{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "B416H6DQDA"
    },
    {
      "field": "uses",
      "target": "HV003CB8EP"
    },
    {
      "field": "uses",
      "target": "NGQMPVNZDP"
    },
    {
      "field": "defines",
      "target": "VGPTRNE3FF"
    }
  ],
  "body": "\n## Summary\n\nThe bifurcation set of a smooth family of functions is a set of parameter values.\nIt contains the values at which some function of the family has a degenerate critical point.\nIt is the central object of [[catastrophe theory|6FKJ6SXP3G]].\nOff this set the critical points move smoothly with the parameters, and across it they are created or destroyed.\nFor the cusp family $x^4/4 + a x^2/2 + b x$ it is the cusp curve $4a^3 + 27b^2 = 0$ in the $(a, b)$-plane.\nFor a family of potentials it describes where sudden jumps of the equilibrium can start.\n\n## Explanations\n\n### L1 human\n\nImagine a landscape that changes shape as you turn some knobs.\nFor most knob settings every valley and every hilltop is a clean, rounded one.\nAt special settings a valley and a hilltop merge into a flat spot and then vanish.\nThe bifurcation set is the map of all such special settings in the space of knobs.\nTake the family $x^3 + a x$ with one knob $a$.\nA valley and a hilltop exist for $a \u003c 0$, they merge at $a = 0$, and none exist for $a \u003e 0$.\nSo the bifurcation set is the single point $a = 0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58K35",
      "58K05"
    ],
    "name": "Bifurcation set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Catastrophe_theory"
      }
    ],
    "tag": "JV45S0KQE6"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/JV45S0KQE6/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/JV45S0KQE6.md",
  "msc": [
    "58K35",
    "58K05"
  ],
  "name": "Bifurcation set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JV4/JV45S0KQE6.md",
  "status": null,
  "tag": "JV45S0KQE6",
  "uri": "m2://JV45S0KQE6",
  "url": "https://tamnd.github.io/math2/wiki/bifurcation-set/",
  "writing": {
    "tag": "JV45S0KQE6",
    "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
      }
    ]
  }
}