scholarly journals A Novel Chaotic System without Equilibrium: Dynamics, Synchronization, and Circuit Realization

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Ahmad Taher Azar ◽  
Christos Volos ◽  
Nikolaos A. Gerodimos ◽  
George S. Tombras ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Unstable Equilibrium ◽  
Physical Realization ◽  
Circuit Realization ◽  
Equilibrium Dynamics ◽  
Dynamical Properties ◽  
Shilnikov Criterion ◽  
New System

A few special chaotic systems without unstable equilibrium points have been investigated recently. It is worth noting that these special systems are different from normal chaotic ones because the classical Shilnikov criterion cannot be used to prove chaos of such systems. A novel unusual chaotic system without equilibrium is proposed in this work. We discover dynamical properties as well as the synchronization of the new system. Furthermore, a physical realization of the system without equilibrium is also implemented to illustrate its feasibility.

scholarly journals A New Four-Scroll Chaotic System with a Self-Excited Attractor and Circuit Implementation

2018 ◽  
Vol 7 (3) ◽  
pp. 1931 ◽  
Author(s):  
Sivaperumal Sampath ◽  
Sundarapandian Vaidyanathan ◽  
Aceng Sambas ◽  
Mohamad Afendee ◽  
Mustafa Mamat ◽  
...  
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Chaotic Model ◽  
New Chaotic System ◽  
Electronic Circuit Simulation ◽  
New System

This paper reports the finding a new four-scroll chaotic system with four nonlinearities. The proposed system is a new addition to existing multi-scroll chaotic systems in the literature. Lyapunov exponents of the new chaotic system are studied for verifying chaos properties and phase portraits of the new system via MATLAB are unveiled. As the new four-scroll chaotic system is shown to have three unstable equilibrium points, it has a self-excited chaotic attractor. An electronic circuit simulation of the new four-scroll chaotic system is shown using MultiSIM to check the feasibility of the four-scroll chaotic model.

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

2019 ◽  
Vol 14 (7) ◽  
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.

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

Electronics ◽  
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.

scholarly journals A New Chaotic System with a Pear-shaped Equilibrium and its Circuit Simulation

2018 ◽  
Vol 8 (6) ◽  
pp. 4951 ◽  
Author(s):  
Aceng Sambas ◽  
Sundarapandian Vaidyanathan ◽  
Mustafa Mamat ◽  
Muhammad Afendee Mohamed ◽  
Mada Sanjaya WS
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Equilibrium Curve ◽  
New Chaotic System ◽  
Electronic Circuit Simulation ◽  
New System

This paper reports the finding a new chaotic system with a pear-shaped equilibrium curve and makes a valuable addition to existing chaotic systems with infinite equilibrium points in the literature. The new chaotic system has a total of five nonlinearities. Lyapunov exponents of the new chaotic system are studied for verifying chaos properties and phase portraits of the new system are unveiled. An electronic circuit simulation of the new chaotic system with pear-shaped equilibrium curve is shown using Multisim to check the model feasibility.

scholarly journals Dynamical Analysis of a Novel 4-D Hyper-Chaotic System With One Non-Hyperbolic Equilibrium Point and Application in Secure Communication

2020 ◽  
Vol 9 (4) ◽  
pp. 74-99
Author(s):  
Pushali Trikha ◽  
Lone Seth Jahanzaib
Keyword(s):  
Chaotic System ◽  
Secure Communication ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Dynamical Analysis ◽  
The Novel ◽  
Bifurcation Diagrams ◽  
Dynamical Properties ◽  
Hyperbolic Equilibrium ◽  
Hyperbolic Equilibrium Point

In this article, a novel hyper-chaotic system has been introduced and its dynamical properties (i.e., phase plots, time series, lyapunov exponents, bifurcation diagrams, equilibrium points, Poincare sections, etc.) have been studied. Also, the novel chaotic systems have been synchronized using novel synchronization technique multi-switching compound difference synchronization and its application have been shown in the field of secure communication. Numerical simulations have been undertaken to validate the efficacy of the synchronization in secure communication.

