{
  "id": "1807.07832",
  "version": "v1",
  "published": "2018-07-20T13:33:05.000Z",
  "updated": "2018-07-20T13:33:05.000Z",
  "title": "Arnold diffusion in multidimensional a priori unstable Hamiltonian systems",
  "authors": [
    "Mars Davletshin",
    "Dmitry Treschev"
  ],
  "comment": "47 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study the Arnold diffusion in a priori unstable near-integrable systems in a neighbourhood of a resonance of low order. We consider a non-autonomous near-integrable Hamiltonian system with $n+1/2$ degrees of freedom, $n\\ge 2$. Let the Hamilton function $H$ of depend on the parameter $\\varepsilon$, for $\\varepsilon=0$ the system is integrable and has a homoclinic asymptotic manifold $\\Gamma$. Our main result is that for small generic perturbation in an $\\varepsilon$-neighborhood of $\\Gamma$ there exist trajectories the projections of which on the space of actions cross the resonance. By ``generic perturbations'' we mean an open dense set in the space of $C^r$-smooth functions $\\frac{d}{d\\varepsilon}\\big|_{\\varepsilon=0} H$, $r=r_0,r_0+1,\\ldots,\\infty,\\omega$. Combination of this result with results of \\cite{DT} answers the main questions on the Arnold diffusion in a priori unstable case: the diffusion takes place for generic perturbation, diffusion trajectories can go along any smooth curve in the action space with average velocity of order $\\varepsilon/|\\log \\varepsilon|$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-20T13:33:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J40",
      "70H08",
      "70H14"
    ],
    "keywords": [
      "priori unstable hamiltonian systems",
      "arnold diffusion",
      "multidimensional",
      "homoclinic asymptotic manifold",
      "open dense set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}