混沌、超混沌电路的控制与同步研究
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摘要
本文主要研究了非线性电路中混沌、超混沌的控制与同步，以及基于超混沌同步的保密通信系统的设计，主要完成了以下五项工作，获得了很好的结果。
首先，通过对多涡卷蔡氏电路中非线性特性函数的相位进行开关调制，把多涡卷蔡氏电路由混沌状态控制到其平衡态和周期轨道，并用控制后系统的最大Lyapunov指数随时间的转变阐述了其控制机制。
其次，采用系统参数开关调制策略，研究了两个超混沌电路的控制问题。通过改变外部控制参数，如调制幅度，把超混沌电路控制到稳定的平衡状态和nP周期轨道。这种超混沌控制方法为将超混沌电路应用于特殊信号发生器或信号变换器提供了一种可能途径。
第三，采用单向耦合方法实现了两个四阶蔡氏超混沌电路的同步，根据Lyapunov稳定性定理，得到了两个系统达到同步时耦合系数的阈值。耦合可以是连续的，也可以是离散的，可以是多变量的，也可以是单变量的。
第四，采用单变量单向耦合方法实现了两个超混沌振荡器的同步，不但有数值研究结果，而且给出了电路实验仿真结果。电路实验仿真中出现一种新的同步现象，即超混沌振荡器的一个电流达到广义同步时，其它的电压和电流还可以实现精确同步，我们对此同步现象的物理机制进行了定性分析。耦合可以是连续的，也可以是离散的。由于只需单变量耦合，所以易于工程上实现。
第五，研究了一种基于状态观测器的超混沌保密通信系统的设计方法，其设计策略是将接收端超混沌系统作为发送端超混沌系统的状态观测器。本设计方法具有一系列的优点。
In this paper,we mainly studied on control and synchronization in chaotic and hyperchaotic circuits as well as on secure communication based on hyperchaos synchronization. Five pieces of research results are finished. Better results are obtained.
At first,a method for controlling chaos is studied,which is based on phase switching modulation. The N-scroll Chua's circuit is controlled to its fixed point as well as periodic orbits by using switching modulation on one its phase. Its mechanism of control is analysed using the change of the max Lyapunov exponent when the system is controlled.
In the second place,controlling using strategy of system parameter switching modulation in two hyperchaotic circuits is studied. The hyperchaotic circuits are controlled to their stable equilibriums as well as nP periodic orbits by change the external controlling parameter,such as the magnitude of modulation. It is possible that hyperchaotic circuits are controlled to special signal producer and signal exchanger by using this method of hyperchaos control.
The third,Synchronization in unidirectionally coupled 4-order hyperchaotic Chua's circuits is studied. Synchronization of hyperchaotic systems can be achieved using only one variable unidirectionally coupled at discrete times. We obtain the threshold of couple coefficient based on Lyapunov stability theory. The theoretical analysis and numerical simulations show that this method is correct and efficient.
The fourth,Synchronization in coupled hyperchaotic oscillator is studied. Synchronization of hyperchaotic oscillators can be achieved using only one variable unidirectionally coupled. The main advantage of this method is that it can be realized easily in engineer. The numerical studies and circuit experimental simulations show that this method is correct and efficient. In our circuit experimental simulations,the other variables of hyperchaotic systems synchronize precisely when a variable synchronizes generally.
Finally,a general methodology for designing hyperchaotic cryptosystems is developed. The basic idea is to make the decrypter a nonlinear observer for the state of the encrypter. The utilization of hyperchaotic cryptosystems,as well as the increased complexity of the transmitted signal,seems to make a further contribution to the development of the communication systems with higher security.
