A chaotic system with an infinite number of equilibrium points located on a line and on a hyperbola and its fractional-order form

A new 4D fractional-order chaotic system, which has an infinite number of equilibrium points, is introduced. There is no-chaotic behavior for its corresponded integer-order system. We obtain that the largest Lyapunov exponent of this 4D fractional-order chaotic system is 0.8939 and yield the chaotic attractor. A chaotic synchronization scheme is presented for this 4D fractional-order chaotic system. Numerical simulations is verified the effectiveness of the proposed scheme.

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A New Multi-wing Chaotic System and Its Fractional-Order Form

Theory, Methodology, Tools and Applications for Modeling and Simulation of Complex Systems - Communications in Computer and Information Science ◽

10.1007/978-981-10-2669-0_11 ◽

2016 ◽

pp. 98-107

Author(s):

Zengqiang Chen ◽

Leilei Zhou ◽

Jian Ma ◽

Zhonglin Wang ◽

Qing Zhang

Keyword(s):

Fractional Order ◽

Chaotic System ◽

Order Form

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A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Entropy ◽

10.3390/e20080564 ◽

2018 ◽

Vol 20 (8) ◽

pp. 564 ◽

Cited By ~ 36

Author(s):

Jesus Munoz-Pacheco ◽

Ernesto Zambrano-Serrano ◽

Christos Volos ◽

Sajad Jafari ◽

Jacques Kengne ◽

...

Keyword(s):

Fractional Order ◽

Chaotic System ◽

Chaotic Attractor ◽

Equilibrium Points ◽

Fractional Order System ◽

Order System ◽

Chaotic Attractors ◽

Equilibrium System ◽

Hidden Attractors ◽

Hidden Chaotic Attractor

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

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Bursting, Dynamics, and Circuit Implementation of a New Fractional-Order Chaotic System With Coexisting Hidden Attractors

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4043003 ◽

2019 ◽

Vol 14 (7) ◽

Cited By ~ 3

Author(s):

Meng Jiao Wang ◽

Xiao Han Liao ◽

Yong Deng ◽

Zhi Jun Li ◽

Yi Ceng Zeng ◽

...

Keyword(s):

Numerical Simulation ◽

Fractional Order ◽

Chaotic System ◽

Chaotic Systems ◽

Equilibrium Points ◽

Neuroendocrine Cells ◽

Order System ◽

Hidden Attractors ◽

Main Mode ◽

Circuit Realization

Systems with hidden attractors have been the hot research topic of recent years because of their striking features. Fractional-order systems with hidden attractors are newly introduced and barely investigated. In this paper, a new 4D fractional-order chaotic system with hidden attractors is proposed. The abundant and complex hidden dynamical behaviors are studied by nonlinear theory, numerical simulation, and circuit realization. As the main mode of electrical behavior in many neuroendocrine cells, bursting oscillations (BOs) exist in this system. This complicated phenomenon is seldom found in the chaotic systems, especially in the fractional-order chaotic systems without equilibrium points. With the view of practical application, the spectral entropy (SE) algorithm is chosen to estimate the complexity of this fractional-order system for selecting more appropriate parameters. Interestingly, there is a state variable correlated with offset boosting that can adjust the amplitude of the variable conveniently. In addition, the circuit of this fractional-order chaotic system is designed and verified by analog as well as hardware circuit. All the results are very consistent with those of numerical simulation.

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A New Fractional-Order Chaotic System with Its Analysis, Synchronization, and Circuit Realization for Secure Communication Applications

Mathematics ◽

10.3390/math9202593 ◽

2021 ◽

Vol 9 (20) ◽

pp. 2593

Author(s):

Zain-Aldeen S. A. Rahman ◽

Basil H. Jasim ◽

Yasir I. A. Al-Yasir ◽

Yim-Fun Hu ◽

Raed A. Abd-Alhameed ◽

...

