scholarly journals Qualitative Study of a 4D Chaos Financial System

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Fuchen Zhang ◽  
Gaoxiang Yang ◽  
Yong Zhang ◽  
Xiaofeng Liao ◽  
Guangyun Zhang
Keyword(s):  
Hausdorff Dimension ◽  
Chaos Synchronization ◽  
Chaos Control ◽  
Dynamical Behavior ◽  
Lyapunov Stability Theory ◽  
Attraction Domain ◽  
Ultimate Boundedness ◽  
Lyapunov Dimension ◽  
Simulation Results ◽  
Ultimate Bound

Some dynamics of a new 4D chaotic system describing the dynamical behavior of the finance are considered. Ultimate boundedness and global attraction domain are obtained according to Lyapunov stability theory. These results are useful in estimating the Lyapunov dimension of attractors, Hausdorff dimension of attractors, chaos control, and chaos synchronization. We will also present some simulation results. Furthermore, the volumes of the ultimate bound set and the global exponential attractive set are obtained.

Dynamics of a New 5D Hyperchaotic System of Lorenz Type

2018 ◽  
Vol 28 (03) ◽  
pp. 1850036 ◽  
Author(s):  
Fuchen Zhang ◽  
Rui Chen ◽  
Xingyuan Wang ◽  
Xiusu Chen ◽  
Chunlai Mu ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Optimization Theory ◽  
Lyapunov Stability Theory ◽  
Hyperchaotic System ◽  
Ultimate Boundedness ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Hyperchaotic Systems ◽  
Ultimate Bound

Ultimate boundedness of chaotic dynamical systems is one of the fundamental concepts in dynamical systems, which plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors and the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, chaos synchronization. However, it is often difficult to obtain the bounds of the hyperchaotic systems due to the complex algebraic structure of the hyperchaotic systems. This paper has investigated the boundedness of solutions of a nonlinear hyperchaotic system. We have obtained the global exponential attractive set and the ultimate bound set for this system. To obtain the ellipsoidal ultimate bound, the ultimate bound of the proposed system is theoretically estimated using Lagrange multiplier method, Lyapunov stability theory and optimization theory. To show the ultimate bound region, numerical simulations are provided.

Dynamical Behaviors of a Modified Lorenz–Stenflo Chaotic System

2017 ◽  
Vol 27 (05) ◽  
pp. 1750074 ◽  
Author(s):  
Fuchen Zhang ◽  
Xingyuan Wang ◽  
Xiaofeng Liao ◽  
Guangyun Zhang ◽  
Chunlai Mu
Keyword(s):  
Chaos Synchronization ◽  
Chaos Control ◽  
Optimization Theory ◽  
Lyapunov Stability Theory ◽  
Theoretical Support ◽  
Dynamical Behaviors ◽  
Special Cases ◽  
Attractive Sets ◽  
Ultimate Bound ◽  
Theoretical Results

In this paper, the ultimate bound and globally exponentially attractive sets of a modified Lorenz–Stenflo system are studied based on the Lyapunov stability theory and optimization theory. Comparing with the best results in the current literature, our new results include the existing results as special cases. Furthermore, the new results offer a theoretical support to studying the Hausdorff dimension of attractor of this modified Lorenz–Stenflo system. These theoretical results are also important and useful for chaos control and chaos synchronization.

scholarly journals Microcontroller Implementation, Chaos Control, Synchronization and Antisynchronization of Josephson Junction Model

2021 ◽  
Vol 1 (2) ◽  
pp. 198-208
Author(s):  
Rolande Tsapla Fotsa ◽  
André Rodrigue Tchamda ◽  
Alex Stephane Kemnang Tsafack ◽  
Sifeu Takougang Kingni
Keyword(s):  
Josephson Junction ◽  
Chaos Control ◽  
Control Method ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Hurwitz Stability ◽  
Feedback Control Method ◽  
Different Shapes ◽  
Simulation Results

