{
  "id": "1501.05935",
  "version": "v1",
  "published": "2015-01-23T20:40:50.000Z",
  "updated": "2015-01-23T20:40:50.000Z",
  "title": "On symplectic dynamics near a homoclinic orbit to 1-elliptic fixed point",
  "authors": [
    "L. Lerman",
    "A. Markova"
  ],
  "comment": "39 pages, 4 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study the orbit behavior of a four dimensional smooth symplectic diffeomorphism $f$ near a homoclinic orbit $\\Gamma$ to an 1-elliptic fixed point under some natural genericity assumptions. 1-elliptic fixed point has two real eigenvalues out of unit circle and two others on the unit circle. Thus there is a smooth 2-dimensional center manifold $W^c$ where the restriction of the diffeomorphism has the elliptic fixed point supposed to be generic (no strong resonances and first Birkhoff coefficient is nonzero). Moser's theorem guarantees the existence of a positive measure set of KAM invariant curves. $W^c$ itself is a normally hyperbolic manifold in the whole phase space and due to Fenichel results every point on $W^c$ has 1-dimensional stable and unstable smooth invariant curves forming two smooth foliations. In particular, each KAM invariant curve has stable and unstable smooth 2-dimensional invariant manifolds being Lagrangian. The related stable and unstable manifolds of $W^c$ are 3-dimensional smooth manifolds which are supposed to be transverse along homoclinic orbit $\\Gamma$. One of our theorems presents conditions under which each KAM invariant curve on $W^c$ in a sufficiently small neighborhood of $\\Gamma$ has four transverse homoclinic orbits. Another result ensures that under some Birkhoff genericity assumption for the restriction of $f$ on $W^c$ saddle periodic orbits in resonance zones also have homoclinic orbits though its transversality or tangency cannot be verified directly.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-23T20:40:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J10",
      "37J40",
      "37J45"
    ],
    "keywords": [
      "homoclinic orbit",
      "fixed point",
      "kam invariant curve",
      "symplectic dynamics",
      "smooth invariant curves forming"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}