{
  "id": "math/0103180",
  "version": "v1",
  "published": "2001-03-27T13:38:32.000Z",
  "updated": "2001-03-27T13:38:32.000Z",
  "title": "The Period Function of Second Order Differential Equations",
  "authors": [
    "A. Raouf Chouikha"
  ],
  "comment": "25 pages",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "We interest in the behaviour of the period function for equations of the type $u'' + g(u) = 0$ and $u'' + f(u)u' + g(u) = 0$ with a center at the origin 0. $g$ is a function of class $C^k$. For the conservative case, if $k \\geq 2$ one shows that the Opial criterion is the better one among those for which these the necessary condition $g''(0) = 0$ holds. In the case where $f$ is of class $C^1$ and $k \\geq 3$, the Lienard equations $ u'' + f(u) u' + g(u) = 0$ may have a monotonic period function if $g'(0) g^{(3)}(0) - {5/3} {g''}^{2}(0) - {2/3} {f'}^{2}(0) g'(0) \\neq 0$ in a neighborhood of 0. {\\it Key Words and phrases:} period function, monotonicity, isochronicity, Lienard equation, polynomial systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-03-27T13:38:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C25",
      "34C35"
    ],
    "keywords": [
      "second order differential equations",
      "lienard equation",
      "monotonic period function",
      "opial criterion",
      "necessary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......3180R"
    }
  }
}