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.

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.

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¯.

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.

Some properties of chain recurrent sets in a nonautonomous discrete dynamical system

2015 ◽  
Vol 6 (3) ◽  
Author(s):  
Dhaval Thakkar ◽  
Ruchi Das
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Discrete System ◽  
Discrete Dynamical System ◽  
Compact Metric Space ◽  
Chain Recurrence ◽  
Shadowing Property

AbstractIn this paper, we define chain recurrence and study properties of chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. We also study chain recurrent sets in a nonautonomous discrete system having shadowing property.

scholarly journals Thickly Syndetical Sensitivity of Topological Dynamical System

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Heng Liu ◽  
Li Liao ◽  
Lidong Wang
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Topological Dynamical System ◽  
Weak Mixing ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Devaney Chaos ◽  
Weakly Mixing

Consider the surjective continuous mapf:X→X, whereXis a compact metric space. In this paper we give several stronger versions of sensitivity, such as thick sensitivity, syndetic sensitivity, thickly syndetic sensitivity, and strong sensitivity. We establish the following. (1) If(X,f)is minimal and sensitive, then(X,f)is syndetically sensitive. (2) Weak mixing implies thick sensitivity. (3) If(X,f)is minimal and weakly mixing, then it is thickly syndetically sensitive. (4) If(X,f)is a nonminimalM-system, then it is thickly syndetically sensitive. Devaney chaos implies thickly periodic sensitivity. (5) We give a syndetically sensitive system which is not thickly sensitive. (6) We give thickly syndetically sensitive examples but not cofinitely sensitive ones.

Transitivity, weakly mixing property and chaos

2016 ◽  
Vol 30 (02) ◽  
pp. 1550274 ◽  
Author(s):  
Lidong Wang ◽  
Jianhua Liang ◽  
Yiyi Wang ◽  
Xuelian Sun
Keyword(s):  
Dynamical System ◽  
Periodic Point ◽  
Metric Space ◽  
Strong Sense ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing ◽  
Topologically Mixing ◽  
Mixing Property

Let [Formula: see text] be a compact metric space without isolated points and let [Formula: see text] be a continuous map. In this paper, if [Formula: see text] is a transitive dynamical system with a repelling periodic point, then [Formula: see text] is chaotic in the sense of Kato. In addition, if [Formula: see text] is weakly topologically mixing, then [Formula: see text] is chaotic in the strong sense of Kato.

On the polynomial entropy for morse gradient systems

2019 ◽  
Vol 69 (3) ◽  
pp. 611-624
Author(s):  
Jelena Katić ◽  
Milan Perić
Keyword(s):  
Metric Space ◽  
Singular Set ◽  
Compact Metric Space ◽  
Easy Method ◽  
Gradient Systems ◽  
Continuous Map ◽  
Negative Gradient

Abstract We adapt the construction from [HAUSEUX, L.—LE ROUX, F.: Polynomial entropy of Brouwer homeomorphisms, arXiv:1712.01502 (2017)] to obtain an easy method for computing the polynomial entropy for a continuous map of a compact metric space with finitely many non-wandering points. We compute the maximal cardinality of a singular set of Morse negative gradient systems and apply this method to compute the polynomial entropy for Morse gradient systems on surfaces.

scholarly journals A remark on positive topological entropy ofN-buffer switched flow networks

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-99 ◽  
Author(s):  
Xiao-Song Yang
Keyword(s):  
Topological Entropy ◽  
Metric Space ◽  
Elementary Proof ◽  
Flow Networks ◽  
Compact Metric Space ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Simple Theorem

We present a simpler elementary proof on positive topological entropy of theN-buffer switched flow networks based on a new simple theorem on positive topological entropy of continuous map on compact metric space.

Chain recurrence and attraction in non-compact spaces

1991 ◽  
Vol 11 (4) ◽  
pp. 709-729 ◽  
Author(s):  
Mike Hurley
Keyword(s):  
Metric Space ◽  
Basins Of Attraction ◽  
The Other ◽  
Compact Metric Space ◽  
Chain Recurrence ◽  
Fundamental Interest ◽  
Compact Spaces ◽  
Continuous Map ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractIn the study of a dynamical systemf:X→Xgenerated by a continuous mapfon a compact metric spaceX, thechain recurrent setis an object of fundamental interest. This set was defined by C. Conley, who showed that it has two rather different looking, but equivalent, definitions: one given in terms of ‘approximate orbits’ through individual points (pseudo-orbits, or ε-chains), and the other given in terms of the global structure of the class of ‘attractors’ and ‘basins of attraction’ off. The first of these definitions generalizes directly to dynamical systems on any metric space, compact or not. The main purpose of this paper is to extend the second definition to non-compact spaces in such a way that it remains equivalent to the first.

Horseshoes and the Conley index spectrum

2000 ◽  
Vol 20 (2) ◽  
pp. 365-377 ◽  
Author(s):  
MARIA C. CARBINATTO ◽  
JAROSLAW KWAPISZ ◽  
KONSTANTIN MISCHAIKOW
Keyword(s):  
Metric Space ◽  
Conley Index ◽  
Spectral Information ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Full Shift ◽  
Shift Dynamics ◽  
Isolating Neighborhood

Given a continuous map on a locally compact metric space and an isolating neighborhood which is decomposed into two disjoint isolating neighborhoods, it is shown that the spectral information of the associated Conley indices is sufficient to conclude the existence of a semi-conjugacy onto the full shift dynamics on two symbols.

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