Sensitivity and strong sensitivity on induced dynamical systems

Given a metric space X, we consider the family of all normal upper semicontinuous fuzzy sets on X, denoted by F(X), and a discrete dynamical system (X, f). In this paper, we study when (F(X), fb) is (strongly) sensitive, where fb is the Zadeh’s extension of f and 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 (K(X), f) is equivalent to the sensitivity in fb: F(X) → F(X) with respect to the uniform metric, the Skorokhod metric and the sendograph metric. We also show that the following conditions are equivalent: a) (X, f) is strongly sensitive; b) (F(X), fb) is strongly sensitive, where F(X) is equipped with the uniform metric; c) (F(X), fb) is strongly sensitive, where F(X) is equipped with the Skorokhod metric; d) (F(X), fb) is strongly sensitive, where F(X) is equipped with the sendograph metric.