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Zbl 1229.93107
Kupka, Jiř{\'\i}
On fuzzifications of discrete dynamical systems.
(English)
[J] Inf. Sci. 181, No. 13, 2858-2872 (2011). ISSN 0020-0255

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.
MSC 2000:
*93C42 Fuzzy control
93C25 Control systems in abstract spaces

Keywords: fuzzy discrete dynamical system; fuzzification; Zadeh's extension principle; endograph topology; sendograph topology; levelwise topology; conjugacy; semiconjugacy

Citations: Zbl 0827.58037

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