摘要
混沌控制与同步是将混沌系统应用于保密通信的前提。一方面,在工业系统通信中,由于利用混沌进行通信有保密性强,抗干扰能力强等特点,所以研究混沌控制与同步并使其应用于通信系统有一定的实际意义；另一方面,生物系统中也存在着混沌现象,如大脑系统、视觉系统等,因此分析生物体神经元之间的同步,以研究其通信机制有一定的理论意义。本文主要应用几种非线性理论的控制方法,通过理论推导研究了一类典型混沌系统同步、反同步,以及耦合神经元之间的同步,并对推导结果做了数值模拟,以进一步说明结果的正确性和有效性,本文的主要工作如下：
(1)研究了基于直接构造法的同构混沌系统和异构混沌系统的混沌同步问题。首先,采用直接构造法为响应系统设计适当的控制器；然后将控制器作用下的误差系统转化为三对角结构,得到误差系统状态在原点渐进稳定,因此实现了驱动系统与响应系统达到同步,并通过数值仿真进一步证明了该方法的有效性。
(2)研究了分数阶超混沌Chen系统的动力学行为。结果表明：当阶数大于或等于0.95时分数阶超混沌Chen系统处于超混沌状态；且随阶数的增大,混沌性会越来越明显。以阶数取0.95及0.98为例,采用主动控制方法,设计了使得两个分数阶超混沌Chen系统同步的控制策略。数值模拟结果证实了该策略对一类分数阶超混沌Chen系统有效。
(3)研究了两个超混沌Lorenz系统的反同步,以及超混沌Lorenz系统和超混沌Chen系统的异结构反同步问题。对于两个参数已知的超混沌Lorenz系统,使用了非线性控制方法；而对于两个参数未知的超混沌Lorenz系统和超混沌Chen系统,运用了自适应控制法设计合适的Lyapunov函数及参数自适应率,并成功辨识出了系统参数。数值模拟验证了所设计方案的有效性。
(4)研究了外部电刺激下未知参数耦合FitzHugh-Nagumo (FHN)神经元的混沌同步问题。基于Lyapunov稳定性理论,设计了自适应控制器及参数更新率,使得当单个神经元混沌时,不论其耦合强度多大都能使耦合神经元达到同步,并且对近似误差,离子通道噪声及外部干扰等有较好的鲁棒性,并能成功辨识出未知参数。数值模拟结果证实了所设计的控制器的有效性。
Chaos control and synchronization of chaotic systems is the premise for applying them in communication systems. On the one hand, due to the strong confidentiality and anti-interference ability of chaos communication, studying of chaos control and synchronization has some practical significance, when chaos is used in the industrial systems' communication; On the other hand, there are also chaotic phenomena in biological system, such as the brain system and visual system. In order to learn the communication mechanism of neurons, analyzing of the synchronization between them has some theoretical significance. In this paper, based on nonlinear control methods, synchronization and anti-synchronization of a class of typical chaotic systems is studied via theoretical analysis, and also the synchronization between coupled neurons. Finally, the results are illustrated by numerical simulation. The major work of this paper is summarized as follows:
(1) Synchronization of the chaotic and hyper-chaotic systems with identical or different structures is studied, which is based on direct construction method. Firstly, an appropriate controller for the response system is designed by direct construction method. And then transform the error system with the controller into triangular structure, so it can be obtained that the error system is asymptotically stable at the origin, which shows the drive system and response system achieve synchronization. Through numerical simulation, the effectiveness of this method is proved further.
(2) The dynamics of fractional order hyper-chaotic Chen system is studied. The results show:when the order equal to or greater than 0.95, the fractional order hyper-chaotic Chen system is hyper-chaotic;and with the order increasing, the chaos becomes evident increasingly. The author designs a control strategy to realize self-synchronization of two fractional order hyper-chaotic Chen systems by active control method, case in the order is 0.95 and 0.98, respectively. The results of numerical simulation show the effectiveness of the strategy to a class of fractional order hyper-chaotic Chen systems.
(3) The anti-synchronization of two hyper-chaotic Lorenz systems is studied, as well as hyper-chaotic Lorenz system and hyper-chaotic Chen system with different structures. Nonlinear control method is used to the two hyper-chaotic Lorenz systems with certain parameters, and adaptive control method is used to the hyper-chaotic Lorenz system and hyper-chaotic Chen system with uncertain parameters, respectively. The Lyapunov function and parameters'update laws are selected, which can identify the unknown parameters successfully. The numerical simulation shows the effectiveness of the strategies.
(4) The chaotic synchronization of two electrical coupled FitzHugh-Nagumo (FHN) neurons with unknown parameters via adaptive control is investigated. Based on the Lyapunov stability theory, an adaptive controller and parameters'update laws are designed, which can achieve the synchronization of the two gap junction coupled FHN neurons when the individual neuron is chaotic, but without considering the coupling strength. The controller is robust to the uncertainties such as approximate error, ionic channel noise and external disturbances, and the parameters can be identified successfully. The numerical simulation results confirm the effectiveness of the designed controller.
引文
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