scholarly journals Design of Positive, Negative, and Alternating Sign Generalized Logistic Maps

2015 ◽  
Vol 2015 ◽  
pp. 1-23 ◽  
Author(s):  
Wafaa S. Sayed ◽  
Ahmed G. Radwan ◽  
Hossam A. H. Fahmy
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Degrees Of Freedom ◽  
Chaotic Maps ◽  
Logistic Map ◽  
Maximum Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Financial Modeling ◽  
Design Flexibility ◽  
Special Case

The discrete logistic map is one of the most famous discrete chaotic maps which has widely spread applications. This paper investigates a set of four generalized logistic maps where the conventional map is a special case. The proposed maps have extra degrees of freedom which provide different chaotic characteristics and increase the design flexibility required for many applications such as quantitative financial modeling. Based on the maximum chaotic range of the output, the proposed maps can be classified as positive logistic map, mostly positive logistic map, negative logistic map, and mostly negative logistic map. Mathematical analysis for each generalized map includes bifurcation diagrams relative to all parameters, effective range of parameters, first bifurcation point, and the maximum Lyapunov exponent (MLE). Independent, vertical, and horizontal scales of the bifurcation diagram are discussed for each generalized map as well as a new bifurcation diagram related to one of the added parameters. A systematic procedure to design two-constraint logistic map is discussed and validated through four different examples.

scholarly journals Modification of the Logistic Map Using Fuzzy Numbers with Application to Pseudorandom Number Generation and Image Encryption

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 474 ◽  
Author(s):  
Lazaros Moysis ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Jesus M. Munoz-Pacheco ◽  
Jacques Kengne ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Image Encryption ◽  
Fuzzy Numbers ◽  
Statistical Tests ◽  
Logistic Map ◽  
Simple Rule ◽  
Pseudorandom Number ◽  
Bifurcation Diagrams ◽  
Triangular Numbers ◽  
Pseudorandom Number Generation

A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. The map is then applied to the problem of pseudo random bit generation, using a simple rule to generate the bit sequence. The resulting random bit generator (RBG) successfully passes the National Institute of Standards and Technology (NIST) statistical tests, and it is then successfully applied to the problem of image encryption.

Analytical Solutions to Multivalued Maps

1997 ◽  
Vol 11 (12) ◽  
pp. 521-530 ◽  
Author(s):  
Jorge A. Gonzalez ◽  
Lindomar B. De Carvalho
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Analytical Solutions ◽  
Chaotic Maps ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Maximum Lyapunov Exponent ◽  
Multivalued Maps ◽  
Return Maps ◽  
Set Of Points

We present explicit solutions for a class of chaotic maps. The return-maps generated by a special class of chaotic functions can be multivalued, or even they can represent an erratic set of points. In some cases the produced time series can have an increasing time-dependent maximum Lyapunov exponent. We discuss some applications of the obtained results. In particular, we present a chaotic lattice model for the investigation of the propagation of carriers in the presence of disorder.

A Novel Four-Order Chaotic System with N-Attractor Multi-Direction Multi-Scroll Attractors

2014 ◽  
Vol 678 ◽  
pp. 81-88
Author(s):  
Wen Shuang Yin ◽  
Dai Jun Wei ◽  
Shi Qiang Chen
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Chaotic System ◽  
Operational Amplifier ◽  
Experimental Results ◽  
Order System ◽  
Chaotic Attractors ◽  
Maximum Lyapunov Exponent ◽  
Nonlinear Functions ◽  
New System

In this paper, a novel four-order system is proposed. It can generate N-attractor multi-direction multi-scroll attractor by adding simple nonlinear functions. We analyze the new system by using means of maximum Lyapunov exponent, bifurcation diagram and Poincaré maps of the system. Moreover, an minimum operational amplifier circuit is designed for realizing 2×(3×3 ×3) scroll chaotic attractors, and experimental results are also obtained, which verify chaos characteristics of the system.

