scholarly journals Some Chaotic Properties of Discrete Fuzzy Dynamical Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yaoyao Lan ◽  
Qingguo Li ◽  
Chunlai Mu ◽  
Hua Huang
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Continuous Map ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Fuzzy Dynamical Systems ◽  
Compact Subsets

Letting(X,d)be a metric space,f:X→Xa continuous map, and(ℱ(X),D)the space of nonempty fuzzy compact subsets ofXwith the Hausdorff metric, one may study the dynamical properties of the Zadeh's extensionf̂:ℱ(X)→ℱ(X):u↦f̂u. In this paper, we present, as a response to the question proposed by Román-Flores and Chalco-Cano 2008, some chaotic relations betweenfandf̂. 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.

scholarly journals Li-Yorke Sensitivity of Set-Valued Discrete Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Heng Liu ◽  
Fengchun Lei ◽  
Lidong Wang
Keyword(s):  
Metric Space ◽  
Short Proof ◽  
Hausdorff Metric ◽  
Discrete Systems ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonempty Compact ◽  
Compact Subsets

Consider the surjective, continuous mapf:X→Xand the continuous mapf¯of𝒦(X)induced byf, whereXis a compact metric space and𝒦(X)is the space of all nonempty compact subsets ofXendowed with the Hausdorff metric. In this paper, we give a short proof that iff¯is Li-Yoke sensitive, thenfis Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity offdoes not imply Li-Yorke sensitivity off¯.

On multi-sensitivity with respect to a vector

2018 ◽  
Vol 32 (15) ◽  
pp. 1850166 ◽  
Author(s):  
Lixin Jiao ◽  
Lidong Wang ◽  
Fengquan Li ◽  
Heng Liu
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Sufficient Conditions ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Continuous Map ◽  
Periodic Sets ◽  
Vector Formula ◽  
Compact Subsets

Consider the surjective continuous map [Formula: see text]: [Formula: see text] defined on a compact metric space X. Let [Formula: see text] be the space of all non-empty compact subsets of X equipped with the Hausdorff metric and define [Formula: see text]: [Formula: see text] by [Formula: see text] for any [Formula: see text]. In this paper, we introduce several stronger versions of sensitivities, such as multi-sensitivity with respect to a vector, [Formula: see text]-sensitivity, strong multi-sensitivity. We obtain some basic properties of the concepts of these sensitivities and discuss the relationships with other sensitivities for continuous self-map on [0,[Formula: see text]1]. Some sufficient conditions for a dynamical system to be [Formula: see text]-sensitive are presented. Also, it is shown that the strong multi-sensitivity of f implies that [Formula: see text] is [Formula: see text]-sensitive. In turn, the [Formula: see text]-sensitivity of [Formula: see text] implies that [Formula: see text] is [Formula: see text]-sensitive. In particular, it is proved that if [Formula: see text] is a multi-transitive map with dense periodic sets, then f is [Formula: see text]-sensitive. Finally, we give a multi-sensitive example which is not [Formula: see text]-sensitive.

Chaos to multiple mappings from a set-valued view

2020 ◽  
Vol 34 (11) ◽  
pp. 2050108
Author(s):  
Hongbo Zeng ◽  
Lidong Wang ◽  
Tao Sun
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Topological Conjugacy ◽  
Compact Metric Space ◽  
Distributional Chaos ◽  
Continuous Maps ◽  
Weakly Mixing

Let [Formula: see text] be a compact metric space and [Formula: see text] be an [Formula: see text]-tuple of continuous maps from [Formula: see text] to itself. In this paper, we investigate the multiple mappings dynamical system [Formula: see text] with Hausdorff metric Li–Yorke chaos, distributional chaos and distributional chaos in a sequence properties from a set-valued view. On the basis of this research, we draw main conclusions as follows: (i) two topological conjugacy dynamical systems to multiple mappings have simultaneously Hausdorff metric Li–Yorke chaos or distributional chaos. (ii) Hausdorff metric Li–Yorke [Formula: see text]-chaos is equivalent to Hausdorff metric distributional [Formula: see text]-chaos in a sequence. (iii) By giving two examples, we show that there is non-mutual implication between that the multiple mappings [Formula: see text] is Hausdorff metric Li–Yorke chaos and that each element [Formula: see text] [Formula: see text] in [Formula: see text] is Li–Yorke chaos. (iv) For the multiple mappings, weakly mixing implies the Hausdorff metric strongly Li–Yorke chaos and Hausdorff metric distributional chaos in a sequence.

A Remark on Invariant Scrambled Sets

2012 ◽  
Vol 204-208 ◽  
pp. 4776-4779
Author(s):  
Lin Huang ◽  
Huo Yun Wang ◽  
Hong Ying Wu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Cantor Sets ◽  
Countable Union ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing

By a dynamical system we mean a compact metric space together with a continuous map . This article is devoted to study of invariant scrambled sets. A dynamical system is a periodically adsorbing system if there exists a fixed point and a periodic point such that and are dense in . We show that every topological weakly mixing and periodically adsorbing system contains an invariant and dense Mycielski scrambled set for some , where has no isolated points. A subset is a Myceilski set if it is a countable union of Cantor sets.

