{
  "id": "1402.7189",
  "version": "v2",
  "published": "2014-02-28T10:35:47.000Z",
  "updated": "2015-06-03T07:40:29.000Z",
  "title": "Periodic orbits near a bifurcating slow manifold",
  "authors": [
    "Kristian Uldall Kristiansen"
  ],
  "comment": "To appear in JDE",
  "categories": [
    "math.DS"
  ],
  "abstract": "This paper studies a class of $1\\frac12$-degree-of-freedom Hamiltonian systems with a slowly varying phase that unfolds a Hamiltonian pitchfork bifurcation. The main result of the paper is that there exists an order of $\\ln^2\\epsilon^{-1}$-many periodic orbits that all stay within an $\\mathcal O(\\epsilon^{1/3})$-distance from the union of the normally elliptic slow manifolds that occur as a result of the bifurcation. Here $\\epsilon\\ll 1$ measures the time scale separation. These periodic orbits are predominantly unstable. The proof is based on averaging of two blowup systems, allowing one to estimate the effect of the singularity, combined with results on asymptotics of the second Painleve equation. The stable orbits of smallest amplitude that are {persistently} obtained by these methods remain slightly further away from the slow manifold being distant by an order $\\mathcal O(\\epsilon^{1/3}\\ln^{1/2}\\ln \\epsilon^{-1})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-28T10:35:47.000Z",
      "abstract": "This paper studies a class of one-degree-of-freedom Hamiltonian systems with a slowly varying phase. The slowly varying phase is assumed to unfold a Hamiltonian pitchfork bifurcation. The main result of the paper is that there exists an order of $\\ln^2\\epsilon^{-1}$-many periodic orbits that all stay within an $\\mathcal O(\\epsilon^{1/3})$-distance from the union of the normally elliptic slow manifolds that occur as a result of the bifurcation. As they pass through the bifurcation the time scales are comparable. Here $\\epsilon\\ll 1$ measures the time scale separation. These periodic orbits are typically ``moderately'' unstable. This is in contrast with the periodic orbits that remain an $\\mathcal O(1)$-distance from the slow manifold. The effect of approaching the normally elliptic slow manifold is therefore to reduce the stability region. The smallest stable orbits that are \\textit{persistently} obtained remain further away from the slow manifold being distant by an order $\\mathcal O(\\epsilon^{1/3}\\ln^{1/2}\\ln \\epsilon^{-1})$. The proofs of these statements are based on averaging of two blow-up systems, allowing one to estimate the effect of the singularity, combined with results on asymptotics of the second Painleve equation.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-03T07:40:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periodic orbits",
      "bifurcating slow manifold",
      "normally elliptic slow manifold",
      "slowly varying phase",
      "time scale separation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.7189U"
    }
  }
}