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On fuzzifications of discrete dynamical systems. (English) Zbl 1229.93107
Summary: Let $X$ denote a locally compact metric space and $\varphi:X \rightarrow X$ be a continuous map. In the 1970s, Zadeh presented an extension principle helping us to fuzzify the dynamical system $(X,\varphi)$, i.e., to obtain a map $\Phi$ for the space of fuzzy sets on $X$. We extend an idea mentioned in [{\it P. Diamond} and {\it 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.

93C42Fuzzy control systems
93C25Control systems in abstract spaces
Full Text: DOI
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