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On Fuzzifications of Discrete Dynamical Systems
, 2008
"... Let X denote a locally compact metric space and ϕ: X → X be a continuous map. In the 1970s L. 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, A. Pokrovskii, ..."
Abstract

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Let X denote a locally compact metric space and ϕ: X → X be a continuous map. In the 1970s L. 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, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets and Systems 61 (1994)] and we generalize Zadeh’s original extension principle. In this paper we study basic properties, such as the continuity of socalled gfuzzifications. We also show that, for any gfuzzification: (i) a uniformly convergent sequence of uniformly convergent maps on X induces a uniformly convegent sequence of continuous maps on the space of fuzzy sets, and (ii) a conjugacy (a semiconjugacy, resp.) between two discrete dynamical systems can be extended to a conjugacy (a semiconjugacy, resp.) between fuzzified dynamical systems. Moreover, at the end of this paper we show that there are connections between gfuzzifications and crisp dynamical systems via setvalued dynamical systems and skewproduct (triangular) maps. Throughout this paper we consider different topological structures in the space of fuzzy sets; namely, the sendograph, endograph and levelwise topologies.
Dynamical properties in the space of fuzzy numbers
, 2011
"... This paper is a contribution to the theoretical foundations of the theory of fuzzy dynamical systems. More precisely, we study relations between a given discrete dynamical system on the space X and its fuzzy counterpart Zadeh’s extension defined on the space of fuzzy sets on X. We provide a short ..."
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This paper is a contribution to the theoretical foundations of the theory of fuzzy dynamical systems. More precisely, we study relations between a given discrete dynamical system on the space X and its fuzzy counterpart Zadeh’s extension defined on the space of fuzzy sets on X. We provide a short discussion and a brief survey of some recent results devoted to various (especially chaotic) dynamical properties, a special attention is paid to fuzzifications on the space of fuzzy numbers, i.e., to fuzzy sets with connected αcuts.