{
  "id": "0709.4643",
  "version": "v1",
  "published": "2007-09-28T19:44:17.000Z",
  "updated": "2007-09-28T19:44:17.000Z",
  "title": "Periodic solutions of periodically perturbed planar autonomous systems: A topological approach",
  "authors": [
    "Mikhail Kamenskii",
    "Oleg Makarenkov",
    "Paolo Nistri"
  ],
  "journal": "Adv. Differential Equations, 11 (2006), 399-418",
  "categories": [
    "math.CA"
  ],
  "abstract": "Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x_0 of least period T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In the case when T_0/T_1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation e>0 sufficiently small, the existence of klT_0-periodic solutions x_e of the perturbed system which converge to the trajectory x_1 of the limit cycle as e->0. Moreover, we state conditions under which T=klT_0 is the least period of the periodic solutions x_e. We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T_0/T_1 is an irrational number we show the nonexistence, whenever T>0 and e>0, of T-periodic solutions x_e of the perturbed system converging to x_1. The employed methods are based on the topological degree theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-28T19:44:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A34",
      "34C25",
      "34D10"
    ],
    "keywords": [
      "periodically perturbed planar autonomous systems",
      "periodic solutions",
      "topological approach",
      "limit cycle",
      "nonlinear planar autonomous system"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.4643K"
    }
  }
}