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On turbulent, erratic and other dynamical properties of Zadeh’s extensions

Author

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  • Román-Flores, H.
  • Chalco-Cano, Y.
  • Silva, G.N.
  • Kupka, Jiří

Abstract

Let (X,d) be a compact metric space and f : X→X a continuous function. Consider the hyperspace (K(X),H) of all nonempty compact subsets of X endowed with the Hausdorff metric induced by d, and let (F(X),d∞) be the metric space of all nonempty compact fuzzy set on X equipped with the supremum metric d∞ which is calculated as the supremum of the Hausdorff distances of the corresponding level sets. If f¯ is the natural extension of f to (K(X),H) and fˆ is the Zadeh’s extension of f to (F(X),d∞), then the aim of this paper is to study the dynamics of f¯ and fˆ when f is turbulent (erratic, respectively).

Suggested Citation

  • Román-Flores, H. & Chalco-Cano, Y. & Silva, G.N. & Kupka, Jiří, 2011. "On turbulent, erratic and other dynamical properties of Zadeh’s extensions," Chaos, Solitons & Fractals, Elsevier, vol. 44(11), pages 990-994.
    • Handle: RePEc:eee:chsofr:v:44:y:2011:i:11:p:990-994
    DOI: 10.1016/j.chaos.2011.08.004
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    References listed on IDEAS

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    1. Román-Flores, Heriberto & Chalco-Cano, Y., 2005. "Robinson’s chaos in set-valued discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 33-42.
    2. Kwietniak, Dominik & Oprocha, Piotr, 2007. "Topological entropy and chaos for maps induced on hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 76-86.
    3. Ma, Xianfeng & Hou, Bingzhe & Liao, Gongfu, 2009. "Chaos in hyperspace system," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 653-660.
    4. Banks, John, 2005. "Chaos for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 681-685.
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