scholarly journals Fractional Form of a Chaotic Map without Fixed Points: Chaos, Entropy and Control

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 720 ◽  
Author(s):  
Adel Ouannas ◽  
Xiong Wang ◽  
Amina-Aicha Khennaoui ◽  
Samir Bendoukha ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Fixed Point ◽  
Fractional Order ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Approximate Entropy ◽  
Linearization Method ◽  
Entropy Measure ◽  
One Dimensional ◽  
Use Phase ◽  
And Control

In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed point. In our investigation, we use phase plots and bifurcation diagrams to examine the dynamics of the fractional map and assess the effect of varying the fractional order. We also use the approximate entropy measure to quantify the level of chaos in the fractional map. In addition, we propose a one-dimensional stabilization controller and establish its asymptotic convergence by means of the linearization method.

scholarly journals Chaotic Map with No Fixed Points: Entropy, Implementation and Control

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 279 ◽  
Author(s):  
Van Huynh ◽  
Adel Ouannas ◽  
Xiong Wang ◽  
Viet-Thanh Pham ◽  
Xuan Nguyen ◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Experimental Observation ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Return Map ◽  
Control Schemes ◽  
And Control

A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents’ diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map.

A Chaotification model based on sine and cosecant functions for enhancing chaos

2021 ◽  
Author(s):  
Yuqing Li ◽  
Xing He ◽  
Dawen Xia
Keyword(s):  
Fixed Point ◽  
Lyapunov Exponent ◽  
Chaotic Maps ◽  
Dynamic Properties ◽  
Chaotic Map ◽  
Sample Entropy ◽  
One Dimensional ◽  
Model Based ◽  
Chaotic Region ◽  
Definition Of

Chaotic maps with higher chaotic complexity are urgently needed in many application scenarios. This paper proposes a chaotification model based on sine and cosecant functions (CMSC) to improve the dynamic properties of existing chaotic maps. CMSC can generate a new map with higher chaotic complexity by using the existing one-dimensional (1D) chaotic map as a seed map. To discuss the performance of CMSC, the chaos properties of CMSC are analyzed based on the mathematical definition of the Lyapunov exponent (LE). Then, three new maps are generated by applying three classical 1D chaotic maps to CMSC respectively, and the dynamic behaviors of the new maps are analyzed in terms of fixed point, bifurcation diagram, sample entropy (SE), etc. The results of the analysis demonstrate that the new maps have a larger chaotic region and excellent chaotic characteristics.

Optimal Control of a Chaotic Map: Fixed Point Stabilization and Attractor Confinement

1997 ◽  
Vol 07 (02) ◽  
pp. 437-446 ◽  
Author(s):  
C. Piccardi ◽  
L. L. Ghezzi
Keyword(s):  
Optimal Control ◽  
Fixed Point ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Chaotic Attractor ◽  
Chaotic Map ◽  
Optimal Controller ◽  
Control Of Chaos ◽  
One Dimensional ◽  
Suppression Of Chaos

Optimal control is applied to a chaotic system. Reference is made to a well-known one-dimensional map. Firstly, attention is devoted to the stabilization of a fixed point. An optimal controller is obtained and compared with other controllers which are popular in the control of chaos. Secondly, allowance is made for uncertainty and emphasis is placed on the reduction rather than the suppression of chaos. The aim becomes that of confining a chaotic attractor within a prescribed region of the state space. A controller fulfilling this task is obtained as the solution of a min-max optimal control problem.

scholarly journals Chaos Control of a Fractional-Order Financial System

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
Mohammed Salah Abd-Elouahab ◽  
Nasr-Eddine Hamri ◽  
Junwei Wang
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Financial System ◽  
Linearization Method ◽  
Nonlinear Feedback Control ◽  
Nonlinear Feedback ◽  
Chaotic Motions ◽  
Control Scheme ◽  
Fractional System

Fractional-order financial system introduced by W.-C. Chen (2008) displays chaotic motions at order less than 3. In this paper we have extended the nonlinear feedback control in ODE systems to fractional-order systems, in order to eliminate the chaotic behavior. The results are proved analytically by applying the Lyapunov linearization method and stability condition for fractional system. Moreover numerical simulations are shown to verify the effectiveness of the proposed control scheme.

