Sensitive dependence and transitivity of fuzzified dynamical systems

Abstract

This paper proves that a set-valued dynamical system is sensitively dependent on initial conditions (resp., $\mathscr{F}$-sensitive, multi-sensitive) if and only if its $g$-fuzzification is sensitively dependent on initial conditions (resp., $\mathscr{F}$-sensitive, multi-sensitive), where $\mathscr{F}$ is a Furstenberg family. As an application, it is shown that there exists a sensitive dynamical system whose $g$-fuzzification does not have such sensitive dependence for any $g$ in a certain domain. Moreover, a sufficient condition ensuring that the $g$-fuzzification of every nontrivial dynamical system is not transitive is obtained. These give an answer to a question posed in \cite[J. Kupka, Information Sciences, {\bf 279} (2014): 642--653]{Kupka2014}.