{
  "id": "1405.0866",
  "version": "v2",
  "published": "2014-05-05T11:58:13.000Z",
  "updated": "2014-12-04T16:52:58.000Z",
  "title": "A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results",
  "authors": [
    "Marian Gidea",
    "Rafael de la Llave",
    "Tere Seara"
  ],
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "nlin.CD"
  ],
  "abstract": "We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on the scattering map (outer) dynamics and on the recurrence property of the (inner) dynamics restricted to a normally hyperbolic invariant manifold. We apply topological methods to find trajectories that follow these two dynamics. This method differs, in several crucial aspects, from earlier works. There are virtually no assumptions on the inner dynamics, as the method does not use at all the invariant objects for the inner dynamics (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets). The method applies when the unperturbed Hamiltonian is not necessarily convex, and of arbitrary degrees of freedom. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems. We include several applications, such as bridging large gaps in a priori unstable models in any dimension, and establishing diffusion in cases when the inner dynamics in a non-twist map.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-05T11:58:13.000Z",
      "abstract": "We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on the scattering map (outer) dynamics and on the recurrence property of the (inner) dynamics restricted to a normally hyperbolic invariant manifold. We apply topological methods to find trajectories that follow these two dynamics. This method differs, in several crucial aspects, from earlier works, as it does not use at all invariant objects for the inner dynamics (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets), and applies to perturbations of integrable Hamiltonians that are not convex. We also include several non-trivial applications, such as bridging large gaps in a priori unstable models in any dimension. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, or to establish diffusion in generic systems.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-04T16:52:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J40",
      "37C50",
      "37C29",
      "37B30"
    ],
    "keywords": [
      "general mechanism",
      "qualitative results",
      "lower dimensional hyperbolic tori",
      "normally hyperbolic invariant manifold",
      "generic systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0866G"
    }
  }
}