Sensitive dependence and transitivity of fuzzified dynamical systems

TL;DR

This paper establishes a theoretical link between the sensitive dependence properties of dynamical systems and their fuzzified versions, providing conditions under which sensitivity and transitivity are preserved or not.

Contribution

It proves the equivalence of sensitivity properties between dynamical systems and their $g$-fuzzifications and addresses an open question on transitivity in fuzzified systems.

Findings

Sensitivity of a system is equivalent to its fuzzification's sensitivity.

Existence of sensitive systems whose fuzzifications lack sensitivity.

Conditions under which fuzzified systems are not transitive.

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}.