{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "ZBZBKWWEBC"
    }
  ],
  "body": "\n## Summary\n\nA normal form of a differential equation near an [[equilibrium point|0N8NGKDZDZ]] is a simplest version of the equation.\nIt is obtained by a change of coordinates close to the identity.\nThe change removes, degree by degree, every nonlinear term that can be removed, and only the resonant terms remain.\nThe method goes back to Henri Poincaré.\nIt is a central tool in bifurcation theory, where normal forms describe the generic local behaviour, as in the Hopf bifurcation.\n\n## Explanations\n\n### L1 human\n\nNear an equilibrium an equation looks like its [[linearisation|09ZWYRQ41J]] plus small nonlinear corrections.\nMany of these corrections are only an artefact of the coordinates and disappear in better coordinates.\nThe normal form is what remains after removing them, and it shows the essential behaviour.\nFor $x' = x + x^2$ near $0$, the new coordinate $u = x / (1 + x)$ gives $u' = u$, so the quadratic term disappears completely.\nThe terms that this method cannot remove are called resonant; they come from special relations among the eigenvalues of the linear part.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "34C20",
      "37G05"
    ],
    "name": "Normal form",
    "same_as": {
      "wikidata": "Q18354536",
      "wikipedia": "Normal form (dynamical systems)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Normal_form_(dynamical_systems)"
      }
    ],
    "tag": "TJ04Y2Z5Q3"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/TJ04Y2Z5Q3/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/TJ04Y2Z5Q3.md",
  "msc": [
    "34C20",
    "37G05"
  ],
  "name": "Normal form",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TJ0/TJ04Y2Z5Q3.md",
  "status": null,
  "tag": "TJ04Y2Z5Q3",
  "uri": "m2://TJ04Y2Z5Q3",
  "url": "https://math2.tamnd.com/wiki/normal-form/",
  "writing": {
    "tag": "TJ04Y2Z5Q3",
    "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
      }
    ]
  }
}