{
  "id": "1801.09244",
  "version": "v1",
  "published": "2018-01-28T15:45:58.000Z",
  "updated": "2018-01-28T15:45:58.000Z",
  "title": "A periodic solution of period two of a delay differential equation",
  "authors": [
    "Yukihiko Nakata"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we prove that the following delay differential equation \\[ \\frac{d}{dt}x(t)=rx(t)\\left(1-\\int_{0}^{1}x(t-s)ds\\right), \\] has a periodic solution of period two for $r>\\frac{\\pi^{2}}{2}$ (when the steady state, $x=1$, is unstable). In order to find the periodic solution, we study an integrable system of ordinary differential equations, following the idea by Kaplan and Yorke \\cite{Kaplan=000026Yorke:1974}. The periodic solution is expressed in terms of the Jacobi elliptic functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-28T15:45:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "delay differential equation",
      "periodic solution",
      "ordinary differential equations",
      "jacobi elliptic functions",
      "steady state"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}