{
  "id": "1611.04807",
  "version": "v1",
  "published": "2016-11-15T12:36:35.000Z",
  "updated": "2016-11-15T12:36:35.000Z",
  "title": "Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov-Schmidt reduction",
  "authors": [
    "Murilo R. Cândido",
    "Jaume Llibre",
    "Douglas D. Novaes"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form $$g(z,\\varepsilon)=g_0(z)+\\sum_{i=1}^k \\varepsilon^i g_i(z)+\\mathcal{O}(\\varepsilon^{k+1}),$$ for $|\\varepsilon|\\neq0$ sufficiently small. Here $g_i:\\mathcal{D}\\rightarrow\\mathbb{R}^n$, for $i=0,1,\\ldots,k$, are smooth functions being $\\mathcal{D}\\subset \\mathbb{R}^n$ an open bounded set. Then we use this result to compute the bifurcation functions which controls the periodic solutions of the following $T$-periodic smooth differential system $$ x'=F_0(t,x)+\\sum_{i=1}^k \\varepsilon^i F_i(t,x)+\\mathcal{O}(\\varepsilon^{k+1}), \\quad (t,z)\\in\\mathbb{S}^1\\times\\mathcal{D}. $$ It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions $\\mathcal{Z}$, $\\textrm{dim}(\\mathcal{Z})\\leq n$. We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-15T12:36:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C29",
      "34C25",
      "37G15"
    ],
    "keywords": [
      "higher order perturbed differential systems",
      "periodic solutions",
      "lyapunov-schmidt reduction",
      "bifurcation functions",
      "persistence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}