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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.
Mathematical AnalysisSlezská univerzita v Opavě
, 2000
"... Chaos in discrete dynamical systems ..."
PERIODIC POINTS OF LATITUDINAL SPHERE MAPS
"... Abstract. For the maps of the twodimensional sphere into itself that preserve the latitude foliation and are differentiable at the poles, lower estimates of the number of fixed points for the maps and their iterates are obtained. Those estimates also show that the growth rate of the number of fixed ..."
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Abstract. For the maps of the twodimensional sphere into itself that preserve the latitude foliation and are differentiable at the poles, lower estimates of the number of fixed points for the maps and their iterates are obtained. Those estimates also show that the growth rate of the number of fixed points of the iterates is larger than or equal to the logarithm of the absolute value of the degree of the map. 1.