Asymptotic average shadowing property, almost specification property and distributional chaos

2016 ◽  
Vol 30 (03) ◽  
pp. 1650001 ◽  
Author(s):  
Lidong Wang ◽  
Xiang Wang ◽  
Fengchun Lei ◽  
Heng Liu
Keyword(s):  
Dynamical System ◽  
Orbit Closure ◽  
Shadowing Property ◽  
Distributional Chaos ◽  
Specification Property ◽  
Asymptotic Average Shadowing

It is proved that a nontrivial compact dynamical system with asymptotic average shadowing property (AASP) displays uniformly distributional chaos or distributional chaos in a sequence. Moreover, distributional chaos in a system with AASP can be uniform and dense in the measure center, that is, there is an uncountable uniformly distributionally scrambled set consisting of such points that the orbit closure of every point contains the measure center. As a corollary, the similar results hold for the system with almost specification property.

Sensitivity and specification property of fuzzified dynamical systems

2018 ◽  
Vol 32 (23) ◽  
pp. 1850268
Author(s):  
Nan Li ◽  
Lidong Wang ◽  
Fengchun Lei
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Original System ◽  
Shadowing Property ◽  
Specification Property

The main purpose of this paper is to further explore the complexity of fuzzified dynamical systems. Especially, we study several kinds of specification properties of Zadeh’s extension. Among other things, we discuss the “stronger” sensitivity on product dynamical systems of g-fuzzification. There are two major ingredients. Firstly, it is proved that the specification (respectively almost specification) property of the original system and its Zadeh’s extension is equivalent, when the original system has the shadowing property. Moreover, we study the [Formula: see text]-sensitivity (respectively multi-sensitivity) of g-fuzzification and its induced product dynamical system.

A Class of Hyperspace Systems and Distributional Chaos

2014 ◽  
Vol 670-671 ◽  
pp. 1570-1572
Author(s):  
Wei Wang ◽  
Xiao Gang Zhu
Keyword(s):  
Dynamical System ◽  
Difference Equation ◽  
Agricultural Production ◽  
Irrigation District ◽  
Corn Yield ◽  
Primitive Substitution ◽  
Yield Analysis ◽  
Distributional Chaos ◽  
Dynamical Properties ◽  
Ecological Difference

Research on dynamical system has penetrated into many problems of agricultural production, such as prediction of corn yield, analysis on operational situation of irrigation district and research on ecological difference equation. In this paper, we investigated dynamical properties for non-primitive substitution and the set-valued maps induced by the substitution. We proved In two cases that the hyperspace systems induced by non-primitive substitution are not distributional chaotic.

scholarly journals A Note on Ergodicity of Systems with the Asymptotic Average Shadowing Property

2011 ◽  
Vol 2011 ◽  
pp. 1-6 ◽  
Author(s):  
Risong Li ◽  
Xiaoliang Zhou
Keyword(s):  
Metric Space ◽  
Compact Metric Space ◽  
Shadowing Property ◽  
Topologically Transitive ◽  
Stable Map ◽  
Asymptotic Average Shadowing

We prove that if a continuous, Lyapunov stable mapffrom a compact metric spaceXinto itself is topologically transitive and has the asymptotic average shadowing property, thenXis consisting of one point. As an application, we prove that the identity mapiX:X→Xdoes not have the asymptotic average shadowing property, whereXis a compact metric space with at least two points.

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.

Generalized Specification Property and Distributional Chaos

2003 ◽  
Vol 13 (07) ◽  
pp. 1683-1694 ◽  
Author(s):  
F. Balibrea ◽  
B. Schweizer ◽  
A. Sklar ◽  
J. Smítal
Keyword(s):  
Compact Interval ◽  
Limit Set ◽  
Distributional Chaos ◽  
Specification Property ◽  
Continuous Map ◽  
Basic Set ◽  
Made In

Let f be a continuous map from a compact interval into itself. Continuing the work begun by Schweizer and Smítal [1994], we prove that the restriction of f to any basic set (i.e. any nonsolenoidal, infinite, maximal ω-limit set) satisfies a generalization of the specification property. We apply this generalization to establish several conjectures made in the abovementioned paper, e.g. the fact that distributional chaos is stable.

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