NUMERICAL APPROXIMATION OF THE PERCENTAGE OF ORDER FOR ONE-DIMENSIONAL MAPS

2005 ◽  
Vol 08 (01) ◽  
pp. 15-32 ◽  
Author(s):  
G. LIVADIOTIS
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Numerical Approximation ◽  
Sampling Method ◽  
Logistic Map ◽  
Unimodal Map ◽  
One Dimensional ◽  
One Dimensional Maps

The percentage of organized motion of the chaotic zone (which shall from now on be referred to as percentage of order) for the logistic, the sine-square and the 4-exponent map, is calculated. The calculations are reached via a sampling method that incorporates the Lyapunov exponent. Although these maps are specially selected examples of one-dimensional ones, the conclusions can also be applied to any other one-dimensional map. Since the metric characteristics of a bifurcation diagram of a unimodal map, such as the referred percentage of order, are dependent on the order of the maximum, this dependence is verified for several maps. Once the chaotic zone can be separated into regions between the sequential band mergings, the percentage of order corresponding to each region is calculated for the logistic map. In each region, the resultant area occupied by order, or the supplementary area occupied by chaos, participates in a sequence similar to Feigenbaum's one, which converges to the same respective Feigenbaum's constant.

scholarly journals Neimark–Sacker Bifurcation of a Two-Dimensional Discrete-Time Chemical Model

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
A. Q. Khan
Keyword(s):  
Lyapunov Exponent ◽  
Equilibrium Point ◽  
Discrete Time ◽  
Oscillator Model ◽  
Chemical Model ◽  
Maximum Lyapunov Exponent ◽  
Two Dimensional ◽  
Unique Equilibrium ◽  
Bifurcation Diagrams ◽  
Local Dynamics

In this paper, the local dynamics and Neimark–Sacker bifurcation of a two-dimensional glycolytic oscillator model in the interior of ℝ+2 are explored. It is investigated that for all α and β, the model has a unique equilibrium point: Pxy+α/β+α2,α. Further about Pxy+α/β+α2,α, local dynamics and the existence of bifurcation are explored. It is investigated about Pxy+α/β+α2,α that the glycolytic oscillator model undergoes no bifurcation except the Neimark–Sacker bifurcation. Some simulations are given to verify the obtained results. Finally, bifurcation diagrams and the corresponding maximum Lyapunov exponent are presented for the glycolytic oscillator model.

Hidden and Nonstandard Bifurcation Diagram of an Alternate Quadratic System

2016 ◽  
Vol 26 (02) ◽  
pp. 1650036 ◽  
Author(s):  
G. Pastor ◽  
M. Romera ◽  
M.-F. Danca ◽  
A. Martin ◽  
A. B. Orue ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Graphical Analysis ◽  
Quadratic System ◽  
Bifurcation Diagrams ◽  
Quadratic Systems ◽  
Interval Maps ◽  
System A ◽  
Parameter Values

Alternate quadratic systems [Formula: see text] where [Formula: see text] and [Formula: see text] are different parameters, seem to be interval maps in a range of the parameter values. However, after a careful graphical analysis of their bifurcation diagrams we conclude that this is true only for system B, but not for system A. In system A we find a hidden and nonstandard bifurcation diagram (“hidden” because it is not visible at normal resolution and “nonstandard” because the bifurcation diagram is empty for some ranges of the parameter values). The different behavior of the underlying critical polynomial in the range of parameter values in both alternate quadratic systems explains why the hidden and nonstandard bifurcation diagram is present in system A and not in system B. The analysis of the Lyapunov exponent also shows both the existence and the different behavior of the hidden bifurcation diagram of system A.

CHAOTIC FEATURE OF MARTIN PROCESS IMPOSED ON THE COSINE FUNCTION

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 191-195
Author(s):  
CHUANHOU GAO ◽  
ZHIMIN ZHOU ◽  
JIUSUN ZENG ◽  
JIMING CHEN
Keyword(s):  
Phase Diagram ◽  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Chaotic Attractor ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Cosine Function ◽  
Maximum Lyapunov Exponent ◽  
System Parameters ◽  
Stable Focus

By analyzing the phase diagram of Martin process on the cosine function, it is shown that with the change of system parameters the system will eventually converge to a chaotic attractor. The process is repeated and stable focus, period doubling bifurcation occurs during this process. Further computation gives the maximum Lyapunov exponent of the system and meanwhile, the bifurcation diagram is drawn. Thus it is proved from theory that the system exhibits strong chaotic properties.

