{
  "id": "1406.0208",
  "version": "v1",
  "published": "2014-06-01T21:26:37.000Z",
  "updated": "2014-06-01T21:26:37.000Z",
  "title": "Cubic perturbations of elliptic Hamiltonian vector fields of degree three",
  "authors": [
    "Lubomir Gavrilov",
    "Iliya D. Iliev"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The purpose of the present paper is to study the limit cycles of one-parameter perturbed plane Hamiltonian vector field $X_\\varepsilon$ $$ X_\\varepsilon : \\left\\{ \\begin{array}{llr} \\dot{x}=\\;\\; H_y+\\varepsilon f(x,y)\\\\ \\dot{y}=-H_x+\\varepsilon g(x,y), \\end{array} \\;\\;\\;\\;\\; H~=\\frac{1}{2} y^2~+U(x) \\right. $$ which bifurcate from the period annuli of $X_0$ for sufficiently small $\\varepsilon$. Here $U$ is a univariate polynomial of degree four without symmetry, and $f, g$ are arbitrary cubic polynomials in two variables. We take a period annulus and parameterize the related displacement map $d(h,\\varepsilon)$ by the Hamiltonian value $h$ and by the small parameter $\\varepsilon$. Let $M_k(h)$ be the $k$-th coefficient in its expansion with respect to $\\varepsilon$. We establish the general form of $M_k$ and study its zeroes. We deduce that the period annuli of $X_0$ can produce for sufficiently small $\\varepsilon$, at most 5, 7 or 8 zeroes in the interior eight-loop case, the saddle-loop case, and the exterior eight-loop case respectively. In the interior eight-loop case the bound is exact, while in the saddle-loop case we provide examples of Hamiltonian fields which produce 6 small-amplitude limit cycles. Polynomial perturbations of $X_0$ of higher degrees are also studied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-01T21:26:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C07"
    ],
    "keywords": [
      "elliptic hamiltonian vector fields",
      "cubic perturbations",
      "period annulus",
      "interior eight-loop case",
      "perturbed plane hamiltonian vector field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.0208G"
    }
  }
}