{
  "id": "0807.3022",
  "version": "v1",
  "published": "2008-07-18T17:53:42.000Z",
  "updated": "2008-07-18T17:53:42.000Z",
  "title": "On the Global Attractor of Delay Differential Equations with Unimodal Feedback",
  "authors": [
    "Eduardo Liz",
    "Gergely Röst"
  ],
  "comment": "10 pages, submitted to Discrete and Continuous Dynamical Systems-Series A (DCDS)",
  "journal": "Discrete and Continuous Dynamical Systems-Series A, 24(4), 2009, 1215 - 1224",
  "doi": "10.3934/dcds.2009.24.1215",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "We give bounds for the global attractor of the delay differential equation $x'(t) =-\\mu x(t)+f(x(t-\\tau))$, where $f$ is unimodal and has negative Schwarzian derivative. If $f$ and $\\mu$ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where f is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the d5A5Aelay is sufficiently small, then all solution enter the domain where $f'$ is negative. Our theorems then are illustrated by numerical examples using Nicholson's blowflies equation and the Mackey-Glass equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-07-18T17:53:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34K20",
      "34D45"
    ],
    "keywords": [
      "delay differential equation",
      "global attractor",
      "unimodal feedback",
      "nicholsons blowflies equation",
      "solution enter"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.3022L"
    }
  }
}