{
  "id": "2206.10040",
  "version": "v1",
  "published": "2022-06-20T23:03:44.000Z",
  "updated": "2022-06-20T23:03:44.000Z",
  "title": "Arnold Tongues in Area-Preserving Maps",
  "authors": [
    "Jing Zhou",
    "Mark Levi"
  ],
  "categories": [
    "math.DS",
    "math-ph",
    "math.CA",
    "math.MP"
  ],
  "abstract": "In the early 60's J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. 20 years later V. Arnold rediscovered a similar phenomenon on sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of objects where a similarly flavored behavior takes place: area-preserving maps of the cylinder. Speaking loosely, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to ``drift\". The observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-20T23:03:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37N05",
      "37C25",
      "34C15"
    ],
    "keywords": [
      "arnold tongues",
      "area-preserving maps",
      "higher-frequency harmonics",
      "mathieu-type equations lose sharpness",
      "periodic orbits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}