{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $f, g \\colon \\R^2 \\to \\R$ be [[smooth|0V33BNKZ9S]], and for $\\varepsilon \u003e 0$ consider the planar slow–fast [[autonomous system|0673D85QKZ]]\n$$\\varepsilon x' = f(x, y), \\qquad y' = g(x, y).$$\nHere $x$ is the fast variable and $y$ is the slow variable.\nThe *critical manifold* is the curve $S = \\set{(x, y) : f(x, y) = 0}$.\nIts points with $\\partial f / \\partial x \u003c 0$ form the attracting part of $S$, and its points with $f = \\partial f / \\partial x = 0$ are the fold points.\nA *relaxation oscillation* is a family of [[periodic orbits|0AA97XA70K]] $\\Gamma_\\varepsilon$, for small $\\varepsilon \u003e 0$, such that, as $\\varepsilon \\to 0$:\n\n1. $\\Gamma_\\varepsilon$ converges, in the Hausdorff distance, to a closed curve $\\Gamma_0$;\n2. $\\Gamma_0$ is a union of arcs of the attracting part of $S$, traversed in the direction of the slow flow $y' = g(x, y)$;\n3. the other pieces of $\\Gamma_0$ are horizontal segments $y = \\text{const}$ that start at fold points;\n4. $\\Gamma_0$ contains at least one horizontal segment.\n\n## Remarks\n\nAlong the arcs of $S$ the motion takes a time of order $1$, and along the horizontal segments, the jumps, it takes a time of order $\\varepsilon$.\n\nExample: the van der Pol equation $u'' - \\mu(1 - u^2) u' + u = 0$ with large $\\mu$.\nWith $F(x) = x^3/3 - x$, $\\varepsilon = 1/\\mu^2$ and a rescaled time, it becomes $\\varepsilon x' = y - F(x)$, $y' = -x$.\nThe critical manifold is the cubic $y = F(x)$; its attracting part is $\\abs{x} \u003e 1$ and its fold points are $(\\pm 1, \\mp 2/3)$.\nThe limit curve $\\Gamma_0$ runs down the right branch from $(2, 2/3)$ to $(1, -2/3)$, jumps to $(-2, -2/3)$, runs up the left branch to $(-1, 2/3)$, and jumps back to $(2, 2/3)$.\nThe period of the van der Pol oscillation is about $(3 - 2 \\ln 2)\\, \\mu$ for large $\\mu$.\n\nNon-example: the harmonic oscillator $x' = y$, $y' = -x$ has periodic orbits, but no small parameter and no fast variable, so it is not of the slow–fast form.\n\nSome authors use the term more loosely, for any oscillation with slow phases and fast jumps.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "2NHWCM0DK8",
    "kind": "definition",
    "msc": [
      "34C26",
      "34E15"
    ],
    "name": "Relaxation oscillation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Relaxation_oscillator"
      }
    ],
    "tag": "JCMXX84BWT",
    "uses": [
      "0AA97XA70K",
      "0V33BNKZ9S",
      "0673D85QKZ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/JCMXX84BWT/",
  "links": [
    {
      "field": "defines",
      "target": "2NHWCM0DK8"
    },
    {
      "field": "uses",
      "target": "0AA97XA70K"
    },
    {
      "field": "uses",
      "target": "0V33BNKZ9S"
    },
    {
      "field": "uses",
      "target": "0673D85QKZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JCMXX84BWT.md",
  "msc": [
    "34C26",
    "34E15"
  ],
  "name": "Relaxation oscillation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JCM/JCMXX84BWT.md",
  "status": null,
  "tag": "JCMXX84BWT",
  "uri": "m2://JCMXX84BWT",
  "url": "https://math2.tamnd.com/wiki/relaxation-oscillation-definition/",
  "writing": {
    "tag": "JCMXX84BWT",
    "kind": "definition",
    "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
      }
    ]
  }
}