scholarly journals Further Discussions of the Complex Dynamics of a 2D Logistic Map: Basins of Attraction and Fractal Dimensions

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2001
Author(s):  
Sameh S. Askar ◽  
Abdulrahman Al-Khedhairi
Keyword(s):  
Phase Plane ◽  
Complex Dynamics ◽  
Logistic Map ◽  
Fractal Dimensions ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Complex Dynamic ◽  
2D Logistic Map ◽  
Two Parameters ◽  
Periodic Cycles

In this paper, we study the complex dynamic characteristics of a simple nonlinear logistic map. The map contains two parameters that have complex influences on the map’s dynamics. Assuming different values for those parameters gives rise to strange attractors with fractal dimensions. Furthermore, some of these chaotic attractors have heteroclinic cycles due to saddle-fixed points. The basins of attraction for some periodic cycles in the phase plane are divided into three regions of rank-1 preimages. We analyze those regions and show that the map is noninvertible and includes Z0,Z2 and Z4 regions.

A MODIFIED GENERALIZED LORENZ-TYPE SYSTEM AND ITS CANONICAL FORM

2009 ◽  
Vol 19 (06) ◽  
pp. 1931-1949 ◽  
Author(s):  
QIGUI YANG ◽  
KANGMING ZHANG ◽  
GUANRONG CHEN
Keyword(s):  
Special Form ◽  
Topological Structure ◽  
Complex Dynamics ◽  
Type System ◽  
Chaotic Attractors ◽  
Hopf Bifurcations ◽  
Compound Structure ◽  
Two Parameters ◽  
First Time ◽  
New System

In this paper, a modified generalized Lorenz-type system is introduced, which is state-equivalent to a simple and special form, and is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, two classes of new chaotic attractors are found for the first time in the literature, which are similar but nonequivalent in topological structure. To further understand the complex dynamics of the new system, some basic properties such as Lyapunov exponents, Hopf bifurcations and compound structure of the attractors are analyzed and demonstrated with careful numerical simulations.

CHARACTERIZATION OF CHAOTIC ATTRACTORS INSIDE BAND-MERGING SCENARIO IN A ZAD-CONTROLLED BUCK CONVERTER

2012 ◽  
Vol 22 (10) ◽  
pp. 1230034
Author(s):  
JOHN ALEXANDER TABORDA ◽  
FABIOLA ANGULO ◽  
GERARD OLIVAR
Keyword(s):  
Control Strategy ◽  
Control Parameter ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Buck Converter ◽  
Chaotic Attractors ◽  
Complex Dynamic ◽  
Self Similar ◽  
Dynamic Links

Zero Average Dynamics (ZAD) control strategy has been developed, applied and widely analyzed in the last decade. Numerous and interesting phenomena have been studied in systems controlled by ZAD strategy. In particular, the ZAD-controlled buck converter has been a source of nonlinear and nonsmooth phenomena, such as period-doubling, merging bands, period-doubling bands, torus destruction, fractal basins of attraction or codimension-2 bifurcations. In this paper, we report a new bifurcation scenario found inside band-merging scenario of ZAD-controlled buck converter. We use a novel qualitative framework named Dynamic Linkcounter (DLC) approach to characterize chaotic attractors between consecutive crisis bifurcations. This approach complements the results that can be obtained with Bandcounter approaches. Self-similar substructures denoted as Complex Dynamic Links (CDLs) are distinguished in multiband chaotic attractors. Geometrical changes in multiband chaotic attractors are detected when the control parameter of ZAD strategy is varied between two consecutive crisis bifurcations. Linkcount subtracting staircases are defined inside band-merging scenario.

scholarly journals Stochastic sensitivity of quasiperiodic and chaotic attractors of the discrete Lotka-Volterra model

2020 ◽  
Vol 55 ◽  
pp. 19-32
Author(s):  
A.V. Belyaev ◽  
T.V. Perevalova
Keyword(s):  
Period Doubling ◽  
Basins Of Attraction ◽  
Invariant Curve ◽  
Volterra Model ◽  
Chaotic Attractors ◽  
Deterministic System ◽  
System A ◽  
Stochastic Sensitivity ◽  
Two Parameters ◽  
Random States