Constructing a Chaotic System with an Infinite Number of Equilibrium Points

2016 ◽  
Vol 26 (13) ◽  
pp. 1650225 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Sajad Jafari ◽  
Tomasz Kapitaniak
Keyword(s):  
Chaotic System ◽  
Infinite Number ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Hidden Attractors ◽  
Memristive Device ◽  
New System

The chaotic systems with hidden attractors, such as chaotic systems with a stable equilibrium, chaotic systems with infinite equilibria or chaotic systems with no equilibrium have been investigated recently. However, the relationships between them still need to be discovered. This work explains how to transform a system with one stable equilibrium into a new system with an infinite number of equilibrium points by using a memristive device. Furthermore, some other new systems with infinite equilibria are also constructed to illustrate the introduced methodology.

scholarly journals New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 951 ◽  
Author(s):  
Xinhe Zhu ◽  
Wei-Shih Du
Keyword(s):  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Current Knowledge ◽  
Equilibrium Points ◽  
Closed Curve ◽  
Phase Portraits ◽  
Hidden Attractors ◽  
Closed Curves ◽  
Dynamical Properties ◽  
New System

In this work, we introduce a chaotic system with infinitely many equilibrium points laying on two closed curves passing the same point. The proposed system belongs to a class of systems with hidden attractors. The dynamical properties of the new system were investigated by means of phase portraits, equilibrium points, Poincaré section, bifurcation diagram, Kaplan–Yorke dimension, and Maximal Lyapunov exponents. The anti-synchronization of systems was obtained using the active control. This study broadens the current knowledge of systems with infinite equilibria.

scholarly journals Composite One- to Six-Scroll Hidden Attractors in a Memristor-Based Chaotic System and Their Circuit Implementation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Ying Li ◽  
Xiaozhu Xia ◽  
Yicheng Zeng ◽  
Qinghui Hong
Keyword(s):  
Fixed Point ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Circuit Implementation ◽  
Hidden Attractors ◽  
Coexisting Attractors ◽  
Extreme Multistability ◽  
Different Types ◽  
Hardware Circuit ◽  
New System

Chaotic systems with hidden multiscroll attractors have received much attention in recent years. However, most parts of hidden multiscroll attractors previously reported were repeated by the same type of attractor, and the composite of different types of attractors appeared rarely. In this paper, a memristor-based chaotic system, which can generate composite attractors with one up to six scrolls, is proposed. These composite attractors have different forms, similar to the Chua’s double scroll and jerk double scroll. Through theoretical analysis, we find that the new system has no fixed point; that is to say, all of the composite multiscroll attractors are hidden attractors. Additionally, some complicated dynamic behaviors including various hidden coexisting attractors, extreme multistability, and transient transition are explored. Moreover, hardware circuit using discrete components is implemented, and its experimental results supported the numerical simulations results.

scholarly journals A New Hamiltonian Chaotic System with Coexisting Chaotic Orbits and its Dynamical Analysis

2018 ◽  
Vol 7 (4) ◽  
pp. 2430
Author(s):  
Sundarapandian Vaidyanathan ◽  
Aceng Sambas ◽  
Sen Zhang ◽  
Mohamad Afendee Mohamed ◽  
Mustafa Mamat
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Research Work ◽  
Classical Mechanics ◽  
Dynamical Analysis ◽  
Galactic Centre ◽  
Chaotic Orbits ◽  
Nonlinear Motion ◽  
Dynamical Properties

Hamiltonian chaotic systems are conservative chaotic systems which arise in many applications in Classical Mechanics. A famous Hamiltonian chaotic system is the H´enon-Heiles system (1964), which was modeled by H´enon and Heiles, describing the nonlinear motion of a star around a galactic centre with the motion restricted to a plane. In this research work, by modifying the dynamics of the H´enon-Heiles system (1964), we obtain a new Hamiltonian chaotic system with coexisting chaotic orbits. We describe the dynamical properties of the new Hamiltonian chaotic system.

scholarly journals A New Family of Chaotic Systems with Different Closed Curve Equilibrium

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 94 ◽  
Author(s):  
Xinhe Zhu ◽  
Wei-Shih Du
Keyword(s):  
Infinite Number ◽  
Chaotic Systems ◽  
Equilibrium Points ◽  
Closed Curve ◽  
Phase Portraits ◽  
General Knowledge ◽  
Hidden Attractors ◽  
The Past ◽  
New Family ◽  
Dynamical Properties

Chaotic systems with hidden attractors, infinite number of equilibrium points and different closed curve equilibrium have received much attention in the past six years. In this work, we introduce a new family of chaotic systems with different closed curve equilibrium. Using the methods of equilibrium points, phase portraits, maximal Lyapunov exponents, Kaplan–Yorke dimension, and eigenvalues, we analyze the dynamical properties of the proposed systems and extend the general knowledge of such systems.

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