On the dynamics and control of a new fractional difference chaotic map

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Samir Bendoukha
Keyword(s):  
Chaotic Map ◽  
Discrete Systems ◽  
Fractional Difference ◽  
Difference Map ◽  
Dynamics And Control ◽  
Generalized Hénon Map ◽  
Use Phase ◽  
The Rich ◽  
The Stability ◽  
And Control

Abstract In this paper, we propose and study a fractional Caputo-difference map based on the 2D generalized Hénon map. By means of numerical methods, we use phase plots and bifurcation diagrams to investigate the rich dynamics of the proposed map. A 1D synchronization controller is proposed similar to that of Pecora and Carrol, whereby we assume knowledge of one of the two states at the slave and replicate the second state. The stability theory of fractional discrete systems is used to guarantee the asymptotic convergence of the proposed controller and numerical simulations are employed to confirm the findings.

scholarly journals A New Image Encryption Algorithm Based on Composite Chaos and Hyperchaos Combined with DNA Coding

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 171 ◽  
Author(s):  
Yujie Wan ◽  
Shuangquan Gu ◽  
Baoxiang Du
Keyword(s):  
Image Encryption ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Security Analysis ◽  
High Sensitivity ◽  
Experimental Simulation ◽  
One Dimensional ◽  
Dna Coding ◽  
Key Space ◽  
Encryption Method

In order to obtain chaos with a wider chaotic scope and better chaotic behavior, this paper combines the several existing one-dimensional chaos and forms a new one-dimensional chaotic map by using a modular operation which is named by LLS system and abbreviated as LLSS. To get a better encryption effect, a new image encryption method based on double chaos and DNA coding technology is proposed in this paper. A new one-dimensional chaotic map is combined with a hyperchaotic Qi system to encrypt by using DNA coding. The first stage involves three rounds of scrambling; a diffusion algorithm is applied to the plaintext image, and then the intermediate ciphertext image is partitioned. The final encrypted image is formed by using DNA operation. Experimental simulation and security analysis show that this algorithm increases the key space, has high sensitivity, and can resist several common attacks. At the same time, the algorithm in this paper can reduce the correlation between adjacent pixels, making it close to 0, and increase the information entropy, making it close to the ideal value and achieving a good encryption effect.

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>

scholarly journals Chaotic behavior analysis and control of a toxin producing phytoplankton and zooplankton system based on linear feedback

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 3779-3789 ◽  
Author(s):  
Yadong Liu ◽  
Wenjun Liu
Keyword(s):  
Feedback Control ◽  
Fractional Order ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Linear Feedback ◽  
Linear Feedback Control ◽  
The Stability ◽  
And Control ◽  
Fractional Orders

In this paper, we study the dynamic behavior and control of the fractional-order nutrientphytoplankton-zooplankton system. First, we analyze the stability of the fractional-order nutrient-plankton system and get the critical stable value of fractional orders. Then, by applying the linear feedback control and Routh-Hurwitz criterion, we yield the sufficient conditions to stabilize the system to its equilibrium points. Finally, Under a modified fractional-order Adams-Bashforth-Monlton algorithm, we simulate the results respectively.

scholarly journals Oscillatory and complex behaviour of Caputo-Fabrizio fractional order HIV-1 infection model

2021 ◽  
Vol 7 (3) ◽  
pp. 4778-4792
Author(s):  
Shabir Ahmad ◽  
◽  
Aman Ullah ◽  
Mohammad Partohaghighi ◽  
Sayed Saifullah ◽  
...  
Keyword(s):  
Fixed Point ◽  
Fractional Order ◽  
Chaotic Behavior ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Picard Iteration ◽  
Infection Model ◽  
Theory Approach ◽  
Complex Behaviour ◽  
Hiv 1

<abstract><p>HIV-1 infection is a dangerous diseases like Cancer, AIDS, etc. Many mathematical models have been introduced in the literature, which are investigated with different approaches. In this article, we generalize the HIV-1 model through nonsingular fractional operator. The non-integer mathematical model of HIV-1 infection under the Caputo-Fabrizio derivative is presented in this paper. The concept of Picard-Lindelof and fixed-point theory are used to address the existence of a unique solution to the HIV-1 model under the suggested operator. Also, the stability of the suggested model is proved through the Picard iteration and fixed point theory approach. The model's approximate solution is constructed through three steps Adams-Bashforth numerical method. Numerical simulations are provided for different values of fractional-order to study the complex dynamics of the model. Lastly, we provide the oscillatory and chaotic behavior of the proposed model for various fractional orders.</p></abstract>

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