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 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.

scholarly journals Dynamics, Circuit Design, and Synchronization of a New Chaotic System with Closed Curve Equilibrium

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Xiong Wang ◽  
Viet-Thanh Pham ◽  
Christos Volos
Keyword(s):  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Closed Curve ◽  
Phase Portraits ◽  
Chaotic Flows ◽  
Theoretical System ◽  
New Chaotic System ◽  
Electronic Implementation ◽  
Line Equilibrium ◽  
New System

After the report of chaotic flows with line equilibrium, there has been much attention to systems with uncountable equilibria in the past five years. This work proposes a new system with an infinite number of equilibrium points located on a closed curve. It is worth noting that the new system generates chaotic behavior as well as hidden attractors. Dynamics of the system with closed curve equilibrium have been investigated by using phase portraits, bifurcation diagram, maximal Lyapunov exponents, and Kaplan–York dimension. In addition, we introduce an electronic implementation of the theoretical system to verify its feasibility. Antisynchronization ability of the new system with infinite equilibria is studied by applying an adaptive control. This study suggests that there exist other chaotic systems with uncountable equilibria in need of further investigation.

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.

scholarly journals S-Box Based Image Encryption Application Using a Chaotic System without Equilibrium

2019 ◽  
Vol 9 (4) ◽  
pp. 781 ◽  
Author(s):  
Xiong Wang ◽  
Ünal Çavuşoğlu ◽  
Sezgin Kacar ◽  
Akif Akgul ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Image Encryption ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Chaotic Behavior ◽  
Encryption Algorithm ◽  
Phase Portraits ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
Image Encryption Algorithm

Chaotic systems without equilibrium are of interest because they are the systems with hidden attractors. A nonequilibrium system with chaos is introduced in this work. Chaotic behavior of the system is verified by phase portraits, Lyapunov exponents, and entropy. We have implemented a real electronic circuit of the system and reported experimental results. By using this new chaotic system, we have constructed S-boxes which are applied to propose a novel image encryption algorithm. In the designed encryption algorithm, three S-boxes with strong cryptographic properties are used for the sub-byte operation. Particularly, the S-box for the sub-byte process is selected randomly. In addition, performance analyses of S-boxes and security analyses of the encryption processes have been presented.

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.

Simplest Megastable Chaotic Oscillator

2019 ◽  
Vol 29 (13) ◽  
pp. 1950187 ◽  
Author(s):  
Sajad Jafari ◽  
Karthikeyan Rajagopal ◽  
Tasawar Hayat ◽  
Ahmed Alsaedi ◽  
Viet-Thanh Pham
Keyword(s):  
Nonlinear Dynamics ◽  
Bifurcation Diagram ◽  
Chaotic Systems ◽  
Basin Of Attraction ◽  
Chaotic Oscillator ◽  
Hidden Attractors ◽  
Coexisting Attractors ◽  
Extreme Multistability ◽  
Dynamical Properties ◽  
New System

Recently, chaotic systems with hidden attractors and multistability have been of great interest in the field of chaos and nonlinear dynamics. Two special categories of systems with multistability are systems with extreme multistability and systems with megastability. In this paper, the simplest (yet) megastable chaotic oscillator is designed and introduced. Dynamical properties of this new system are completely investigated through tools like bifurcation diagram, Lyapunov exponents, and basin of attraction. It is shown that between its countable infinite coexisting attractors, only one is self-excited and the rest are hidden.

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 A Chaotic System with an Infinite Number of Equilibrium Points: Dynamics, Horseshoe, and Synchronization

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sundarapandian Vaidyanathan ◽  
Xiong Wang
Keyword(s):  
Chaotic System ◽  
Infinite Number ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Equilibrium Analysis ◽  
Chaotic Attractors ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
Topological Horseshoes

Discovering systems with hidden attractors is a challenging topic which has received considerable interest of the scientific community recently. This work introduces a new chaotic system having hidden chaotic attractors with an infinite number of equilibrium points. We have studied dynamical properties of such special system via equilibrium analysis, bifurcation diagram, and maximal Lyapunov exponents. In order to confirm the system’s chaotic behavior, the findings of topological horseshoes for the system are presented. In addition, the possibility of synchronization of two new chaotic systems with infinite equilibria is investigated by using adaptive control.

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 A New 3-D Chaotic System with Conch-Shaped Equilibrium Curve and its Circuit Implementation

2018 ◽  
Vol 7 (3) ◽  
pp. 1410 ◽  
Author(s):  
Mustafa Mamat ◽  
Sundarapandian Vaidyanathan ◽  
Aceng Sambas ◽  
Mohamad Afendee ◽  
Sivaperumal Sampath ◽  
...  
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Equilibrium Curve ◽  
Closed Curves ◽  
New Chaotic System ◽  
Electronic Circuit Simulation

This paper reports the finding a new chaotic system with a conch-shaped equilibrium curve. The proposed system is a new addition to existing chaotic systems with closed curves of equilibrium points in the literature. Lyapunov exponents of the new chaotic system are studiedfor verifying chaos properties and phase portraits of the new system via MATLAB are unveiled. An electronic circuit simulation of the new chaotic system with conch-shaped equilibrium curve is shown using MultiSIM to check the model feasibility.

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