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.

scholarly journals Omega-limit sets for shift spaces and unimodal maps

2016 ◽  
Vol 209 ◽  
pp. 33-45 ◽  
Author(s):  
Lori Alvin ◽  
Nicholas Ormes
Keyword(s):  
Unimodal Maps ◽  
Limit Sets ◽  
Shift Spaces

Allee’s Effect Bifurcation in Generalized Logistic Maps

2019 ◽  
Vol 29 (03) ◽  
pp. 1950039
Author(s):  
J. Leonel Rocha ◽  
Abdel-Kaddous Taha
Keyword(s):  
Allee Effect ◽  
Dynamical Behavior ◽  
Complete Classification ◽  
Unimodal Maps ◽  
New Class ◽  
New Type ◽  
Local And Global Bifurcations ◽  
One Dimensional Maps ◽  
Necessary And Sufficient

This paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee’s effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.

scholarly journals The Dynamics of a Four-Step Feedback Procedure to Control Chaos

2021 ◽  
Author(s):  
Jose S. Cánovas
Keyword(s):  
Topological Entropy ◽  
Dynamical Behavior ◽  
Logistic Map ◽  
Simple Complex ◽  
Step Procedure ◽  
Parrondo’S Paradox

Abstract In this paper we make a description of the dynamics of a four-step procedure to control the dynamics of the logistic map. Some massive calculations are made for computing the topological entropy with prescribed accuracy. This provides us the parameter regions where the model has a complicated dynamical behavior. Our computations also show the dynamic Parrondo's paradox ``simple+simple=complex'', which should be taking into account to avoid undesirable dynamics.

scholarly journals Topological properties of Lorenz maps derived from unimodal maps

2020 ◽  
Vol 26 (8) ◽  
pp. 1174-1191 ◽  
Author(s):  
Ana Anušić ◽  
Henk Bruin ◽  
Jernej Činč
Keyword(s):  
Topological Properties ◽  
Unimodal Maps ◽  
Lorenz Maps

Knaster-like continua and complex dynamics

1993 ◽  
Vol 13 (4) ◽  
pp. 627-634 ◽  
Author(s):  
Robert L. Devaney
Keyword(s):  
Complex Dynamics ◽  
Julia Set ◽  
Dynamical Behavior ◽  
Invariant Sets ◽  
Entire Plane ◽  
Limit Sets

AbstractIn this paper we discuss the topology and dynamics ofEλ(z) = λezwhen λ is real and λ > 1/e. It is known that the Julia set ofEλis the entire plane in this case. Our goal is to show that there are certain natural invariant subsets forEλwhich are topologically Knaster-like continua. Moreover, the dynamical behavior on these invariant sets is quite tame. We show that the only trivial kinds of α- and ω-limit sets are possible.

scholarly journals Monotonicity and non-monotonicity regions of topological entropy for Lorenz-like families with infinite derivatives

2020 ◽  
Vol 5 (2) ◽  
pp. 293-306
Author(s):  
M.I. Malkin ◽  
K.A. Safonov
Keyword(s):  
Normal Form ◽  
Topological Entropy ◽  
Lorenz Attractor ◽  
Homoclinic Loop ◽  
One Dimensional ◽  
Study Behavior ◽  
Saddle Value ◽  
Lorenz Attractors ◽  
Two Parameter ◽  
Lorenz Maps

AbstractWe study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps Tc,ɛ(x) = (−1 + c|x|1−ɛ) · sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

DEVIL'S STAIRCASE OF GAP MAPS

2003 ◽  
Vol 13 (01) ◽  
pp. 115-122 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
CHENG-HSIUNG HSU ◽  
SONG-SUN LIN
Keyword(s):  
Topological Entropy ◽  
Devil's Staircase ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Devil’S Staircase ◽  
Staircase Structure ◽  
Entropy Functions ◽  
Kneading Theory

This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.

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.

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