a chaotic map is [[Devaney chaos|21B8AHPD4P]].\nClassical examples are the [[Lorenz system|WYRC41CQN9]] in meteorology and the [[logistic map|QN9X9BNKTX]] in population dynamics.\n\n## Explanations\n\n### L1 human\n\nA chaotic system follows exact rules, with no chance involved, and yet its future is practically unpredictable.\nThe reason is that any small error in the present, such as rounding a measurement, is magnified again and again.\nThe doubling map $x \\mapsto 2x \\bmod 1$ shows this well: it shifts the binary digits of $x$ one place to the left.\nTwo starting values that agree in their first $50$ binary digits give orbits that look unrelated after about $50$ steps.\nWeather is the most famous example, which is why forecasts become unreliable after a week or two.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "37D45",
      "37N99",
      "34C28"
    ],
    "name": "Chaos theory",
    "same_as": {
      "wikidata": "Q166314",
      "wikipedia": "Chaos theory"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Chaos_theory"
      }
    ],
    "tag": "DBKXCGDKTD"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/DBKXCGDKTD/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/DBKXCGDKTD.md",
  "msc": [
    "37D45",
    "37N99",
    "34C28"
  ],
  "name": "Chaos theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DBK/DBKXCGDKTD.md",
  "status": null,
  "tag": "DBKXCGDKTD",
  "uri": "m2://DBKXCGDKTD",
  "url": "https://tamnd.github.io/math2/wiki/chaos-theory/",
  "writing": {
    "tag": "DBKXCGDKTD",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}