{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "3CVERVJRV9"
    },
    {
      "field": "uses",
      "target": "9BPTKH2HYJ"
    },
    {
      "field": "uses",
      "target": "CJDS1BGDVH"
    }
  ],
  "body": "\n## Summary\n\nAn equilibrium of an autonomous difference equation $x_{n+1} = f(x_n)$ is a point $x^*$ with $f(x^*) = x^*$, that is, a [[fixed point|R6QVS6P21N]] of $f$.\nStarting there, the solution stays there forever, so equilibria are the constant solutions.\nThey are the discrete-time analogue of the [[equilibrium points|0N8NGKDZDZ]] of differential equations.\nThe first question about a nonlinear difference equation is usually where its equilibria are and whether nearby solutions approach them.\n\n## Explanations\n\n### L1 human\n\nA difference equation tells you how a state changes from one time step to the next.\nAn equilibrium is a state that the rule leaves unchanged.\nOn a graph of $f$, the equilibria are where the graph crosses the line $y = x$.\nFor the population rule $x_{n+1} = 2 x_n (1 - x_n)$, the equilibria are $0$ and $1/2$: a population of $1/2$ stays at $1/2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "39A05",
      "37C25"
    ],
    "name": "Equilibrium of a difference equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fixed_point_(mathematics)"
      }
    ],
    "specializes": [
      "R6QVS6P21N"
    ],
    "tag": "D62PEKHRV5"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/D62PEKHRV5/",
  "links": [
    {
      "field": "specializes",
      "target": "R6QVS6P21N"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/D62PEKHRV5.md",
  "msc": [
    "39A05",
    "37C25"
  ],
  "name": "Equilibrium of a difference equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D62/D62PEKHRV5.md",
  "status": null,
  "tag": "D62PEKHRV5",
  "uri": "m2://D62PEKHRV5",
  "url": "https://tamnd.github.io/math2/wiki/equilibrium-of-a-difference-equation/",
  "writing": {
    "tag": "D62PEKHRV5",
    "kind": "concept",
    "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
      }
    ]
  }
}