Martelli Chaotic Properties of a Generalized Form of Zadeh’s Extension Principle

Abstract

Let $X$ denote a compact metric space and let $f : X\to X$ be a continuous map. It is known that a discrete dynamical system ($X,f$) naturally induces its fuzzified counterpart, that is, a discrete dynamical system on the space of fuzzy compact subsets of $X$. In 2011, a new generalized form of Zadeh’s extension principle, so-called $g$-fuzzification, had been introduced by Kupka 2011. In this paper, we study the relations between Martelli’s chaotic properties of the original and $g$-fuzzified system. More specifically, we study the transitivity, sensitivity, and stability of the orbits in system ($X,f$) and its connections with the same ones in its $g$-fuzzified system.