{
  "id": "1612.07254",
  "version": "v1",
  "published": "2016-12-21T17:51:36.000Z",
  "updated": "2016-12-21T17:51:36.000Z",
  "title": "Quantitative stability of certain families of periodic solutions in the Sitnikov problem",
  "authors": [
    "Jorge Galán",
    "Daniel Núñez",
    "Andrés Rivera"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The Sitnikov problem is a special case of the restricted three-body problem where the primaries moves in elliptic orbits of the two-body problem with eccentricity $e\\in [0,1[$ and the massless body moves on a straight line perpendicular to the plane of motion of the primaries through their barycenter. It is well known that for the circular case ($e=0$) and a given $N\\in \\mathbb{N}$ there are a finite number of nontrivial symmetric $2N\\pi$ periodic solutions all of them parabolic and unstable (in the Lyapunov sense) if we consider the corresponding autonomous equation like a $2\\pi$-periodic equation. Using the method of global continuation of Leray-Schauder, J.Llibre and R.Ortega (J.Llibre $\\&$ R. Ortega, 2008) proved that these families of periodic solutions can be continued from the known $2N\\pi$-periodic solutions in the circular case for nonnecessarily small values of the eccentricity $e$ and in some cases for all values of $e\\in \\, [0,1[.$ However this approach does not say anything about the stability properties of this periodic solutions. In this document we present a new method that quantifies the mentioned bifurcating families and them stabilities properties at least in first approximation. Our approach proposes two general methods: The first one is to estimate the growing of the canonical solutions for one-parametric differential equation of the form \\[ \\ddot{x}+a(t,\\lambda)x=0, \\] with $a\\in C^{1}([0,T] \\times [0,\\Lambda])$. The second one gives stability criteria for one-parametric Hill's equation of the form \\[ \\ddot{x}+q(t,\\lambda)x=0, \\quad (\\ast) \\] where $q(\\cdot,\\lambda)$ is $T$-periodic and $q\\in C^{3}(\\mathbb{R}\\times [0,\\Lambda])$, such that for $\\lambda=0$ the equation $(*)$ is parabolic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-21T17:51:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "70F07",
      "37N05",
      "34B15",
      "37G15"
    ],
    "keywords": [
      "periodic solutions",
      "sitnikov problem",
      "quantitative stability",
      "circular case",
      "one-parametric differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}