Extreme Chaos and Transitivity

2003 ◽  
Vol 13 (07) ◽  
pp. 1695-1700 ◽  
Author(s):  
Marta Babilonová-Štefánková
Keyword(s):  
Topological Entropy ◽  
Distributional Chaos ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Almost Everywhere ◽  
Provided Examples

In the eighties, Misiurewicz, Bruckner and Hu provided examples of functions chaotic in the sense of Li and Yorke almost everywhere. In this paper we show that similar results are true for distributional chaos, introduced in [Schweizer & Smítal, 1994]. In fact, we show that any bitransitive continuous map of the interval is conjugate to a map uniformly distributionally chaotic almost everywhere. Using a result of A. M. Blokh we get as a consequence that for any map f with positive topological entropy there is a k such that fk is semiconjugate to a continuous map uniformly distributionally chaotic almost everywhere, and consequently, chaotic in the sense of Li and Yorke almost everywhere.

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.

scholarly journals A characterization of chaos

1986 ◽  
Vol 34 (2) ◽  
pp. 283-292 ◽  
Author(s):  
K. Janková ◽  
J. Smítal
Keyword(s):  
Topological Entropy ◽  
Real Interval ◽  
Complex Behaviour ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Continuous Mappings

Consider the continuous mappings f from a compact real interval to itself. We show that when f has a positive topological entropy (or equivalently, when f has a cycle of order ≠ 2n, n = 0, 1, 2, …) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a certain level ɛ > 0. Recent results of the second author then imply that any continuous map has exactly one of the following properties: It is either strongly chaotic or every trajectory is approximable by cycles. Also some other conditions characterizing chaos are given.

Distributional Chaos of Cournot maps

2001 ◽  
Vol 1 (2) ◽  
Author(s):  
José Salvador Cáovas Peña ◽  
Gabriel Soler López ◽  
Manuel Ruiz Marín
Keyword(s):  
Topological Entropy ◽  
Two Dimensional ◽  
Cournot Duopoly ◽  
Distributional Chaos ◽  
Positive Topological Entropy ◽  
Economic Process

AbstractWe study the notion of distributional chaos for a class of two-dimensional maps called Cournot maps, which models a competitive economic process (Cournot duopoly). Among other results, we prove that a Cournot map has positive topological entropy if and only if it is distributionally chaotic.

scholarly journals Distribution of Maps with Transversal Homoclinic Orbits in a Continuous Map Space

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Qiuju Xing ◽  
Yuming Shi
Keyword(s):  
Banach Space ◽  
Topological Entropy ◽  
Homoclinic Orbits ◽  
Positive Topological Entropy ◽  
Continuous Map ◽  
Bounded Set ◽  
Transversal Homoclinic Orbits ◽  
Continuous Maps ◽  
Dense Set

This paper is concerned with distribution of maps with transversal homoclinic orbits in a continuous map space, which consists of continuous maps defined in a closed and bounded set of a Banach space. By the transversal homoclinic theorem, it is shown that the map space contains a dense set of maps that have transversal homoclinic orbits and are chaotic in the sense of both Li-Yorke and Devaney with positive topological entropy.

Topological Entropy and Mixing Invariant Extremal Distributional Chaos

2017 ◽  
Vol 27 (09) ◽  
pp. 1750139 ◽  
Author(s):  
Lidong Wang ◽  
Nan Li ◽  
Fengchun Lei ◽  
Zhenyan Chu
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Distributional Chaos ◽  
Complex Dynamical System ◽  
Positive Topological Entropy

We show that there exists a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set. Meanwhile, we prove that such a complex dynamical system has zero topological entropy. Finally, we extend the method of constructing the “strong” distributionally scrambled set and show that the new dynamical system has a positive topological entropy.

scholarly journals ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

scholarly journals On α-Limit Sets in Lorenz Maps

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1153
Author(s):  
Łukasz Cholewa ◽  
Piotr Oprocha
Keyword(s):  
Topological Entropy ◽  
Dynamical Behavior ◽  
Unimodal Maps ◽  
Limit Sets ◽  
Lorenz Maps ◽  
Provided Examples

The aim of this paper is to show that α-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on [0,1]. On the basis of provided examples, we also present how the performed study on the structure of α-limit sets is closely connected with the calculation of the topological entropy.

CHAOS ON ONE-DIMENSIONAL COMPACT METRIC SPACES

2012 ◽  
Vol 22 (10) ◽  
pp. 1250259 ◽  
Author(s):  
ZDENĚK KOČAN
Keyword(s):  
Topological Entropy ◽  
Metric Spaces ◽  
Chaotic Behavior ◽  
Homoclinic Trajectory ◽  
One Dimensional ◽  
Distributional Chaos ◽  
Compact Metric Spaces ◽  
Continuous Maps

We consider various kinds of chaotic behavior of continuous maps on compact metric spaces: the positivity of topological entropy, the existence of a horseshoe, the existence of a homoclinic trajectory (or perhaps, an eventually periodic homoclinic trajectory), three levels of Li–Yorke chaos, three levels of ω-chaos and distributional chaos of type 1. The relations between these properties are known when the space is an interval. We survey the known results in the case of trees, graphs and dendrites.

scholarly journals Ivanov’s Theorem for Admissible Pairs Applicable to Impulsive Differential Equations and Inclusions on Tori

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1602
Author(s):  
Jan Andres ◽  
Jerzy Jezierski
Keyword(s):  
Differential Equations ◽  
Topological Entropy ◽  
Lower Estimate ◽  
Deterministic Chaos ◽  
Impulsive Differential Equations ◽  
Nielsen Numbers ◽  
Positive Topological Entropy ◽  
Admissible Pairs ◽  
Impulsive Dynamics ◽  
Differential Equations And Inclusions

The main aim of this article is two-fold: (i) to generalize into a multivalued setting the classical Ivanov theorem about the lower estimate of a topological entropy in terms of the asymptotic Nielsen numbers, and (ii) to apply the related inequality for admissible pairs to impulsive differential equations and inclusions on tori. In case of a positive topological entropy, the obtained result can be regarded as a nontrivial contribution to deterministic chaos for multivalued impulsive dynamics.

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