{
  "id": "2105.09785",
  "version": "v1",
  "published": "2021-05-20T14:33:57.000Z",
  "updated": "2021-05-20T14:33:57.000Z",
  "title": "Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: Coefficient properties",
  "authors": [
    "David Marín",
    "Jordi Villadelprat"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider a $\\mathscr C^\\infty$ family of planar vector fields $\\{X_{\\hat\\mu}\\}_{\\hat\\mu\\in\\hat W}$ having a hyperbolic saddle and we study the Dulac map $D(s;\\hat\\mu)$ and the Dulac time $T(s;\\hat\\mu)$ from a transverse section at the stable separatrix to a transverse section at the unstable separatrix, both at arbitrary distance from the saddle. Since the hyperbolicity ratio $\\lambda$ of the saddle plays an important role, we consider it as an independent parameter, so that $\\hat\\mu=(\\lambda,\\mu)\\in \\hat W=(0,+\\infty)\\times W$, where $W$ is an open subset of $\\mathbb R^N.$ For each $\\hat\\mu_0\\in\\hat W$ and $L>0$, the functions $D(s;\\hat\\mu)$ and $T(s;\\hat\\mu)$ have an asymptotic expansion at $s=0$ and $\\hat\\mu\\approx\\hat\\mu_0$ with the remainder being uniformly $L$-flat with respect to the parameters. The principal part of both asymptotic expansions is given in a monomial scale containing a deformation of the logarithm, the so-called Ecalle-Roussarie compensator. In this paper we are interested in the coefficients of these monomials, which are functions depending on $\\hat\\mu$ that can be shown to be $\\mathscr C^\\infty$ in their respective domains and \"universally\" defined, meaning that their existence is stablished before fixing the flatness $L$ and the unfolded parameter $\\hat\\mu_0.$ Each coefficient has its own domain and it is of the form $((0,+\\infty)\\setminus D)\\times W$, where~$D$ a discrete set of rational numbers at which a resonance of the hyperbolicity ratio $\\lambda$ occurs. In our main result we give the explicit expression of some of these coefficients and to this end a fundamental tool is the employment of a sort of incomplete Mellin transform. With regard to these coefficients we also prove that they have poles of order at most two at $D\\times W$ and we give the corresponding residue, that plays an important role when compensators appear in the principal part.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-20T14:33:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C07",
      "34C20",
      "34C23"
    ],
    "keywords": [
      "asymptotic expansion",
      "hyperbolic saddle",
      "dulac map",
      "coefficient properties",
      "important role"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}