Keyword(s):

Fractional Order ◽

Chaotic System ◽

Real World ◽

Communication System ◽

Secure Communication ◽

Communication Systems ◽

Equilibrium Points ◽

Lyapunov Theory ◽

Uncertain Parameter ◽

Real World Applications

This article presents a novel four-dimensional autonomous fractional-order chaotic system (FOCS) with multi-nonlinearity terms. Several dynamics, such as the chaotic attractors, equilibrium points, fractal dimension, Lyapunov exponent, and bifurcation diagrams of this new FOCS, are studied analytically and numerically. Adaptive control laws are derived based on Lyapunov theory to achieve chaos synchronization between two identical new FOCSs with an uncertain parameter. For these two identical FOCSs, one represents the master and the other is the slave. The uncertain parameter in the slave side was estimated corresponding to the equivalent master parameter. Next, this FOCS and its synchronization were realized by a feasible electronic circuit and tested using Multisim software. In addition, a microcontroller (Arduino Due) was used to implement the suggested system and the developed synchronization technique to demonstrate its digital applicability in real-world applications. Furthermore, based on the developed synchronization mechanism, a secure communication scheme was constructed. Finally, the security analysis metric tests were investigated through histograms and spectrograms analysis to confirm the security strength of the employed communication system. Numerical simulations demonstrate the validity and possibility of using this new FOCS in high-level security communication systems. Furthermore, the secure communication system is highly resistant to pirate attacks. A good agreement between simulation and experimental results is obtained, showing that the new FOCS can be used in real-world applications.

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High-Security Image Encryption Based on a Novel Simple Fractional-Order Memristive Chaotic System with a Single Unstable Equilibrium Point

Electronics ◽

10.3390/electronics10243130 ◽

2021 ◽

Vol 10 (24) ◽

pp. 3130

Author(s):

Zain-Aldeen S. A. Rahman ◽

Basil H. Jasim ◽

Yasir I. A. Al-Yasir ◽

Raed A. Abd-Alhameed

Keyword(s):

Equilibrium Point ◽

Image Encryption ◽

Fractional Order ◽

Chaotic System ◽

Chaotic Systems ◽

Equilibrium Points ◽

Unstable Equilibrium ◽

Phase Portraits ◽

Time Efficiency ◽

Unstable Equilibrium Point

Fractional-order chaotic systems have more complex dynamics than integer-order chaotic systems. Thus, investigating fractional chaotic systems for the creation of image cryptosystems has been popular recently. In this article, a fractional-order memristor has been developed, tested, numerically analyzed, electronically realized, and digitally implemented. Consequently, a novel simple three-dimensional (3D) fractional-order memristive chaotic system with a single unstable equilibrium point is proposed based on this memristor. This fractional-order memristor is connected in parallel with a parallel capacitor and inductor for constructing the novel fractional-order memristive chaotic system. The system’s nonlinear dynamic characteristics have been studied both analytically and numerically. To demonstrate the chaos behavior in this new system, various methods such as equilibrium points, phase portraits of chaotic attractor, bifurcation diagrams, and Lyapunov exponent are investigated. Furthermore, the proposed fractional-order memristive chaotic system was implemented using a microcontroller (Arduino Due) to demonstrate its digital applicability in real-world applications. Then, in the application field of these systems, based on the chaotic behavior of the memristive model, an encryption approach is applied for grayscale original image encryption. To increase the encryption algorithm pirate anti-attack robustness, every pixel value is included in the secret key. The state variable’s initial conditions, the parameters, and the fractional-order derivative values of the memristive chaotic system are used for contracting the keyspace of that applied cryptosystem. In order to prove the security strength of the employed encryption approach, the cryptanalysis metric tests are shown in detail through histogram analysis, keyspace analysis, key sensitivity, correlation coefficients, entropy analysis, time efficiency analysis, and comparisons with the same fieldwork. Finally, images with different sizes have been encrypted and decrypted, in order to verify the capability of the employed encryption approach for encrypting different sizes of images. The common cryptanalysis metrics values are obtained as keyspace = 2648, NPCR = 0.99866, UACI = 0.49963, H(s) = 7.9993, and time efficiency = 0.3 s. The obtained numerical simulation results and the security metrics investigations demonstrate the accuracy, high-level security, and time efficiency of the used cryptosystem which exhibits high robustness against different types of pirate attacks.

