{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0EERDDPSVT"
    },
    {
      "field": "uses",
      "target": "0N7ATVF2X8"
    },
    {
      "field": "uses",
      "target": "0T3GK0XCBC"
    }
  ],
  "body": "\n## Summary\n\nThe phase plane of a planar [[autonomous system|07BNRFQ0PJ]] $x' = f(x, y)$, $y' = g(x, y)$ is the plane with coordinates $(x, y)$, the two state variables.\nEach solution traces a curve in this plane, its [[trajectory|0BVT6VSB3S]]. The vector $(f(x, y), g(x, y))$ is tangent to the trajectory through $(x, y)$.\nIt is the two-dimensional case of the phase space.\n\n## Remarks\n\nA second-order equation $x'' = h(x, x')$ gives a planar system with $y = x'$. Its phase plane has the position and the velocity as coordinates.\nFor a linear system $(x, y)' = A (x, y)$ with a $2 \\times 2$ real matrix $A$, the eigenvalues of $A$ determine the shape of the trajectories. See the [[classification of planar linear systems|0N7ATVF2X8]].\n\n## Explanations\n\n### L1 human\n\nThe phase plane shows the two state variables at once, without the time axis.\nEach point is a state, and the system moves from state to state along a curve.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "concept",
    "msc": [
      "34C05"
    ],
    "name": "Phase plane",
    "same_as": {
      "wikidata": "Q2088081",
      "wikipedia": "Phase plane"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Phase_plane"
      },
      {
        "locator": "Example of a linear system",
        "url": "https://en.wikipedia.org/wiki/Phase_plane"
      }
    ],
    "tag": "05VDGH129F",
    "uses": [
      "07BNRFQ0PJ",
      "0BVT6VSB3S",
      "0A6QZ367GX"
    ]
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/05VDGH129F/",
  "links": [
    {
      "field": "uses",
      "target": "07BNRFQ0PJ"
    },
    {
      "field": "uses",
      "target": "0BVT6VSB3S"
    },
    {
      "field": "uses",
      "target": "0A6QZ367GX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/05VDGH129F.md",
  "msc": [
    "34C05"
  ],
  "name": "Phase plane",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/05V/05VDGH129F.md",
  "status": null,
  "tag": "05VDGH129F",
  "uri": "m2://05VDGH129F",
  "url": "https://math2.tamnd.com/wiki/phase-plane/",
  "writing": {
    "tag": "05VDGH129F",
    "kind": "concept",
    "level": "W1",
    "next": [
      {
        "what": "edge defines in",
        "note": "at least one definition",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of in",
        "note": "at least two examples",
        "have": 0,
        "want": 2
      },
      {
        "what": "edge not_instance_of in",
        "note": "at least one non-example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge specializes both",
        "note": "the more general notions and the special cases",
        "have": 0,
        "want": 1
      }
    ]
  }
}