{
  "id": "1212.1150",
  "version": "v2",
  "published": "2012-12-05T20:28:09.000Z",
  "updated": "2013-01-28T14:39:05.000Z",
  "title": "A strong form of Arnold diffusion for two and a half degrees of freedom",
  "authors": [
    "Vadim Kaloshin",
    "Ke Zhang"
  ],
  "comment": "207 pages, 21 figures. This paper includes and supercedes arXiv:1202.1032. arXiv admin note: text overlap with arXiv:math/0412300 by other authors",
  "categories": [
    "math.DS"
  ],
  "abstract": "In the present paper we prove a strong form of Arnold diffusion. Let $\\T^2$ be the two torus and $B^2$ be the unit ball around the origin in $\\R^2$. Fix $\\rho>0$. Our main result says that for a \"generic\" time-periodic perturbation of an integrable system of two degrees of freedom $H_\\epsilon = H_0 + \\epsilon H_1(\\theta, p, t)$ with a strictly convex $H_0$, there exists a $\\rho$-dense orbit $(\\theta_{\\epsilon},p_{\\epsilon},t)(t)$ in $\\T^2 \\times B^2 \\times \\T$, namely, a $\\rho$-neighbourhood of the orbit contains $\\T^2 \\times B^2 \\times \\T$. Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are usage of crumpled normally hyperbolic invariant cylinders from [13], flower and simple normally hyperbolic invariant manifolds from [47] as well as their kissing property at a strong double resonance. This allows us to build a \"connected\" net of 3-dimensional normally hyperbolic invariant manifolds. To construct diffusing orbits along this net we employ a version of Mather variational method [54] equipped with weak KAM theory [34], proposed by Bernard in [9].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-28T14:39:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J40",
      "37J45",
      "37J50",
      "37D10"
    ],
    "keywords": [
      "arnold diffusion",
      "strong form",
      "half degrees",
      "normally hyperbolic invariant cylinders",
      "variational method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 207,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.1150K"
    }
  }
}