On Devaney chaotic induced fuzzy and set-valued dynamical systems

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 Román-Flores and Chalco-Cano [Some chaotic properties of Zadehʹs extension, Chaos, Solitons and Fractals 35(3) (2008) 452–459].