{
  "id": "0709.1116",
  "version": "v2",
  "published": "2007-09-07T16:52:57.000Z",
  "updated": "2007-09-09T12:05:33.000Z",
  "title": "Bifurcation of the ACT map",
  "authors": [
    "Bau-Sen Du",
    "Ming-Chia Li",
    "Mikhail Malkin"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper, we study the Arneodo-Coullet-Tresser map $ F(x,y,z)=(ax-b(y-z), bx+a(y-z), cx-dx^k+e z)$ where $a,b,c,d,e$ are real with $bd\\neq 0$ and $k>1$ is an integer. We obtain stability regions for fixed points of $F$ and symmetric period-2 points while $c$ and $e$ vary as parameters. Varying $a$ and $e$ as parameters, we show that there is a hyperbolic invariant set on which $F$ is conjugate to the full shift on two or three symbols. We also show that chaotic behaviors of $F$ while $c$ and $d$ vary as parameters and $F$ is near an anti-integrable limit. Some numerical results indicates $F$ has Hopf bifurcation, strange attractors, and nested structure of invariant tori.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-09-09T12:05:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37G10",
      "37C25",
      "37C70",
      "37E99"
    ],
    "keywords": [
      "act map",
      "hyperbolic invariant set",
      "parameters",
      "arneodo-coullet-tresser map",
      "chaotic behaviors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.1116D"
    }
  }
}