WEAKLY MIXING IMPLIES DISTRIBUTIONAL CHAOS IN A SEQUENCE

2010 ◽  
Vol 24 (14) ◽  
pp. 1595-1600 ◽  
Author(s):  
LIDONG WANG ◽  
YU'E YANG ◽  
ZHENYAN CHU ◽  
GONGFU LIAO
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Distributional Chaos ◽  
Weakly Mixing ◽  
Separable Metric Space

Let X be a separable metric space containing at least two points, and f:X→X be continuous. In this paper, we consider dynamical systems on X, and prove that topologically weakly mixing implies distributional chaos in a sequence.

scholarly journals A note on compact sets in spaces of subsets

1988 ◽  
Vol 38 (3) ◽  
pp. 393-395 ◽  
Author(s):  
Phil Diamond ◽  
Peter Kloeden
Keyword(s):  
Metric Space ◽  
Complete Metric Space ◽  
Hausdorff Metric ◽  
Compact Sets ◽  
Selection Theorem ◽  
Starshaped Sets ◽  
Nonempty Compact ◽  
Compact Subsets

A simple characterisation is given of compact sets of the space K(X), of nonempty compact subsets of a complete metric space X, with the Hausdorff metric dH. It is used to give a new proof of the Blaschke selection theorem for compact starshaped sets.

scholarly journals Dynamics of induced systems

2016 ◽  
Vol 37 (7) ◽  
pp. 2034-2059 ◽  
Author(s):  
ETHAN AKIN ◽  
JOSEPH AUSLANDER ◽  
ANIMA NAGAR
Keyword(s):  
Phase Space ◽  
Uniform Convergence ◽  
Metric Space ◽  
Cantor Set ◽  
Main Theme ◽  
Hausdorff Topology ◽  
Continuous Maps ◽  
Dynamical Properties ◽  
Compact Subsets

In this paper we study the dynamical properties of actions on the space of compact subsets of the phase space. More precisely, if$X$is a metric space, let$2^{X}$denote the space of non-empty compact subsets of$X$provided with the Hausdorff topology. If$f$is a continuous self-map on$X$, there is a naturally induced continuous self-map$f_{\ast }$on$2^{X}$. Our main theme is the interrelation between the dynamics of$f$and$f_{\ast }$. For such a study, it is useful to consider the space${\mathcal{C}}(K,X)$of continuous maps from a Cantor set$K$to$X$provided with the topology of uniform convergence, and$f_{\ast }$induced on${\mathcal{C}}(K,X)$by composition of maps. We mainly study the properties of transitive points of the induced system$(2^{X},f_{\ast })$both topologically and dynamically, and give some examples. We also look into some more properties of the system$(2^{X},f_{\ast })$.

scholarly journals Martelli Chaotic Properties of a Generalized Form of Zadeh’s Extension Principle

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Yaoyao Lan ◽  
Chunlai Mu
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Discrete Dynamical System ◽  
Extension Principle ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Compact Subsets

LetXdenote a compact metric space and letf : X→Xbe a continuous map. It is known that a discrete dynamical system (X,f) naturally induces its fuzzified counterpart, that is, a discrete dynamical system on the space of fuzzy compact subsets ofX. In 2011, a new generalized form of Zadeh’s extension principle, so-calledg-fuzzification, had been introduced by Kupka 2011. In this paper, we study the relations between Martelli’s chaotic properties of the original andg-fuzzified system. More specifically, we study the transitivity, sensitivity, and stability of the orbits in system (X,f) and its connections with the same ones in itsg-fuzzified system.

scholarly journals Orbital shadowing, internal chain transitivity and -limit sets

2016 ◽  
Vol 38 (1) ◽  
pp. 143-154 ◽  
Author(s):  
CHRIS GOOD ◽  
JONATHAN MEDDAUGH
Keyword(s):  
Metric Space ◽  
Hausdorff Metric ◽  
Compact Metric Space ◽  
Limit Sets ◽  
Continuous Map ◽  
Orbital Shadowing ◽  
Limit Shadowing ◽  
Chain Transitivity

Let $f:X\rightarrow X$ be a continuous map on a compact metric space, let $\unicode[STIX]{x1D714}_{f}$ be the collection of $\unicode[STIX]{x1D714}$-limit sets of $f$ and let $\mathit{ICT}(f)$ be the collection of closed internally chain transitive subsets. Provided that $f$ has shadowing, it is known that the closure of $\unicode[STIX]{x1D714}_{f}$ in the Hausdorff metric coincides with $\mathit{ICT}(f)$. In this paper, we prove that $\unicode[STIX]{x1D714}_{f}=\mathit{ICT}(f)$ if and only if $f$ satisfies Pilyugin’s notion of orbital limit shadowing. We also characterize those maps for which $\overline{\unicode[STIX]{x1D714}_{f}}=\mathit{ICT}(f)$ in terms of a variation of orbital shadowing.

Weakly Almost Periodic Points and Some Chaotic Properties of Dynamical Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550115 ◽  
Author(s):  
Jiandong Yin ◽  
Zuoling Zhou
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Full Measure ◽  
Almost Periodic ◽  
Countable Base ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Weakly Almost Periodic

Let X be a compact metric space and f : X → X be a continuous map. In this paper, ergodic chaos and strongly ergodic chaos are introduced, and it is proven that f is strongly ergodically chaotic if f is transitive but not minimal and has a full measure center. In addition, some sufficient conditions for f to be Ruelle–Takens chaotic are presented. For instance, we prove that f is Ruelle–Takens chaotic if f is transitive and there exists a countable base [Formula: see text] of X such that for each i > 0, the meeting time set N(Ui, Ui) for Ui with respect to itself has lower density larger than [Formula: see text].

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