Estimating the Globally Exponentially Attractive Set and Positively Invariant Set for the Qi Chaotic System and its Applications to Chaos Synchronization

2010 ◽  
Author(s):  
Jigui Jian ◽  
Zhengwen Tu ◽  
Yanjun Shen
Keyword(s):  
Chaotic System ◽  
Chaos Synchronization ◽  
Invariant Set ◽  
Positively Invariant Set

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.

scholarly journals Ultimate Bound of a 3D Chaotic System and Its Application in Chaos Synchronization

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jiezhi Wang ◽  
Qing Zhang ◽  
Zengqiang Chen ◽  
Hang Li
Keyword(s):  
Chaotic System ◽  
Chaos Synchronization ◽  
Three Dimensional ◽  
Boundary Region ◽  
Cross Product ◽  
State Equation ◽  
Linear Coefficient ◽  
Analytical Expression ◽  
Product Term ◽  
Cross Product Term

Two ellipsoidal ultimate boundary regions of a special three-dimensional (3D) chaotic system are proposed. To this chaotic system, the linear coefficient of theith state variable in theith state equation has the same sign; it also has two one-order terms and one quadratic cross-product term in each equation. A numerical solution and an analytical expression of the ultimate bounds are received. To get the analytical expression of the ultimate boundary region, a new result of one maximum optimization question is proved. The corresponding ultimate boundary regions are demonstrated through numerical simulations. Utilizing the bounds obtained, a linear controller is proposed to achieve the complete chaos synchronization. Numerical simulation exhibits the feasibility of the designed scheme.

scholarly journals Low Model Analysis and Synchronous Simulation of the Wave Mechanics

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Wenyuan Duan ◽  
Heyuan Wang ◽  
Meng Kan
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Chaotic System ◽  
Invariant Set ◽  
Positive Definite ◽  
Chaotic Attractors ◽  
Wave Mechanics ◽  
Wave Dynamics ◽  
Simulation Results ◽  
Positive Invariant Set

The dynamic behavior of a chaotic system in the internal wave dynamics and the problem of the tracing and synchronization are investigated, and the numerical simulation is carried out in this paper. The globally exponentially attractive set and positive invariant set of the chaotic system are studied via constructing the positive definite and radial unbounded Lyapunov function. There are no equilibrium positions, periodic solutions, quasi-period motions, wandering recovering motions, and other chaotic attractors of the system out of the globally exponentially attractive set. Strange attractors can only locate in the globally exponentially attractive set. A feedback controller is designed for the chaotic system to realize the control of the unstable point. The second method of Lyapunov is used to discuss theoretically the rationality of the design of the controller. The driving-response synchronization method is used to realize the globally exponential synchronization. The numerical simulation is carried out by MATLAB software, and the simulation results show that the method is effective.

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