On fuzzifications of discrete dynamical systems

  • Authors:

  • Jiří Kupka

  • Affiliations:

  • Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 33 Ostrava, Czech Republic

  • Venue:
  • Information Sciences: an International Journal
  • Year:
  • 2011

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Abstract

Let X denote a locally compact metric space and @f : X-X be a continuous map. In the 1970s Zadeh presented an extension principle helping us to fuzzify the dynamical system (X,@f), i.e., to obtain a map @F for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets Syst. 61 (1994) 277-283] to generalize Zadeh's original extension principle. In this paper we study basic properties of so-called g-fuzzifications, such as their continuity properties. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on X induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems. Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, endograph and levelwise topologies.