{
  "id": "1409.2166",
  "version": "v1",
  "published": "2014-09-07T21:04:57.000Z",
  "updated": "2014-09-07T21:04:57.000Z",
  "title": "Chaos in Dynamics of a Family of Transcendental Meromorphic Functions",
  "authors": [
    "M. Sajid",
    "G. P. Kapoor"
  ],
  "comment": "18 pages",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "The characterization and properties of Julia sets of one parameter family of transcendental meromorphic functions $\\zeta_\\lambda(z)=\\lambda \\frac{z}{z+1} e^{-z}$, $\\lambda >0$, $z\\in \\mathbb{C}$ is investigated in the present paper. It is found that bifurcations in the dynamics of $\\zeta_\\lambda(x)$, $x\\in {\\mathbb{R}}\\setminus \\{-1\\}$, occur at several parameter values and the dynamics of the family becomes chaotic when the parameter $\\lambda$ crosses certain values. The Lyapunov exponent of $\\zeta_\\lambda(x)$ for certain values of the parameter $\\lambda$ is computed for quantifying the chaos in its dynamics. The characterization of the Julia set of the function $\\zeta_\\lambda(z)$ as complement of the basin of attraction of an attracting real fixed point of $\\zeta_\\lambda(z)$ is found here and is applied to computationally simulate the images of the Julia sets of $\\zeta_\\lambda(z)$. Further, it is established that the Julia set of $\\zeta_\\lambda(z)$ for $\\lambda>(\\sqrt{2}+1) e^{\\sqrt{2}}$ contains the complement of attracting periodic orbits of $\\zeta_\\lambda(x)$. Finally, the results on the dynamics of functions $\\lambda \\tan z$, $\\lambda \\in {\\mathbb{\\hat{C}}}\\setminus\\{0\\}$, $E_{\\lambda}(z) = \\lambda \\frac{e^{z} -1}{z}$, $\\lambda > 0$ and $f_{\\lambda}=\\lambda f(z)$, $\\lambda>0$, where $f(z)$ has certain properties, are compared with the results found in the present paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-07T21:04:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A18",
      "30D05",
      "37C25",
      "37D45",
      "37F50",
      "58K05"
    ],
    "keywords": [
      "transcendental meromorphic functions",
      "julia set",
      "properties",
      "parameter values",
      "attracting real fixed point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.2166S"
    }
  }
}