Sensitivity and strong sensitivity on induced dynamical systems

Abstract

Given a metric space X, we consider the family of all normal upper semicontinuous fuzzy sets on X, denoted by $mathcal{F}(X)$, and a discrete dynamical system $(X,f)$. In this paper, we study when $(mathcal{F}(X), widehat{f})$ is (strongly) sensitive, where $widehat{f}$ is the Zadeh's extension of f and $mathcal{F}(X)$ is equipped with different metrics: The uniform metric, the Skorokhod metric, the sendograph metric and the endograph metric. We prove that the sensitivity in the induced dynamical system $(mathcal{K}(X),overline{f})$ is equivalent to the sensitivity in $ widehat{f} :mathcal{F}(X)to mathcal{F}(X) $ with respect to the uniform metric, the Skorokhod metric and the sendograph metric. We also show that the following conditions are equivalent:item {rm a)} $(X,f)$ is strongly sensitive;item {rm b)} $(mathcal{F}(X), widehat{f})$ is strongly sensitive, where $mathcal{F}(X)$ is equipped with the uniform metric;item {rm c)} $(mathcal{F}(X), widehat{f})$ is strongly sensitive, where $mathcal{F}(X)$ is equipped with the Skorokhod metric;item {rm d)} $(mathcal{F}(X), widehat{f})$ is strongly sensitive, where $mathcal{F}(X)$ is equipped with the sendograph metric.