{
  "id": "1503.01534",
  "version": "v1",
  "published": "2015-03-05T04:25:58.000Z",
  "updated": "2015-03-05T04:25:58.000Z",
  "title": "Generating hyperbolic singularities in completely integrable systems",
  "authors": [
    "Holger R. Dullin",
    "Álvaro Pelayo"
  ],
  "comment": "24 pages, 9 figures",
  "categories": [
    "math-ph",
    "math.DS",
    "math.MP",
    "math.SG"
  ],
  "abstract": "Let $(M,\\Omega)$ be a connected symplectic 4-manifold and let $F=(J,H) : M \\to \\mathbb{R}^2$ be a completely integrable system on $M$ with only non-degenerate singularities and for which $J : M \\to \\mathbb{R}$ is a proper map. Assume that $F$ does not have singularities with hyperbolic blocks and that $p_1,...,p_n$ are the focus-focus singularities of $F$. For each subset $S=\\{i_1,...,i_j\\}$ we will show how to modify $F$ locally around any $p_i, i \\in S$, in order to create a new integrable system $\\tilde{F}=(J, \\tilde{H}) : M \\to \\mathbb{R}^2$ such that its classical spectrum $\\tilde{F}(M)$ contains $j$ smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of $\\tilde{F}$. Moreover the focus-focus singularities of $\\tilde{F}$ are precisely $p_i$, $i \\in \\{1,...,n\\} \\setminus S$, and each of these $p_i$ is non-degenerate. The proof is based on Eliasson's linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-05T04:25:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J35",
      "37J15",
      "70H06",
      "53D20"
    ],
    "keywords": [
      "integrable system",
      "generating hyperbolic singularities",
      "non-degenerate singularities",
      "focus-focus singularities",
      "non-degenerate transversally hyperbolic singularities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}