﻿ 具有可变系数的三维混沌系统与五维超混沌系统的同步问题研究 On Synchronization of 5D Hyperchaotic System and 3D Chaotic System with Variable Coefficient

Dynamical Systems and Control
Vol.05 No.02(2016), Article ID:17361,7 pages
10.12677/DSC.2016.52005

On Synchronization of 5D Hyperchaotic System and 3D Chaotic System with Variable Coefficient

Hong Niu

College of Electronic Information and Automation, Tianjin University of Science & Technology, Tianjin

Received: Mar. 17th, 2016; accepted: Apr. 15th, 2016; published: Apr. 18th, 2016

ABSTRACT

In this paper, the 3D chaotic system and the 5D hyperchaotic system are synchronized via the center translation method, where the variable coefficient in the nonlinear part of the 3D chaotic system is taken as the uncertainty in the synchronization. The center of the state variables of the response 3D chaotic system is translated to the assigned state variables of the drive 5D hyperchaotic system, such that the model of the error system is the same as that of the response system. Thus, synchronization of different systems is converted to stability control of the error system. This method can effectively simplify the design procedure for synchronization controller, and it can be applied to the study of drive system with uncertainty, so long as the synchronized state variables of the drive system and their derivatives are known or can be estimated by state observer.

Keywords:Synchronization via Center Translation Method, Synchronization of Different Systems, Nonlinear Control, Uncertainty

1. 引言

2. 误差系统建模与同步控制器设计

2.1. 五维超混沌系统

(1)

2.2. 三维混沌系统

(2)

2.3. 误差系统建模

(3)

Figure 1. Lyapunov exponent spectrum of the response system (2) versus increasing H

2.4. 同步控制器设计

。 (4)

， (5)

(6)

3. 数值仿真

， (7)

(8)

4. 算例

。未加入同步控制器和结构补偿器时，响应系统(8)的Lyapunov指数分别为，说明系统是混沌的。此时响应系统(8)与驱动系统(1)各对应状态变量的误差及同步曲线分别如图3(a)和图3(b)中所示。图中横轴t表示式(8)的求解区间，为一无量纲量，后文各图中横轴t的定义与此处相同。从图3中可以看出，状态变量的差异很大。

Figure 2. Sub-Lyapunov exponent spectrum of the response system (8) versus increasing H

(a) (b)

Figure 3. Error and synchronization curves of the corresponding state variables of the response and the drive systems before adding uc and us to the response system: (a) Error curves; (b) Synchronization curves

(a) (b)

Figure 4. Error and synchronization curves of the corresponding state variables of the response and the drive systems after adding uc and us to the response system: (a) Error curves; (b) Synchronization curves

5. 中心平移同步法的优点

6. 结论

On Synchronization of 5D Hyperchaotic System and 3D Chaotic System with Variable Coefficient[J]. 动力系统与控制, 2016, 05(02): 41-47. http://dx.doi.org/10.12677/DSC.2016.52005

1. 1. Liu, Y.Z., Jiang, C.S., Lin, C.S., et al. (2007) Chaos Synchronization between Two Different 4D Hyperchaotic Chen Systems. Chinese Physics, 16, 660-665. http://dx.doi.org/10.1088/1009-1963/16/3/017

2. 2. 牛弘, 张国山. 一类具有可变系数的混沌系统的同步[J]. 物理学报, 2013, 62(13): 105-115.

3. 3. Niu, H., Zhang, G.S. and Wang, J.K. (2014) Chaos Synchronization of Chua’s Circuit and Lorenz System Based on Strictly Positive Realness. Proceedings of the 33th Chinese Control Conference, CCC 2014, Nanjing, 28-30 July 2014, 1972-1976. http://dx.doi.org/10.1109/chicc.2014.6896932

4. 4. Salarieh, H. and Shahrokhi, M. (2008) Adaptive Synchroniza-tion of Two Different Chaotic Systems with Time Varying Unknown Parameters. Chaos, Solitons & Fractals, 37, 125-136. http://dx.doi.org/10.1016/j.chaos.2006.08.038

5. 5. Zhu, C.X. (2009) Adaptive Synchronization of Two Novel Different Hyperchaotic Systems with Partly Uncertain Parameters. Applied Mathematics and Computation, 215, 557-561. http://dx.doi.org/10.1016/j.amc.2009.05.026

6. 6. Fu, G.Y. and Li, Z.S. (2010) Adaptive Synchronization of a Hyperchaotic Lü System Based on Extended Passive Control. Chinese Physics B, 19, 060505-1-5.

7. 7. Kuntanapreeda, S. and Sangpet, T. (2012) Synchronization of Chaotic Systems with Unknown Pa-rameters Using Adaptive Passivity-Based Control. Journal of the Franklin Institute, 349, 2547-2569. http://dx.doi.org/10.1016/j.jfranklin.2012.08.002

8. 8. 张文革, 韩京清. 一类混沌系统的状态观测与控制[J]. 控制与决策, 2000, 15(3): 301-304.

9. 9. 张国山, 李思瑶, 王江. 基于自抗扰控制的2个耦合神经元间的混沌同步[J]. 天津大学学报(自然科学与工程技术版), 2013, 46(3): 263-268.

10. 10. 牛弘. 混沌及超混沌系统的分析、控制、同步与电路实现[D]: [博士学位论文]. 天津: 天津大学, 2014.

11. 11. 张国山, 牛弘. 一个基于Chen系统的新混沌系统的分析与同步[J]. 物理学报, 2012, 61(11): 137-147.

12. 12. 牛弘. 具有可变系数的三维混沌系统的稳定性控制与电路实现[J]. 动力系统与控制, 2016, 5(1): 31-40.

13. 13. Pecora, L.M. and Carroll, T.L. (1990) Synchronization in Chaotic Systems. Physical Review Letters, 64, 821-824. http://dx.doi.org/10.1103/PhysRevLett.64.821

14. 14. 刘秉正, 彭建华. 非线性动力学[M]. 北京: 高等教育出版社, 2007.