On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization

2005 ◽  
Vol 48 (3) ◽  
pp. 304 ◽  
Author(s):  
Xiaoxin LIAO
Keyword(s):  
Chaotic System ◽  
Chaos Control ◽  
Invariant Set ◽  
Lorenz Chaotic System ◽  
Control And Synchronization ◽  
Global Attractive Set ◽  
Positive Invariant Set

scholarly journals Chaos Control and Synchronization via Switched Output Control Strategy

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Runzi Luo ◽  
Haipeng Su ◽  
Yanhui Zeng
Keyword(s):  
Chaotic System ◽  
Control Strategy ◽  
Chaos Control ◽  
Chaotic Systems ◽  
State Variables ◽  
Output Control ◽  
Lorenz Chaotic System ◽  
Control And Synchronization ◽  
Theoretical Results

This paper investigates the control and synchronization of a class of chaotic systems with switched output which is assumed to be switched between the first and the second state variables of chaotic system. Some novel and yet simple criteria for the control and synchronization of a class of chaotic systems are proposed via the switched output. The generalized Lorenz chaotic system is taken as an example to show the feasibility and efficiency of theoretical results.

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.

scholarly journals Parameter Estimation of a Class One-Dimensional Discrete Chaotic System

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Lidong Liu ◽  
Jinfeng Hu ◽  
Huiyong Li ◽  
Jun Li ◽  
Zishu He ◽  
...  
Keyword(s):  
Parameter Estimation ◽  
Chaotic System ◽  
Chaos Control ◽  
Mean Value ◽  
Unknown Parameters ◽  
Tent Map ◽  
One Dimensional ◽  
Chebyshev Map ◽  
Mean Value Method ◽  
Control And Synchronization

It is of vital importance to exactly estimate the unknown parameters of chaotic systems in chaos control and synchronization. In this paper, we present a method for estimating one-dimensional discrete chaotic system based on mean value method (MVM). It is proposed by exploiting the ergodic and synchronization features of chaos. It can effectively estimate the parameter value, and it is more exact than MVM. Finally, numerical simulations on Chebyshev map and Tent map show that the proposed method has better performance of parameter estimation than MVM.

scholarly journals The Robust Control and Synchronization of a Class of Fractional-Order Chaotic Systems with External Disturbances via a Single Output

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Runzi Luo ◽  
Haipeng Su
Keyword(s):  
Robust Control ◽  
Fractional Order ◽  
Chaos Control ◽  
Chaotic Systems ◽  
Sufficient Conditions ◽  
External Disturbances ◽  
Numerical Examples ◽  
Single Output ◽  
Lorenz Chaotic System ◽  
Control And Synchronization

This paper investigates the stabilization and synchronization of a class of fractional-order chaotic systems which are affected by external disturbances. The chaotic systems are assumed that only a single output can be used to design the controller. In order to design the proper controller, some observer systems are proposed. By using the observer systems some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived. Numerical examples are presented by taking the fractional-order generalized Lorenz chaotic system as an example to show the feasibility and validity of the proposed method.

GLOBALLY ATTRACTIVE AND POSITIVE INVARIANT SET OF THE LORENZ SYSTEM

2006 ◽  
Vol 16 (03) ◽  
pp. 757-764 ◽  
Author(s):  
PEI YU ◽  
XIAOXIN LIAO
Keyword(s):  
Lyapunov Function ◽  
Chaos Synchronization ◽  
Chaos Control ◽  
Lorenz System ◽  
Equilibrium Points ◽  
Invariant Set ◽  
The Lorenz System ◽  
Globally Attractive ◽  
Generalized Lyapunov Function ◽  
Positive Invariant Set

In this paper, based on a generalized Lyapunov function, a simple proof is given to improve the estimation of globally attractive and positive invariant set of the Lorenz system. In particular, a new estimation is derived for the variable x. On the globally attractive set, the Lorenz system satisfies Lipschitz condition, which is very useful in the study of chaos control and chaos synchronization. Applications are presented for globally, exponentially tracking periodic solutions, stabilizing equilibrium points and synchronizing two Lorenz systems.

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