scholarly journals Dynamical properties of a novel one dimensional chaotic map

2022 ◽  
Vol 19 (3) ◽  
pp. 2489-2505
Author(s):  
Amit Kumar ◽  
◽  
Jehad Alzabut ◽  
Sudesh Kumari ◽  
Mamta Rani ◽  
...  
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Logistic Map ◽  
Maximal Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
One Dimensional ◽  
Dynamical Properties ◽  
Chaotic Logistic Map

<abstract><p>In this paper, a novel one dimensional chaotic map $ K(x) = \frac{\mu x(1\, -x)}{1+ x} $, $ x\in [0, 1], \mu &gt; 0 $ is proposed. Some dynamical properties including fixed points, attracting points, repelling points, stability and chaotic behavior of this map are analyzed. To prove the main result, various dynamical techniques like cobweb representation, bifurcation diagrams, maximal Lyapunov exponent, and time series analysis are adopted. Further, the entropy and probability distribution of this newly introduced map are computed which are compared with traditional one-dimensional chaotic logistic map. Moreover, with the help of bifurcation diagrams, we prove that the range of stability and chaos of this map is larger than that of existing one dimensional logistic map. Therefore, this map might be used to achieve better results in all the fields where logistic map has been used so far.</p></abstract>

A novel memristor-based chaotic system with line equilibria and its complex dynamics

2021 ◽  
Author(s):  
Dengwei Yan ◽  
Musha Ji’e ◽  
Lidan Wang ◽  
Shukai Duan
Keyword(s):  
Lyapunov Exponent ◽  
Chaotic System ◽  
Dynamic Characteristics ◽  
Complex Dynamics ◽  
Circuit Simulation ◽  
Basins Of Attraction ◽  
Maximum Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Structure Complexity ◽  
Complexity Algorithm

Memristor, as a nonlinear element, provides many advantages thanks to its superior properties to design different chaotic circuits. Thus, a novel four-dimensional double-scroll chaotic system with line equilibria as well as two unstable equilibria based on the flux-memristor model is proposed in this paper. The effects of initial values and parameters on the dynamic characteristics of the system are studied in detail by means of phase diagrams, Lyapunov exponent spectrums, bifurcation diagrams, two-parameter bifurcation diagrams and basins of attraction. Besides, a series of complex phenomena in the system, such as sustained chaos, bistability, transient chaos and coexisting attractors are observed, proving that the chaotic system has rich dynamic characteristics. Also, spectral entropy (SE) complexity algorithm and [Formula: see text] complexity algorithm based on structure complexity are adopted to analyze the complexity of the system. Additionally, PSPICE circuit simulation and Micro-Controller Unit (MCU) hardware experiment are carried out to verify the correctness of theoretical analysis and numerical simulation. Finally, the pulse chaos synchronization is achieved from the perspective of maximum Lyapunov exponent, and numerical simulations demonstrate the occurrence of the proposed system and practicability of the pulse synchronization control.

scholarly journals A Quadratic Fractional Map without Equilibria: Bifurcation, 0–1 Test, Complexity, Entropy, and Control

Electronics ◽  
2020 ◽  
Vol 9 (5) ◽  
pp. 748 ◽  
Author(s):  
Adel Ouannas ◽  
Amina-Aicha Khennaoui ◽  
Shaher Momani ◽  
Giuseppe Grassi ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Test Phase ◽  
Research Topic ◽  
Chaotic Attractors ◽  
Maximum Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Hidden Attractors ◽  
Discrete Time Systems ◽  
And Control ◽  
Time Systems

Fractional calculus in discrete-time systems is a recent research topic. The fractional maps introduced in the literature often display chaotic attractors belonging to the class of “self-excited attractors”. The field of fractional map with “hidden attractors” is completely unexplored. Based on these considerations, this paper presents the first example of fractional map without equilibria showing a number of hidden attractors for different values of the fractional order. The presence of the chaotic hidden attractors is validated via the computation of bifurcation diagrams, maximum Lyapunov exponent, 0–1 test, phase diagrams, complexity, and entropy. Finally, an active controller with the aim for stabilizing the proposed fractional map is successfully designed.

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