{
  "id": "1710.00883",
  "version": "v1",
  "published": "2017-10-02T19:48:18.000Z",
  "updated": "2017-10-02T19:48:18.000Z",
  "title": "Generic 2-parameter perturbations of parabolic singular points of vector fields in C",
  "authors": [
    "Martin Klimes",
    "Christiane Rousseau"
  ],
  "comment": "44 pages, 34 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We describe the equivalence classes of germs of generic $2$-parameter families of complex vector fields $\\dot z = \\omega_\\epsilon(z)$ on $\\mathbb{C}$ unfolding a singular parabolic point of multiplicity $k+1$: $\\omega_0= z^{k+1} +o(z^{k+1})$. The equivalence is under conjugacy by holomorphic change of coordinate and parameter. As a preparatory step, we present the bifurcation diagram of the family of vector fields $\\dot z = z^{k+1} + \\epsilon_1 z + \\epsilon_0$ over $\\mathbb{CP}^1$. This presentation is done using the new tools of periodgon and star domain. We then provide a description of the modulus space and (almost) unique normal forms for the equivalence classes of germs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-02T19:48:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32M25",
      "32S65",
      "34M99"
    ],
    "keywords": [
      "parabolic singular points",
      "equivalence classes",
      "perturbations",
      "singular parabolic point",
      "unique normal forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}