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Zbl 1242.37014
Kupka, Jiř{\'\i}
On Devaney chaotic induced fuzzy and set-valued dynamical systems.
(English)
[J] Fuzzy Sets Syst. 177, No. 1, 34-44 (2011). ISSN 0165-0114

Summary: It is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, i.e., a discrete dynamical system on the space of fuzzy sets. In this paper we study relations between dynamical properties of the original and the fuzzified dynamical system. Especially, we study conditions used in the definition of Devaney chaotic maps, i.e., periodic density and transitivity. Among other things we show that dynamical behavior of the set-valued and fuzzy extensions of the original system mutually inherits some global characteristics and that the space of fuzzy sets admits a transitive fuzzification. This paper contains the solution of the problem that was partially solved by {\it H. Román-Flores} and {\it Y. Chalco-Cano} [Chaos Solitons Fractals 35, No. 3, 452--459 (2008; Zbl 1142.37308)].
MSC 2000:
*37B99 Topological dynamics
37D45 Strange attractors, chaotic dynamics
37C25 Fixed points, periodic points, fixed-point index theory
54A40 Fuzzy topology

Keywords: fuzzy dynamical system; set-valued dynamical system; Zadeh's extension; fuzzification; Devaney chaos; transitivity; exactness; periodic density

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