The aim of the study presented in this article is to analyze the possible dynamic modes of the deterministic and stochastic Lotka-Volterra model. Depending on the two parameters of the system, a map of regimes is constructed. Parametric areas of existence of stable equilibria, cycles, closed invariant curves, and also chaotic attractors are studied. The bifurcations such as the period doubling, Neimark-Sacker and the crisis are described. The complex shape of the basins of attraction of irregular attractors (closed invariant curve and chaos) is demonstrated. In addition to the deterministic system, the stochastic system, which describes the influence of external random influence, is discussed. Here, the key is to find the sensitivity of such complex attractors as a closed invariant curve and chaos. In the case of chaos, an algorithm to find critical lines giving the boundary of a chaotic attractor, is described. Based on the found function of stochastic sensitivity, confidence domains are constructed that allow us to describe the form of random states around a deterministic attractor.

A DOUBLE LOGISTIC MAP

1994 ◽  
Vol 04 (01) ◽  
pp. 145-176 ◽  
Author(s):  
LAURA GARDINI ◽  
RALPH ABRAHAM ◽  
RONALD J. RECORD ◽  
DANIELE FOURNIER-PRUNARET
Keyword(s):  
Fixed Points ◽  
Logistic Map ◽  
Basins Of Attraction ◽  
The Other ◽  
Chaotic Attractors ◽  
Global Bifurcations ◽  
Invariant Curves ◽  
Critical Curves

Several endomorphisms of a plane have been constructed by coupling two logistic maps. Here we study the dynamics occurring in one of them, a twisted version due to J. Dorband, which (like the other models) is rich in global bifurcations. By use of critical curves, absorbing and invariant areas are determined, inside which global bifurcations of the attracting sets (fixed points, closed invariant curves, cycles or chaotic attractors) take place. The basins of attraction of the absorbing areas are determined together with their bifurcations.

Image Encryption Algorithm Based on Substitution Principle and Shuffling Scheme

2020 ◽  
Vol 38 (3B) ◽  
pp. 98-103
Author(s):  
Atyaf S. Hamad ◽  
Alaa K. Farhan
Keyword(s):  
Image Encryption ◽  
Chaotic Map ◽  
Logistic Map ◽  
Encryption Algorithm ◽  
Second Phase ◽  
Cipher Text ◽  
The Face ◽  
Substitution Principle ◽  
2D Logistic Map ◽  
Image Encryption Algorithm

This research presents a method of image encryption that has been designed based on the algorithm of complete shuffling, transformation of substitution box, and predicated image crypto-system. This proposed algorithm presents extra confusion in the first phase because of including an S-box based on using substitution by AES algorithm in encryption and its inverse in Decryption. In the second phase, shifting and rotation were used based on secrete key in each channel depending on the result from the chaotic map, 2D logistic map and the output was processed and used for the encryption algorithm. It is known from earlier studies that simple encryption of images based on the scheme of shuffling is insecure in the face of chosen cipher text attacks. Later, an extended algorithm has been projected. This algorithm performs well against chosen cipher text attacks. In addition, the proposed approach was analyzed for NPCR, UACI (Unified Average Changing Intensity), and Entropy analysis for determining its strength.

In–Out Intermittency with Nested Subspaces in a System of Globally Coupled, Complex Ginzburg–Landau Equations

2021 ◽  
Vol 31 (01) ◽  
pp. 2130001
Author(s):  
Gerhard Dangelmayr ◽  
Iuliana Oprea
Keyword(s):  
Normal Form ◽  
Traveling Waves ◽  
Complex Dynamics ◽  
Invariant Subspaces ◽  
Chaotic Attractors ◽  
Governing Equations ◽  
Ginzburg Landau ◽  
Higher Dimensional ◽  
Ginzburg Landau Equations ◽  
Anisotropic Systems

