{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an interval, let $f \\colon I \\to I$ be a map, and let $x_0 \\in I$.\nLet $x_n = f^n(x_0)$ for $n \\ge 0$, where $f^n$ is the $n$-th [[iterate|AYVTR2ES5B]] of $f$, so that $x_{n+1} = f(x_n)$.\nConsider the points $P_{2n} = (x_n, x_n)$ and $P_{2n+1} = (x_n, x_{n+1})$ for $n \\ge 0$.\nThe *cobweb plot* of the orbit of $x_0$ is the polygonal path $P_0 P_1 P_2 \\cdots$.\nIt is drawn together with the graph of $f$ and the diagonal $\\set{(x, x) : x \\in I}$.\n\n## Remarks\n\nThe points $(x_n, x_{n+1})$ lie on the graph of $f$ and the points $(x_n, x_n)$ lie on the diagonal.\nSo the path alternates vertical segments to the graph and horizontal segments to the diagonal.\nA [[fixed point|R6QVS6P21N]] $p$ of $f$ is a point where the graph meets the diagonal, and the path starting at $x_0 = p$ stays at $(p, p)$.\n\nExample: let $f(x) = x/2 + 1$ on $\\R$ and $x_0 = 0$.\nThe orbit is $0, 1, 3/2, 7/4, \\dots$, with $x_n = 2 - 2^{1-n}$ for $n \\ge 0$.\nThe cobweb path goes from $(0, 0)$ up to $(0, 1)$, across to $(1, 1)$, up to $(1, 3/2)$, and so on, a staircase that climbs to the fixed point $(2, 2)$.\n\nNon-example: the plot of the points $(n, x_n)$ against the time $n$ is a time series plot.\nIt is not a cobweb plot, because it does not use the graph of $f$ and the diagonal.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "B0K8MDA7CJ",
    "kind": "definition",
    "msc": [
      "37E05",
      "26A18",
      "37M99"
    ],
    "name": "Cobweb plot",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cobweb_plot"
      }
    ],
    "tag": "9GWGW4YRD9",
    "uses": [
      "AYVTR2ES5B",
      "R6QVS6P21N"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/9GWGW4YRD9/",
  "links": [
    {
      "field": "defines",
      "target": "B0K8MDA7CJ"
    },
    {
      "field": "uses",
      "target": "AYVTR2ES5B"
    },
    {
      "field": "uses",
      "target": "R6QVS6P21N"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/9GWGW4YRD9.md",
  "msc": [
    "37E05",
    "26A18",
    "37M99"
  ],
  "name": "Cobweb plot",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9GW/9GWGW4YRD9.md",
  "status": null,
  "tag": "9GWGW4YRD9",
  "uri": "m2://9GWGW4YRD9",
  "url": "https://tamnd.github.io/math2/wiki/cobweb-plot-definition/",
  "writing": {
    "tag": "9GWGW4YRD9",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}