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On fuzzifications of discrete dynamical systems. (English) Zbl 1229.93107

Summary: Let X denote a locally compact metric space and φ:XX be a continuous map. In the 1970s, Zadeh presented an extension principle helping us to fuzzify the dynamical system (X,φ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond and A. Pokrovskii, Fuzzy Sets Syst. 61, No. 3, 277–283 (1994; Zbl 0827.58037)] 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, the endograph and levelwise topologies.

MSC:
93C42Fuzzy control systems
93C25Control systems in abstract spaces
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