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Some chaotic properties of discrete fuzzy dynamical systems. (English) Zbl 1263.37031

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 H. Román-Flores and 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.

MSC:

37B99 Topological dynamics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Citations:

Zbl 1142.37308
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References:

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