On Devaney chaotic induced fuzzy and set-valued dynamical systems

Kupka, Jiří

doi:10.1016/j.fss.2011.04.006

Cited by 33 publications

(18 citation statements)

References 20 publications

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“…Banks [1], Liao, Wang and Zhang [27], and Peris [30] solved independently the main problem proposed in [32,12] by giving a characterization of topological transitivity for collective dynamics in terms of the weak mixing property for individual dynamics. Several recent results in this topic can be found in, e.g., [25,36,18,28,37,24,13,19,29,7,26,39].…”

Section: Introductionmentioning

confidence: 99%

Set-Valued Chaos in Linear Dynamics

Bernardes

Peris

Ródenas

2017

Integr. Equ. Oper. Theory

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We study several notions of chaos for hyperspace dynamics associated to continuous linear operators. More precisely, we consider a continuous linear operator T : X → X on a topological vector space X, and the natural hyperspace extensions T and T of T to the spaces K(X) of compact subsets of X and C(X) of convex compact subsets of X, respectively, endowed with the Vietoris topology. We show that, when X is a complete locally convex space (respectively, a locally convex space), then Devaney chaos (respectively, topological ergodicity) is equivalent for the maps T , T and T . Also, under very general conditions, we obtain analogous equivalences for Li-Yorke chaos. Finally, some remarks concerning the topological transitivity and weak mixing properties are included, extending results in [1] and [30].

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Section: Introductionmentioning

confidence: 99%

Set-Valued Chaos in Linear Dynamics

Bernardes

Peris

Ródenas

2017

Integr. Equ. Oper. Theory

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“…Román-Flores and Chalco-Cano [25] studied some chaotic properties (for example, transitivity, sensitive dependence, periodic density) for the Zadeh's extension of a dynamical system. Then, Kupka [15] investigated the relations between Devaney chaos in the original system and in the Zadeh's extension and proved that Zadeh's extension is periodically dense in F(X) (resp. F ≥λ (X) for any λ ∈ (0, 1]) if and only if so is (K(X), T K ) (see Lemma 4).…”

Section: Introductionmentioning

confidence: 99%

Topological Dynamics of Zadeh’s Extension on Upper Semi-Continuous Fuzzy Sets

Ding

et al. 2017

Int. J. Bifurcation Chaos

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In this paper, some characterizations about transitivity, mildly mixing property, a-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh's extensions restricted on some invariant closed subsets of the space of all upper semi-continuous fuzzy sets with the level-wise metric are obtained. In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and a-transitive, equicontinuous, uniformly rigid) if and only if the Zadeh's extension is transitive (resp., mildly mixing, a-transitive, equicontinuous, uniformly rigid).

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“…and the chaotic relationships between f andf has been also recently studied (see [10,3,6], where F(X) is the class of all non empty and compact fuzzy sets on X,f is the Zadeh's extension of f to F(X) and D is the extension of the Hausdorff metric to F(X) which is defined by…”

Section: Preliminaries and Basic Resultsmentioning

confidence: 99%

Transitivity of interval and fuzzy-interval extensions of interval functions

Román-Flores¹,

Chalco-Cano²

2015

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and

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Let I = [a, b] be a real compact interval and f : I → I a continuous function. Define K c ([a, b]) the class of all non empty compact subinterval of [a, b] and letf the natural extension of f to K c ([a, b] ([a, b]) the class of all fuzzy-intervals with support contained in [a, b] and considerf the Zadeh's extension of f to F c ([a, b]). The aim of this paper is to show thatf andf are not transitive for any arbitrary self continuous function f defined on I.

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On Devaney chaotic induced fuzzy and set-valued dynamical systems

Cited by 33 publications

References 20 publications

Set-Valued Chaos in Linear Dynamics

Set-Valued Chaos in Linear Dynamics

Topological Dynamics of Zadeh’s Extension on Upper Semi-Continuous Fuzzy Sets

Transitivity of interval and fuzzy-interval extensions of interval functions

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