Hidden Coexisting Attractors in a Chaotic System Without Equilibrium Point

2018 ◽  
Vol 28 (10) ◽  
pp. 1830033 ◽  
Author(s):  
Wei Zhou ◽  
Guangyi Wang ◽  
Yiran Shen ◽  
Fang Yuan ◽  
Simin Yu
Keyword(s):  
Equilibrium Point ◽  
Chaotic System ◽  
Dynamic Characteristics ◽  
Three Dimensional ◽  
Digital Signal ◽  
Chaotic Signal ◽  
Coexisting Attractors ◽  
Pseudorandom Sequences ◽  
The Lorenz System ◽  
Chaotic Signal Generator

This paper proposes a new three-dimensional chaotic system with no equilibrium point but can generate hidden chaotic attractors. Dynamic characteristics of the system are analyzed in detail by theoretical analysis and simulating experiments, including hidden attractors, transient period and coexisting attractors. Different hidden coexisting attractors exist in this system, which shows abundant and complex dynamic characteristics and can be used to generate pseudorandom sequences for encryption fields. Besides, the presented system is realized by the digital signal processing (DSP) technology to construct a chaotic signal generator, whose statistical properties are tested by National Institute of Standards and Technology (NIST) software. The obtained results are better than that of the Lorenz system and imply the presented system can be used in the encrypted fields.

scholarly journals Coexisting Attractors and Multistability in a Simple Memristive Wien-Bridge Chaotic Circuit

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 678 ◽  
Author(s):  
Yixuan Song ◽  
Fang Yuan ◽  
Yuxia Li
Keyword(s):  
Chaotic System ◽  
Analog Circuit ◽  
Random Sequence ◽  
Mathematical Expression ◽  
Digital Signal ◽  
Approximate Entropy ◽  
Chaotic Circuit ◽  
Coexisting Attractors ◽  
Key Space ◽  
Sequence Generator

In this paper, a new voltage-controlled memristor is presented. The mathematical expression of this memristor has an absolute value term, so it is called an absolute voltage-controlled memristor. The proposed memristor is locally active, which is proved by its DC V–I (Voltage–Current) plot. A simple three-order Wien-bridge chaotic circuit without inductor is constructed on the basis of the presented memristor. The dynamical behaviors of the simple chaotic system are analyzed in this paper. The main properties of this system are coexisting attractors and multistability. Furthermore, an analog circuit of this chaotic system is realized by the Multisim software. The multistability of the proposed system can enlarge the key space in encryption, which makes the encryption effect better. Therefore, the proposed chaotic system can be used as a pseudo-random sequence generator to provide key sequences for digital encryption systems. Thus, the chaotic system is discretized and implemented by Digital Signal Processing (DSP) technology. The National Institute of Standards and Technology (NIST) test and Approximate Entropy analysis of the proposed chaotic system are conducted in this paper.

Jacobi analysis for an unusual 3D autonomous system

2020 ◽  
Vol 17 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Chunsheng Feng ◽  
Qiujian Huang ◽  
Yongjian Liu
Keyword(s):  
Chaotic System ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Chen System ◽  
Parameter Region ◽  
Stable Equilibria ◽  
The Lorenz System ◽  
Deviation Vector

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.

scholarly journals CONTROLLED PARAMETER MODULATIONS IN SECURE DIGITAL SIGNAL TRANSMISSIONS

2007 ◽  
Vol 17 (11) ◽  
pp. 4187-4194 ◽  
Author(s):  
P. PALANIYANDI ◽  
M. LAKSHMANAN
Keyword(s):  
Signal Transmission ◽  
Digital Signal ◽  
Chaotic Signal ◽  
Single Step ◽  
Simple Method ◽  
Parameter Modulation ◽  
Maximum Security ◽  
Modulation Technique ◽  
Modified Method ◽  
The Lorenz System

We propose a simple method for secure digital signal transmission by making some modifications in the single-step parameter modulation technique propo sed earlier to overcome certain inherent deficiencies. In the modified method, the parameter modulation is effectively regulated or controlled by the chaotic signal obtained from the transmitting chaotic system so that it has the maximum security. Then, the same idea is also extended to the multistep parameter modulation technique. It is found that both the methods are secure against ciphertext (return map) and plaintex attacks. We have illustrated these methods by means of the Lorenz system.

A Switchable Chaotic Oscillator with Two Amplitude–Frequency Controllers

2017 ◽  
Vol 26 (10) ◽  
pp. 1750158 ◽  
Author(s):  
Wen Hu ◽  
Akif Akgul ◽  
Chunbiao Li ◽  
Taicheng Zheng ◽  
Peng Li
Keyword(s):  
Chaotic System ◽  
Value Function ◽  
Chaotic Signal ◽  
Signal Generator ◽  
Chaotic Oscillator ◽  
Absolute Value ◽  
Independent Amplitude ◽  
Chaotic Signals ◽  
Chaotic Signal Generator ◽  
Frequency Controller

A simple chaotic system with a single nonquadratic term is developed to be a switchable chaotic signal generator in this paper. Additional nonlinearity of the absolute value function is introduced for reforming the structure without damaging the basic dynamics but yielding a new independent amplitude–frequency controller. A switchable chaotic experimental oscillator is designed afterwards, where two coefficients corresponding to two independent rheostats rescale the amplitude and frequency of the chaotic signals smoothly. To our knowledge, this has never been found in other chaotic oscillators.

