{
  "id": "0902.0681",
  "version": "v1",
  "published": "2009-02-04T09:52:14.000Z",
  "updated": "2009-02-04T09:52:14.000Z",
  "title": "Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor",
  "authors": [
    "Isaac A. Garcia",
    "Hector Giacomini",
    "Maite Grau"
  ],
  "comment": "41 pages, no figures",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \\theta)$, to an equation over a cylinder in which the singular point $p_0$ corresponds to a limit cycle $\\gamma_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $\\gamma_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $\\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-04T09:52:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37G15",
      "37G10",
      "34C07"
    ],
    "keywords": [
      "inverse integrating factor",
      "planar vector fields",
      "generalized hopf bifurcation",
      "limit cycle",
      "focus singular point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.0681G"
    }
  }
}