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

Abstract

LetXdenote a compact metric space and letf : X→Xbe 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 ofX. In 2011, a new generalized form of Zadeh’s extension principle, so-calledg-fuzzification, had been introduced by Kupka 2011. In this paper, we study the relations between Martelli’s chaotic properties of the original andg-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 itsg-fuzzified system.