A new three-dimensional chaotic system without equilibrium points, its dynamical analyses and electronic circuit application

Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.

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Generation of Multi-Scroll Chaotic Attractors from a Jerk Circuit with a Special Form of a Sine Function

Electronics ◽

10.3390/electronics9050842 ◽

2020 ◽

Vol 9 (5) ◽

pp. 842

Author(s):

Pengfei Ding ◽

Xiaoyi Feng

Keyword(s):

Chaotic System ◽

Special Form ◽

Electronic Circuit ◽

Equilibrium Points ◽

Phase Portraits ◽

Chaotic Attractors ◽

Sine Function ◽

Sign Function ◽

System Parameters ◽

Left And Right

A novel chaotic system for generating multi-scroll attractors based on a Jerk circuit using a special form of a sine function (SFSF) is proposed in this paper, and the SFSF is the product of a sine function and a sign function. Although there are infinite equilibrium points in this system, the scroll number of the generated chaotic attractors is certain under appropriate system parameters. The dynamical properties of the proposed chaotic system are studied through Lyapunov exponents, phase portraits, and bifurcation diagrams. It is found that the scroll number of the chaotic system in the left and right part of the x-y plane can be determined arbitrarily by adjusting the values of the parameters in the SFSF, and the size of attractors is dominated by the frequency of the SFSF. Finally, an electronic circuit of the proposed chaotic system is implemented on Pspice, and the simulation results of electronic circuit are in agreement with the numerical ones.

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

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i3.14865 ◽

2018 ◽

Vol 7 (3) ◽

pp. 1931 ◽

Cited By ~ 1

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.

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A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications

Optik ◽

10.1016/j.ijleo.2016.05.010 ◽

2016 ◽

Vol 127 (18) ◽

pp. 7062-7071 ◽

Cited By ~ 62

Author(s):

Akif Akgul ◽

Shafqat Hussain ◽

Ihsan Pehlivan

Keyword(s):

Chaotic System ◽

Electronic Circuit ◽

Three Dimensional ◽

Dynamical Analysis

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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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A New Chaotic Attractor and its Digital Implementation Based on LabVIEW

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.278-280.54 ◽

2013 ◽

Vol 278-280 ◽

pp. 54-57

Author(s):

Hong Yang

Keyword(s):

Chaotic System ◽

Equilibrium Points ◽

Three Dimensional ◽

Theoretical Simulation ◽

Digital Implementation ◽

New Chaotic System ◽

Quadratic Nonlinearities ◽

External Conditions ◽

New Chaotic Attractor ◽

Implementation Method

In this paper, a novel three-dimensional autonomous chaotic system with six terms and two quadratic nonlinearities is presented. Some basic dynamical properties of the new chaotic system are analyzed by means of equilibrium points, eigenvalue structures, Lyapunov exponent and Lyapunov dimension. In order to overcome the external conditions affected by the analog circuit’s chaotic system, digital implementation of the new chaotic system based on LabVIEW is also proposed. The results show that the experimental results by LabVIEW are consistent with the theoretical simulation results by Matlab, and the method is an effective digital implementation method.

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A Novel Cubic–Equilibrium Chaotic System with Coexisting Hidden Attractors: Analysis, and Circuit Implementation

Journal of Circuits System and Computers ◽

10.1142/s0218126618500664 ◽

2017 ◽

Vol 27 (04) ◽

pp. 1850066 ◽

Cited By ~ 36

Author(s):

Viet-Thanh Pham ◽

Christos Volos ◽

Sajad Jafari ◽

Tomasz Kapitaniak

Keyword(s):

Chaotic System ◽

Autonomous System ◽

Chaotic Systems ◽

Electronic Circuit ◽

Complex Dynamics ◽

Three Dimensional ◽

Chaotic Attractors ◽

Circuit Implementation ◽

Hidden Attractors ◽

Dynamical Properties

Chaotic systems with a curve of equilibria have attracted considerable interest in theoretical researches and engineering applications because they are categorized as systems with hidden attractors. In this paper, we introduce a new three-dimensional autonomous system with cubic equilibrium. Fundamental dynamical properties and complex dynamics of the system have been investigated. Of particular interest is the coexistence of chaotic attractors in the proposed system. Furthermore, we have designed and implemented an electronic circuit to verify the feasibility of such a system with cubic equilibrium.

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A New Chaotic System Family Dynamics Analysis and Circuit Implementation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.392.232 ◽

2013 ◽

Vol 392 ◽

pp. 232-236

Author(s):

Shu Min Duan ◽

Guo Zeng Wu

Keyword(s):

Chaotic System ◽

Electronic Circuit ◽

Physical Phenomenon ◽

Three Dimensional ◽

Dynamics Analysis ◽

Circuit Implementation ◽

Parallel Lines ◽

New Chaotic System ◽

Dynamical Properties ◽

Electronic Circuit Implementation

A new three-dimensional chaotic system is presented in this paper. Some basic dynamical Properties of this chaotic system are investigated by means of Poincaré mapping, Lyapunov exponents and bifurcation diagram. The dynamical behaviours of this system are proved not only by performing numerical simulation and brief theoretical analysis but also by conducting an electronic circuit implementation. It is new physical phenomenon that the Poincaré mapping of this system is a group of parallel lines.

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A novel data hiding method by using a chaotic system without equilibrium points

Modern Physics Letters B ◽

10.1142/s0217984919503573 ◽

2019 ◽

Vol 33 (29) ◽

pp. 1950357 ◽

Cited By ~ 3

Author(s):

Akif Akgul ◽

Irene M. Moroz ◽

Ali Durdu

Keyword(s):

Equilibrium Point ◽

Chaotic System ◽

Data Hiding ◽

Measurement Method ◽

Equilibrium Points ◽

Random Number Generator ◽

Three Dimensional ◽

Image Distortion ◽

Heteroclinic Orbits ◽

Secret Data

In this paper, we investigate how special is the choice of parameter values in the three-dimensional nonlinear system, proposed by Akgul and Pehlivan (2016), in producing a system, which exhibits chaos but has no real equilibrium states. Also, a data hiding method with a three-dimensional chaotic system without equilibrium point, developed by Akgul and Pehlivan, is realized. Numerous encryption studies have recently been made based on chaos. Encryption processes that are used with chaos bring about some security deficiencies in some cases. Steganography, unlike encryption studies, helps communicate the secret data by hiding it in an innocent-looking cover in order to avoid detection by third parties at first glance. In this work, a novel chaos-based data hiding method that hides an image with a different color into color images is proposed. Via the proposed method, data are hidden in bit spaces with the help of the chaotic random number generator (RNG). The generated random numbers are found with a chaotic system without equilibrium point, which is new in the literature. Shilnikov method cannot be applied to find whether the system is chaotic or not because they cannot have homoclinic or heteroclinic orbits. Thus, it can be useful in several engineering applications, especially in chaos-based cryptology and coding information. In the study, bits are hidden in pixels indicated by numbers generated by RNG. As the order of the hiding process is made randomly on a chaotic level, it has made data hiding algorithm stronger. The proposed method hides the data in cover image in such a way that it cannot be easily detected. Furthermore, the proposed method has been evaluated with steganalysis methods and image distortion measurement method PSNR. The chaos-based steganography method realized here has produced more best results in image distortion measurement method PSNR than other studies in the literature.

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