Ultimate bound sets of a hyperchaotic system and its application in chaos synchronization

Complexity ◽  
2014 ◽  
Vol 20 (4) ◽  
pp. 30-44 ◽  
Author(s):  
Hassan Saberi Nik ◽  
Sohrab Effati ◽  
JAFAR Saberi-Nadjafi
Keyword(s):  
Chaos Synchronization ◽  
Hyperchaotic System ◽  
Ultimate Bound ◽  
Bound Sets

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.

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.

scholarly journals Analysis of a Lorenz-Like Chaotic System by Lyapunov Functions

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Fuchen Zhang
Keyword(s):  
Chaotic System ◽  
Chaos Synchronization ◽  
Lorenz System ◽  
Chen System ◽  
Lyapunov Function Method ◽  
Exponential Attractivity ◽  
Positively Invariant Set ◽  
Ultimate Bound ◽  
Special Case ◽  
Theoretical Results

In this paper, we investigate the ultimate bound set and positively invariant set of a 3D Lorenz-like chaotic system, which is different from the well-known Lorenz system, Rössler system, Chen system, Lü system, and even Lorenz system family. Furthermore, we investigate the global exponential attractive set of this system via the Lyapunov function method. The rate of the trajectories going from the exterior of the globally exponential attractive set to the interior of the globally exponential attractive set is also obtained for all the positive parameters values a,b,c. The innovation of this paper is that our approach to construct the ultimate bounded and globally exponential attractivity sets assumes that the corresponding sets depend on some artificial parameters (λ and m); that is, for the fixed parameters of the system, we have a series of sets depending on λ and m. The results contain the known result as a special case for the fixed λ and m. The efficiency of the scheme is shown numerically. The theoretical results may find wide applications in chaos control and chaos synchronization.

Chaos synchronization of coupled hyperchaotic system

2009 ◽  
Vol 42 (2) ◽  
pp. 724-730
Author(s):  
Li-Xin Yang ◽  
Yan-Dong Chu ◽  
Jian-Gang Zhang ◽  
Xian-Feng Li
Keyword(s):  
Chaos Synchronization ◽  
Hyperchaotic System

Chaos synchronization in a new 6D hyperchaotic system with self-excited attractors and seventeen terms

2020 ◽  
pp. 2150085 ◽  
Author(s):  
Saad Fawzi AL-Azzawi ◽  
Ahmed S. Al-Obeidi
Keyword(s):  
Chaos Synchronization ◽  
Equilibrium Points ◽  
Linearization Method ◽  
Auxiliary Function ◽  
Hyperchaotic System ◽  
Analytical Technique ◽  
Control Scheme ◽  
Error Dynamics ◽  
The Stability ◽  
Methods Numerical

Based on a state feedback controller, a new 6D hyperchaotic system with real variables and a self-excited attractor is constructed. The dynamic behavior of the new system is investigated in terms of Lyapunov exponents, equilibrium points, and stability. Moreover, chaos synchronization implementation is also presented. A nonlinear control scheme was proposed to find the stability of error dynamics, which reduced the computational complexity of the synchronization algorithm. In comparison to the existing synchronization approaches, the controller in this paper was designed based on the analytical technique (linearization method), which does not require an auxiliary function (Lyapunov function) as in traditional methods. Numerical simulations were carried out by using MATLAB to validate the effectiveness of the analytical technique.

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