The microcontroller implementation, chaos control, synchronization, and antisynchronization of the nonlinear resistive-capacitive-inductive shunted Josephson junction (NRCISJJ) model are reported in this paper. The dynamical behavior of the NRCISJJ model is performed using phase portraits, and time series. The numerical simulation results reveal that the NRCISJJ model exhibits different shapes of hidden chaotic attractors by varying the parameters. The existence of different shapes of hidden chaotic attractors is confirmed by microcontroller results obtained from the microcontroller implementation of the NRCISJJ model. It is theoretically demonstrated that the two designed single controllers can suppress the hidden chaotic attractors found in the NRCISJJ model. Finally, the synchronization and antisynchronization of unidirectional coupled NRCISJJ models are studied by using the feedback control method.  Thanks to the Routh Hurwitz stability criterion, the controllers are designed in order to control chaos in JJ models and achieved synchronization and antisynchronization between coupled NRCISJJ models. Numerical simulations are shown to clarify and confirm the control, synchronization, and antisynchronization.

scholarly journals Projective Synchronization of Chaotic Systems Via Backstepping Design

2013 ◽  
Vol 18 (3) ◽  
pp. 965-973 ◽  
Author(s):  
A. Tarai ◽  
M.A. Khan
Keyword(s):  
Numerical Simulation ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Lyapunov Stability Theory ◽  
Discrete Dynamical Systems ◽  
Projective Synchronization ◽  
Control Law ◽  
Backstepping Design ◽  
Simulation Results ◽  
Duffing Map

Abstract Chaos synchronization of discrete dynamical systems is investigated. An algorithm is proposed for projective synchronization of chaotic 2D Duffing map and chaotic Tinkerbell map. The control law was derived from the Lyapunov stability theory. Numerical simulation results are presented to verify the effectiveness of the proposed algorithm

Synchronization of Incommensurate Fractional-Order Chaotic System Using Adaptive Control

2014 ◽  
Vol 602-605 ◽  
pp. 946-949
Author(s):  
Jing Fang ◽  
Ruo Xun Zhang
Keyword(s):  
Adaptive Control ◽  
Fractional Order ◽  
Chaotic System ◽  
Lyapunov Stability ◽  
Differential Inequality ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Lyapunov Stability Theory ◽  
Adaptive Feedback ◽  
Simulation Results

This paper investigates the synchronization of incommensurate fractional-order chaotic systems, and proposes a modified adaptive-feedback controller for fractional-order chaos synchronization based on Lyapunov stability theory, fractional order differential inequality and adaptive control theory. This synchronization approach that is simple, global and theoretically rigorous enables synchronization of fractional-order chaotic systems be achieved in a systematic way. Simulation results for a fractional-order chaotic system is provided to illustrate the effectiveness of the proposed scheme.

CHAOS SYNCHRONIZATION OF CHEN SYSTEM AND ITS APPLICATION TO SECURE COMMUNICATION

2008 ◽  
Vol 22 (21) ◽  
pp. 3709-3720 ◽  
Author(s):  
XINGYUAN WANG ◽  
XIANGJUN WU ◽  
YIJIE HE ◽  
GULZILA ANIWAR
Keyword(s):  
Lyapunov Stability ◽  
Secure Communication ◽  
Chaos Synchronization ◽  
Control Method ◽  
Lyapunov Stability Theory ◽  
Chen System ◽  
Simulation Results ◽  
Feedback Method ◽  
Active Control Method ◽  
Pc Method

This paper describes the chaos synchronization of two identical Chen systems theoretically and numerically. Based on Lyapunov stability theory, the controllers for achieving synchronization of two identical Chen systems using the PC method, active control method, and feedback method are designed. Numerical simulations show the correctness of the results. Moreover, as an application, the well-known PC method is applied to chaos-synchronization-based secure communication. Simulation results verify the proposed scheme's effectiveness in the communication application and also show its well robustness.

scholarly journals Design of PDC Controllers by Matrix Reversibility for Synchronization ofYinandYangChaotic Takagi-Sugeno Fuzzy Henon Maps

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Chun-Yen Ho ◽  
Hsien-Keng Chen ◽  
Zheng-Ming Ge
Keyword(s):  
Chaos Synchronization ◽  
Matrix Theory ◽  
Chinese Philosophy ◽  
Fuzzy Model ◽  
Invertible Matrix ◽  
Henon Maps ◽  
Hénon Maps ◽  
Feminine Principle ◽  
Simulation Results ◽  
Takagi Sugeno

