{
  "id": "1108.5582",
  "version": "v1",
  "published": "2011-08-29T14:48:23.000Z",
  "updated": "2011-08-29T14:48:23.000Z",
  "title": "Nonpersistence of resonant caustics in perturbed elliptic billiards",
  "authors": [
    "Sonia Pinto-de-Carvalho",
    "Rafael Ramirez-Ros"
  ],
  "comment": "14 pages, 3 figures",
  "categories": [
    "math.DS",
    "nlin.CD"
  ],
  "abstract": "Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics ---the ones whose tangent trajectories are closed polygons--- are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-29T14:48:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "perturbed elliptic billiards",
      "resonant caustics",
      "billiard table",
      "nonpersistence",
      "resonant elliptical caustics persists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.5582P"
    }
  }
}