Chaos and intermittency are studied for the system of globally coupled, complex Ginzburg–Landau equations governing the dynamics of extended, two-dimensional anisotropic systems near an oscillatory (Hopf) instability of a basic state with two pairs of counterpropagating, oblique traveling waves. Parameters are chosen such that the underlying normal form, which governs the dynamics of the spatially constant modes, has two symmetry-conjugated chaotic attractors. Two main states residing in nested invariant subspaces are identified, a state referred to as Spatial Intermittency ([Formula: see text]) and a state referred to as Spatial Persistence ([Formula: see text]). The [Formula: see text]-state consists of laminar phases where the dynamics is close to a normal form attractor, without spatial variation, and switching phases with spatiotemporal bursts during which the system switches from one normal form attractor to the conjugated normal form attractor. The [Formula: see text]-state also consists of two symmetry-conjugated states, with complex spatiotemporal dynamics, that reside in higher dimensional invariant subspaces whose intersection forms the 8D space of the spatially constant modes. We characterize the repeated appearance of these states as (generalized) in–out intermittency. The statistics of the lengths of the laminar phases is studied using an appropriate Poincaré map. Since the Ginzburg–Landau system studied in this paper can be derived from the governing equations for electroconvection in nematic liquid crystals, the occurrence of in–out intermittency may be of interest in understanding spatiotemporally complex dynamics in nematic electroconvection.

scholarly journals A Novel 2D – Grid of Scroll Chaotic Attractor Generated by CNN

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 99 ◽  
Author(s):  
Ahmed M. Ali ◽  
Saif M. Ramadhan ◽  
Fadhil R. Tahir
Keyword(s):  
Theoretical Analysis ◽  
Chaotic Attractor ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Chaotic Attractors ◽  
Electronic Circuits ◽  
Nonlinear Networks ◽  
Design And Implementation ◽  
Simulation Results ◽  
New System

The complex grid of scroll chaotic attractors that are generated through nonlinear electronic circuits have been raised considerably over the last decades. In this paper, it is shown that a subclass of Cellular Nonlinear Networks (CNNs) allows us to generate complex dynamics and chaos in symmetry pattern. A novel grid of scroll chaotic attractor, based on a new system, shows symmetry scrolls about the origin. Also, the equilibrium points are located in a manner such that the symmetry about the line x=y has been achieved. The complex dynamics of system can be generated using CNNs, which in turn are derived from a CNN array (1×3) cells. The paper concerns on the design and implementation of 2×2 and 3×3 2D-grid of scroll via the CNN model. Theoretical analysis and numerical simulations of the derived model are included. The simulation results reveal that the grid of scroll attractors can be successfully reproduced using PSpice.

scholarly journals Temporal dynamics of stomatal conductance of plants under water deficit: can homeostasis be improved by more complex dynamics?

2004 ◽  
Vol 47 (3) ◽  
pp. 423-431 ◽  
Author(s):  
Gustavo Maia Souza ◽  
Ricardo Ferraz de Oliveira ◽  
Victor José Mendes Cardoso
Keyword(s):  
Stomatal Conductance ◽  
Sugar Beet ◽  
Water Deficit ◽  
Time Series Data ◽  
Temporal Dynamics ◽  
Complex Dynamics ◽  
Fractal Dimensions ◽  
Series Data ◽  
Complex Behaviour ◽  
Lz Complexity

In this study we hypothesized that chaotic or complex behavior of stomatal conductance could improve plant homeostasis after water deficit. Stomatal conductance of sunflower and sugar beet leaves was measured in plants grown either daily irrigation or under water deficit using an infrared gas analyzer. All measurements were performed under controlled environmental conditions. In order to measure a consistent time series, data were scored with time intervals of 20s during 6h. Lyapunov exponents, fractal dimensions, KS entropy and relative LZ complexity were calculated. Stomatal conductance in both irrigated and non-irrigated plants was chaotic-like. Plants under water deficit showed a trend to a more complex behaviour, mainly in sunflower that showed better homeostasis than in sugar beet. Some biological implications are discussed.

Relation between Topological Entropies and Fractal Dimensions of Chaotic Attractors of the Hénon Map

2012 ◽  
Vol 569 ◽  
pp. 447-450
Author(s):  
Xiao Zhou Chen ◽  
Liang Lin Xiong ◽  
Long Li
Keyword(s):  
Linear Relation ◽  
Chaotic Dynamics ◽  
Fractal Dimensions ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Hénon Map ◽  
Henon Map ◽  
Efficient Approach ◽  
Integral Relationship

In two-dimensional chaotic dynamics, relationship between fractal dimensions and topological entropies is an important issue to understand the chaotic attractors of Hénon map. we proposed a efficient approach for the estimation of topological entropies through the study on the integral relationship between fractal dimensions and topological entropies. Our result found that there is an approximate linear relation between their topological entropies and fractal dimensions.

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