{
  "id": "2011.06802",
  "version": "v1",
  "published": "2020-11-13T08:10:59.000Z",
  "updated": "2020-11-13T08:10:59.000Z",
  "title": "Versal deformations of vector field singularities",
  "authors": [
    "Mauricio Garay",
    "Duco van Straten"
  ],
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "When a singular point of a vector field passes through resonance, a formal invariant cone appears. In the seventies, Pyartli proved that for $(-1,1)$-resonance the cone is in fact analytic and is the degeneration of a family of invariant cylinders. In his thesis, Stolovitch established a new type of normal form and proved that for a simple resonance and under arithmetic conditions the cone is (the germ of) an analytic variety. In this paper, we prove a versal deformation theorem for analytic vector fields with an isolated singularity over Cantor sets. Our result implies that, under arithmetic conditions, the resonant cone is the degeneration of a set of invariant manifolds like in Pyartli's example. For the multi-Hopf bifurcation, that is for the $(-1,1)^d$-resonance, this implies the existence of vanishing tori carrying quasi-periodic motions generalising previous results of Chenciner and Li.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-13T08:10:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singularity",
      "vector field singularities",
      "versal deformation",
      "carrying quasi-periodic motions generalising",
      "arithmetic conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}