This paper investigates the synchronization ofYinandYangchaotic T-S fuzzy Henon maps via PDC controllers. Based on the Chinese philosophy,Yinis the decreasing, negative, historical, or feminine principle in nature, whileYangis the increasing, positive, contemporary, or masculine principle in nature.YinandYangare two fundamental opposites in Chinese philosophy. The Henon map is an invertible map; so the Henon maps with increasing and decreasing argument can be called theYangandYinHenon maps, respectively. Chaos synchronization ofYinandYangT-S fuzzy Henon maps is achieved by PDC controllers. The design of PDC controllers is based on the linear invertible matrix theory. The T-S fuzzy model ofYinandYangHenon maps and the design of PDC controllers are novel, and the simulation results show that the approach is effective.

A Design Method of LS-SVM Based Stable Adaptive Controller for a Class of Nonlinear Systems

2011 ◽  
Vol 48-49 ◽  
pp. 17-20
Author(s):  
Chun Li Xie ◽  
Tao Zhang ◽  
Dan Dan Zhao ◽  
Cheng Shao
Keyword(s):  
Nonlinear Systems ◽  
Design Method ◽  
Nonlinear Function ◽  
Lyapunov Stability Theory ◽  
Adaptive Controller ◽  
Control Law ◽  
Continuous Systems ◽  
Control Schemes ◽  
Closed Systems ◽  
Simulation Results

A design method of LS-SVM based stable adaptive controller is proposed for a class of nonlinear continuous systems with unknown nonlinear function in this paper. Due to the fact that the control law is derived based on the Lyapunov stability theory, the scheme can not only solve the tracking problem of this class of nonlinear systems, but also it can guarantee the asymptotic stability of the closed systems, which is superior to many LS-SVM based control schemes. The effectiveness of the proposed scheme is demonstrated by simulation results.

CHAOS SYNCHRONIZATION VIA UNIDIRECTIONAL COUPLING AND ITS APPLICATION TO SECURE COMMUNICATION

2009 ◽  
Vol 23 (32) ◽  
pp. 5949-5964 ◽  
Author(s):  
XINGYUAN WANG ◽  
MINGJUN WANG
Keyword(s):  
Secure Communication ◽  
Chaos Synchronization ◽  
Function Method ◽  
The Self ◽  
Largest Lyapunov Exponent ◽  
Global Synchronization ◽  
Unidirectional Coupling ◽  
Response System ◽  
Lyapunov Function Method ◽  
Simulation Results

This paper studies chaos synchronization via unidirectional coupling. The self-synchronization of Lorenz systems, modified coupled dynamos systems and hyperchaotic Chen systems is studied by three methods: the Lyapunov function method, the global synchronization method and the numerical calculation of the largest Lyapunov exponent method. In regard to application to communication, we show that via transmitting single signal the synchronization of the drive system and the response system can be achieved. An example of applying self-synchronization of hyperchaotic Chen systems to chaotic masking secure communication is presented in this paper. Simulation results show the effectiveness of the method.

Optimal Synchronization of a Memristive Chaotic Circuit

2016 ◽  
Vol 26 (06) ◽  
pp. 1650093 ◽  
Author(s):  
Michaux Kountchou ◽  
Patrick Louodop ◽  
Samuel Bowong ◽  
Hilaire Fotsin ◽  
Jurgen Kurths
Keyword(s):  
Numerical Simulations ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Analog Circuit ◽  
Electronic Circuit ◽  
Dynamical Behavior ◽  
Circuit Implementation ◽  
Chaotic Circuit ◽  
Dynamical Properties ◽  
Analog Circuit Implementation

This paper deals with the problem of optimal synchronization of two identical memristive chaotic systems. We first study some basic dynamical properties and behaviors of a memristor oscillator with a simple topology. An electronic circuit (analog simulator) is proposed to investigate the dynamical behavior of the system. An optimal synchronization strategy based on the controllability functions method with a mixed cost functional is investigated. A finite horizon is explicitly computed such that the chaos synchronization is achieved at an established time. Numerical simulations are presented to verify the effectiveness of the proposed synchronization strategy. Pspice analog circuit implementation of the complete master-slave-controller systems is also presented to show the feasibility of the proposed scheme.

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