{
  "id": "2010.11402",
  "version": "v1",
  "published": "2020-10-22T03:12:46.000Z",
  "updated": "2020-10-22T03:12:46.000Z",
  "title": "On invariant tori in some reversible systems",
  "authors": [
    "Lu Chen"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In the present paper, we consider the following reversible system \\begin{equation*} \\begin{cases} \\dot{x}=\\omega_0+f(x,y),\\\\ \\dot{y}=g(x,y), \\end{cases} \\end{equation*} where $x\\in\\mathbf{T}^{d}$, $y\\backsim0\\in \\mathbf{R}^{d}$, $\\omega_0$ is Diophantine, $f(x,y)=O(y)$, $g(x,y)=O(y^2)$ and $f$, $g$ are reversible with respect to the involution G: $(x,y)\\mapsto(-x,y)$, that is, $f(-x,y)=f(x,y)$, $g(-x,y)=-g(x,y)$. We study the accumulation of an analytic invariant torus $\\Gamma_0$ of the reversible system with Diophantine frequency $\\omega_0$ by other invariant tori. We will prove that if the Birkhoff normal form around $\\Gamma_0$ is 0-degenerate, then $\\Gamma_0$ is accumulated by other analytic invariant tori, the Lebesgue measure of the union of these tori being positive and the density of the union of these tori at $\\Gamma_0$ being one. We will also prove that if the Birkhoff normal form around $\\Gamma_0$ is $j$-degenerate ($1\\leq j\\leq d-1$) and condition (1.6) is satisfied, then through $\\Gamma_0$ there passes an analytic subvariety of dimension $d+j$ foliated into analytic invariant tori with frequency vector $\\omega_0$. If the Birkhoff normal form around $\\Gamma_0$ is $d-1$-degenerate, we will prove a stronger result, that is, a full neighborhood of $\\Gamma_0$ is foliated into analytic invariant tori with frequency vectors proportional to $\\omega_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-22T03:12:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "analytic invariant torus",
      "reversible system",
      "birkhoff normal form",
      "frequency vectors proportional",
      "analytic subvariety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}