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A Chaotic System with Infinite Equilibria and Its S-Box Constructing Application

Applied Sciences ◽

10.3390/app8112132 ◽

2018 ◽

Vol 8 (11) ◽

pp. 2132 ◽

Cited By ~ 26

Author(s):

Xiong Wang ◽

Akif Akgul ◽

Unal Cavusoglu ◽

Viet-Thanh Pham ◽

Duy Vo Hoang ◽

...

Keyword(s):

Chaotic System ◽

Infinite Number ◽

Electronic Circuit ◽

Equilibrium Points ◽

Chaotic Attractors ◽

Performance Tests ◽

Generation Algorithm ◽

Performance Results ◽

Number Of Equilibria ◽

Line Equilibrium

Systems with many equilibrium points have attracted considerable interest recently. A chaotic system with a line equilibrium has been studied in this work. The system has infinite equilibria and exhibits coexisting chaotic attractors. The system with an infinite number of equilibria has been realized by an electronic circuit, which confirms the feasibility of the system. Based on such a system, we have developed a new S-Box generation algorithm. With the developed algorithm, two new S-Boxes are produced. Performance tests of S-Boxes are performed. The tests have shown that proposed S-Boxes have good performance results.

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Control and Synchronization of the Fractional-Order Lorenz Chaotic System via Fractional-Order Derivative

Mathematical Problems in Engineering ◽

10.1155/2012/214169 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14 ◽

Cited By ~ 9

Author(s):

Ping Zhou ◽

Rui Ding

Keyword(s):

Fractional Order ◽

Chaotic System ◽

Stability Theory ◽

Chaos Synchronization ◽

Equilibrium Points ◽

Order Derivative ◽

Fractional Order Systems ◽

Fractional Order Derivative ◽

Lorenz Chaotic System ◽

Control And Synchronization

The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.

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A New Chaotic System with Only Nonhyperbolic Equilibrium Points: Dynamics and Its Engineering Application

Complexity ◽

10.1155/2022/4488971 ◽

2022 ◽

Vol 2022 ◽

pp. 1-16

Author(s):

Maryam Zolfaghari-Nejad ◽

Mostafa Charmi ◽

Hossein Hassanpoor

Keyword(s):

Chaotic System ◽

Infinite Number ◽

Equilibrium Points ◽

Engineering Application ◽

Period Doubling ◽

Spectral Entropy ◽

Multiple Attractors ◽

Initial Values ◽

New Chaotic System ◽

New System

In this work, we introduce a new non-Shilnikov chaotic system with an infinite number of nonhyperbolic equilibrium points. The proposed system does not have any linear term, and it is worth noting that the new system has one equilibrium point with triple zero eigenvalues at the origin. Also, the novel system has an infinite number of equilibrium points with double zero eigenvalues that are located on the z -axis. Numerical analysis of the system reveals many strong dynamics. The new system exhibits multistability and antimonotonicity. Multistability implies the coexistence of many periodic, limit cycle, and chaotic attractors under different initial values. Also, bifurcation analysis of the system shows interesting phenomena such as periodic window, period-doubling route to chaos, and inverse period-doubling bifurcations. Moreover, the complexity of the system is analyzed by computing spectral entropy. The spectral entropy distribution under different initial values is very scattered and shows that the new system has numerous multiple attractors. Finally, chaos-based encoding/decoding algorithms for secure data transmission are developed by designing a state chain diagram, which indicates the applicability of the new chaotic system.

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Qualitative Analysis of Class of Fractional-Order Chaotic System via Bifurcation and Lyapunov Exponents Notions

Journal of Mathematics ◽

10.1155/2021/5548569 ◽

2021 ◽

Vol 2021 ◽

pp. 1-18

Author(s):

Ndolane Sene

Keyword(s):

Lyapunov Exponents ◽

Fractional Order ◽

Chaotic System ◽

Numerical Scheme ◽

Initial Conditions ◽

Equilibrium Points ◽

Phase Portraits ◽

Fractional Chaotic System ◽

The Stability ◽

Predictor Corrector

This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.

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