首先，通过对多涡卷蔡氏电路中非线性特性函数的相位进行开关调制，把多涡卷蔡氏电路由混沌状态控制到其平衡态和周期轨道，并用控制后系统的最大Lyapunov指数随时间的转变阐述了其控制机制。
其次，采用系统参数开关调制策略，研究了两个超混沌电路的控制问题。通过改变外部控制参数，如调制幅度，把超混沌电路控制到稳定的平衡状态和nP周期轨道。这种超混沌控制方法为将超混沌电路应用于特殊信号发生器或信号变换器提供了一种可能途径。
第三，采用单向耦合方法实现了两个四阶蔡氏超混沌电路的同步，根据Lyapunov稳定性定理，得到了两个系统达到同步时耦合系数的阈值。耦合可以是连续的，也可以是离散的，可以是多变量的，也可以是单变量的。
第四，采用单变量单向耦合方法实现了两个超混沌振荡器的同步，不但有数值研究结果，而且给出了电路实验仿真结果。电路实验仿真中出现一种新的同步现象，即超混沌振荡器的一个电流达到广义同步时，其它的电压和电流还可以实现精确同步，我们对此同步现象的物理机制进行了定性分析。耦合可以是连续的，也可以是离散的。由于只需单变量耦合，所以易于工程上实现。
第五，研究了一种基于状态观测器的超混沌保密通信系统的设计方法，其设计策略是将接收端超混沌系统作为发送端超混沌系统的状态观测器。本设计方法具有一系列的优点。
In this paper,we mainly studied on control and synchronization in chaotic and hyperchaotic circuits as well as on secure communication based on hyperchaos synchronization. Five pieces of research results are finished. Better results are obtained.
At first,a method for controlling chaos is studied,which is based on phase switching modulation. The N-scroll Chua's circuit is controlled to its fixed point as well as periodic orbits by using switching modulation on one its phase. Its mechanism of control is analysed using the change of the max Lyapunov exponent when the system is controlled.
In the second place,controlling using strategy of system parameter switching modulation in two hyperchaotic circuits is studied. The hyperchaotic circuits are controlled to their stable equilibriums as well as nP periodic orbits by change the external controlling parameter,such as the magnitude of modulation. It is possible that hyperchaotic circuits are controlled to special signal producer and signal exchanger by using this method of hyperchaos control.
The third,Synchronization in unidirectionally coupled 4-order hyperchaotic Chua's circuits is studied. Synchronization of hyperchaotic systems can be achieved using only one variable unidirectionally coupled at discrete times. We obtain the threshold of couple coefficient based on Lyapunov stability theory. The theoretical analysis and numerical simulations show that this method is correct and efficient.
The fourth,Synchronization in coupled hyperchaotic oscillator is studied. Synchronization of hyperchaotic oscillators can be achieved using only one variable unidirectionally coupled. The main advantage of this method is that it can be realized easily in engineer. The numerical studies and circuit experimental simulations show that this method is correct and efficient. In our circuit experimental simulations,the other variables of hyperchaotic systems synchronize precisely when a variable synchronizes generally.
Finally,a general methodology for designing hyperchaotic cryptosystems is developed. The basic idea is to make the decrypter a nonlinear observer for the state of the encrypter. The utilization of hyperchaotic cryptosystems,as well as the increased complexity of the transmitted signal,seems to make a further contribution to the development of the communication systems with higher security.
引文
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[11] 郝柏林，从抛物线谈起——混沌动力学引论，上海：上海科技出版社，1993
[12] 傅新楚等，分叉·混沌·符号动力学，武汉：武汉大学出版社，1993
[13] 李后强，汪富泉，分形理论及其在分子科学中的应用，北京：北京科学出版社，1993
[14] K.M. Cuomo and A. V. Oppenheim, Circuit implementation of Synchronized chaos with application to communications, Phys. Rev. Lett., 1993, 71(1):65-68
[15] 郑伟谋，郝柏林，实用符号动力学，上海：上海科技出版社，1994
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[17] 王光瑞，陈光旨，非线性常微分方程与混沌运动，广西：广西科技出版社，1995
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[20] 倪皖荪，华一满，邓浩等，混沌通讯，物理学进展，1996，16(3)：645-656
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[22] 陈士华，陆君安，混沌动力学初步，武汉：武汉水利电力大学出版社，1999
[23] 胡岗，萧井华，郑志刚，混沌控制，上海：上海科技教育出版社，2000
[24] 何大韧，汪秉宏，汪颖梅等，非线性动力学引论——处处光滑与分段光滑系统的动力学特性，陕西：陕西科学技术出版社，2001
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[26] Pyfragas K, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 1992,170:421-428
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