On Devaney's definition of chaos
American Mathematical Monthly
Chaos, entropy and a generalized extension principle
Fuzzy Sets and Systems
Fuzzy homoclinic orbits and commuting fuzzifications
Fuzzy Sets and Systems
Topological entropy of fuzzified dynamical systems
Fuzzy Sets and Systems
On fuzzifications of discrete dynamical systems
Information Sciences: an International Journal
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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 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 Roman-Flores and Chalco-Cano [Some chaotic properties of Zadeh's extension, Chaos, Solitons and Fractals 35(3) (2008) 452-459].