{
  "id": "1608.08408",
  "version": "v1",
  "published": "2016-08-30T11:42:35.000Z",
  "updated": "2016-08-30T11:42:35.000Z",
  "title": "Arnold diffusion for a complete family of perturbations",
  "authors": [
    "Amadeu Delshams",
    "Rodrigo G. Schaefer"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this work we illustrate the Arnold diffusion in a concrete example---the \\emph{a priori} unstable Hamiltonian system of $2+1/2$ degrees of freedom $H(p,q,I,\\varphi,s) = p^{2}/2+\\cos q -1 +I^{2}/2 + h(q,\\varphi,s;\\varepsilon)$---proving that for \\emph{any} small periodic perturbation of the form $h(q,\\varphi,s;\\varepsilon) = \\varepsilon\\cos q\\left( a_{00} + a_{10}\\cos\\varphi + a_{01}\\cos s \\right)$ ($a_{10}a_{01} \\neq 0$) there is global instability for the action. For the proof we apply a geometrical mechanism based in the so-called Scattering map. This work has the following structure: In a first stage, for a more restricted case ($I^*\\thicksim\\pi/2\\mu$, $\\mu = a_{10}/a_{01}$), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of the instability for any $\\mu$). The bifurcations of the scattering map are also studied as a function of $\\mu$. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-30T11:42:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J40"
    ],
    "keywords": [
      "arnold diffusion",
      "scattering map",
      "complete family",
      "small periodic perturbation",
      "special property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}