{
  "id": "1805.11563",
  "version": "v1",
  "published": "2018-05-29T16:19:11.000Z",
  "updated": "2018-05-29T16:19:11.000Z",
  "title": "Existence of periodic orbits near heteroclinic connections",
  "authors": [
    "Giorgio Fusco",
    "Giovanni F. Gronchi",
    "Matteo Novaga"
  ],
  "comment": "36 pages, 4 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider a potential $W:R^m\\rightarrow R$ with two different global minima $a_-, a_+$ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system \\begin{equation} \\ddot{u}=W_u(u), \\hskip 2cm (1) \\end{equation} has a family of $T$-periodic solutions $u^T$ which, along a sequence $T_j\\rightarrow+\\infty$, converges locally to a heteroclinic solution that connects $a_-$ to $a_+$. We then focus on the elliptic system \\begin{equation} \\Delta u=W_u(u),\\;\\; u:R^2\\rightarrow R^m, \\hskip 2cm (2) \\end{equation} that we interpret as an infinite dimensional analogous of (1), where $x$ plays the role of time and $W$ is replaced by the action functional \\[J_R(u)=\\int_R\\Bigl(\\frac{1}{2}\\vert u_y\\vert^2+W(u)\\Bigr)dy.\\] We assume that $J_R$ has two different global minimizers $\\bar{u}_-, \\bar{u}_+:R\\rightarrow R^m$ in the set of maps that connect $a_-$ to $a_+$. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions $u^L:R^2\\rightarrow R^m$, which is $L$-periodic in $x$, converges to $a_\\pm$ as $y\\rightarrow\\pm\\infty$ and, along a sequence $L_j\\rightarrow+\\infty$, converges locally to a heteroclinic solution that connects $\\bar{u}_-$ to $\\bar{u}_+$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-29T16:19:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periodic orbits",
      "heteroclinic connections",
      "heteroclinic solution",
      "minimization procedure",
      "global minima"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}