Some chaotic properties of discrete fuzzy dynamical systems
Summary: Letting \((X, d)\) be a metric space, \(f : X \rightarrow X\) a continuous map, and \((\mathcal F(X), D)\) the space of nonempty fuzzy compact subsets of \(X\) with the Hausdorff metric, one may study the dynamical properties of the Zadeh's extension \(\hat{f} : \mathcal F(X) \rightarrow \mathcal F(X) : u \mapsto \hat{f}u\). In this paper, we present, as a response to the question proposed by \textit{H. Román-Flores} and \textit{Y. Chalco-Cano} [Chaos Solitons Fractals 35, No. 3, 452--459 (2008; Zbl 1142.37308)], some chaotic relations between \(f\) and \(\hat{f}\). More specifically, we study the transitivity, weakly mixing, periodic density in system \((X, f)\), and its connections with the same ones in its fuzzified system.
scientific article; zbMATH DE number 6134182
scientific article