A Novel 3D Fractional-Order Chaotic System with Multifarious Coexisting Attractors

2019 ◽  
Vol 29 (01) ◽  
pp. 1950004 ◽  
Author(s):  
Chengyi Zhou ◽  
Zhijun Li ◽  
Yicheng Zeng ◽  
Sen Zhang
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Three Dimensional ◽  
Order System ◽  
Attractive Feature ◽  
Huge Amount ◽  
Coexisting Attractors ◽  
Interesting Phenomenon ◽  
Periodic Attractors

A novel three-dimensional fractional-order autonomous chaotic system marked by the ample and complex coexisting attractors is presented. There are a total of seven terms including four nonlinearities in the new system. The evolution of coexisting attractors of the system are numerically investigated by considering both the fractional-order and other system parameters as bifurcation parameters. Numerical simulation results indicate that the system has a huge amount of multifarious coexisting strange attractors for various ranges of parameters, including coexisting point, periodic attractors, multifarious coexisting chaotic, and periodic attractors. Compared with other chaotic systems, the biggest difference and most attractive feature is the capability of the proposed fractional-order system to produce coexisting attractors that undergo a simultaneous displacement phenomenon with variation of a single parameter. Moreover, it is worth noting that constant Lyapunov exponents and the interesting phenomenon of transient coexisting attractors are also observed. Finally, the corresponding implementation circuit is designed. The consistency of the hardware experimental results with numerical simulations verifies the feasibility of the new fractional-order chaotic system.

scholarly journals Investigation of Early Warning Indexes in a Three-Dimensional Chaotic System with Zero Eigenvalues

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 341 ◽  
Author(s):  
Lianyu Chen ◽  
Fahimeh Nazarimehr ◽  
Sajad Jafari ◽  
Esteban Tlelo-Cuautle ◽  
Iqtadar Hussain
Keyword(s):  
Numerical Analysis ◽  
Equilibrium Point ◽  
Chaotic System ◽  
Early Warning ◽  
Three Dimensional ◽  
Period Doubling ◽  
Warning Signals ◽  
Bifurcation Points ◽  
Route To Chaos ◽  
Dynamical Properties

A rare three-dimensional chaotic system with all eigenvalues equal to zero is proposed, and its dynamical properties are investigated. The chaotic system has one equilibrium point at the origin. Numerical analysis shows that the equilibrium point is unstable. Bifurcation analysis of the system shows various dynamics in a period-doubling route to chaos. We highlight that from the evaluation of the entropy, bifurcation points can be predicted by identifying early warning signals. In this manner, bifurcation points of the system are analyzed using Shannon and Kolmogorov-Sinai entropy. The results are compared with Lyapunov exponents.

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

2019 ◽  
Vol 33 (29) ◽  
pp. 1950357 ◽  
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.

IMPULSIVE CONTROL OF CHAOTIC SYSTEM

2002 ◽  
Vol 12 (05) ◽  
pp. 1181-1190 ◽  
Author(s):  
XINZHI LIU ◽  
KOK LAY TEO
Keyword(s):  
Feedback Control ◽  
Control Problem ◽  
Equilibrium Point ◽  
Chaotic System ◽  
Lyapunov Functions ◽  
Lorenz System ◽  
Impulsive Control ◽  
The Lorenz System ◽  
Impulsive Stabilization

This paper studies an impulsive control problem. By utilizing the method of Lyapunov functions, a set of impulsive stabilization criteria are established. These results are then applied to the Lorenz system. It is shown that by using impulsive feedback control, all the solutions of the Lorenz system will converge to an equilibrium point.

ROBUSTNESS AND SIGNAL RECOVERY IN A SYNCHRONIZED CHAOTIC SYSTEM

1993 ◽  
Vol 03 (06) ◽  
pp. 1629-1638 ◽  
Author(s):  
KEVIN M. CUOMO ◽  
ALAN V. OPPENHEIM ◽  
STEVEN H. STROGATZ
Keyword(s):  
Analytical Model ◽  
Chaotic System ◽  
Lorenz System ◽  
Chaotic Signal ◽  
Synchronization Error ◽  
Signal Recovery ◽  
Additive Perturbation ◽  
The Lorenz System

Recent papers have demonstrated that synchronization in the Lorenz system is highly robust to additive perturbation of the drive signal. This property has led to a concept known as chaotic signal masking and recovery. This paper presents experiments and an approximate analytical model that quantify and explain the observed robustness of synchronization in the Lorenz system. In particular, we explain why speech and other narrowband perturbations can be recovered faithfully, even though the synchronization error is comparable in power to the message itself.

scholarly journals A New 4D Chaotic System with Two-Wing, Four-Wing, and Coexisting Attractors and Its Circuit Simulation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Lilian Huang ◽  
Zefeng Zhang ◽  
Jianhong Xiang ◽  
Shiming Wang
Keyword(s):  
Numerical Simulation ◽  
Chaotic System ◽  
Dynamic Characteristics ◽  
Simulation Analysis ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Coexisting Attractors ◽  
System A ◽  
Simulation Results ◽  
New System

In order to further improve the complexity of chaotic system, a new four-dimensional chaotic system is constructed based on Sprott B chaotic system. By analyzing the system’s phase diagrams, symmetry, equilibrium points, and Lyapunov exponents, it is found that the system can generate not only both two-wing and four-wing attractors but also the attractors with symmetrical coexistence, and the dynamic characteristics of the new system constructed are more abundant. In addition, the system is simulated by Multisim software, and the simulation results show that the results of circuit simulation and numerical simulation analysis are basically the same.

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