{
  "id": "2108.08540",
  "version": "v1",
  "published": "2021-08-19T07:49:37.000Z",
  "updated": "2021-08-19T07:49:37.000Z",
  "title": "Averaging and passage through resonances in two-frequency systems near separatrices",
  "authors": [
    "Anatoly Neishtadt",
    "Alexey Okunev"
  ],
  "comment": "49 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study averaging method for time-periodic perturbations of one-frequency Hamiltonian systems such that solutions of the perturbed system cross separatrices of the unperturbed system. The Hamiltonian depends on a parameter that slowly changes for the perturbed system (so Hamiltonian systems with two and a half degree of freedom are included in our class). We prove that for most initial conditions (except a set of measure $O(\\sqrt{\\varepsilon} |\\ln^5 \\varepsilon|)$, here $\\varepsilon$ is the small parameter) the evolution of slow variables is described by the averaged system with accuracy $O(\\sqrt{\\varepsilon} |\\ln \\varepsilon|)$ over time $\\sim \\varepsilon^{-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-19T07:49:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C29"
    ],
    "keywords": [
      "two-frequency systems",
      "resonances",
      "perturbed system cross separatrices",
      "one-frequency hamiltonian systems",
      "study averaging method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}