混沌系统分析与电路设计
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摘要
系统行为的动态复杂性分析与判定,不仅是电路系统中的重要问题,而且是具有深刻科学意义和工程背景的其他许多系统的核心问题。如何判定这些系统复杂行为,揭示其产生机制,进而从根本上实现人为控制,成为当今科学研究发展中十分艰巨的问题;其中,混沌性判定的难度很大,该项研究需要将深刻的动力系统理论巧妙地应用到大量的数值计算、算法设计、拓扑马蹄寻找等工作中。本文围绕这一难题,系统地研究了混沌系统分析问题和混沌电路设计问题,主要内容如下:
1)为了更严格地讨论具有一维拉伸的混沌现象,首先给出一个简单的拓扑马蹄结果,该结果对于计算机寻找拓扑马蹄而言非常适用;然后,在此基础上,提出了关于一维拉伸拓扑马蹄的寻找方法,并为其设计了MATLAB软件;作为应用实例,本文研究了低维Glass网络模型和超导Josephson结电路模型,利用找到的拓扑马蹄分析和证明了这些系统的混沌行为。
2)为了进一步探讨更复杂的混沌现象,即,存在多个方向拉伸的混沌行为,首先提出了若干适用于多维拉伸的拓扑马蹄判定方法;然后,在此基础上,提出了三维空间中二维拉伸马蹄的寻找方法,并为其设计了MATLAB软件;作为应用实例,依次研究了著名的Hénon超混沌映射、R(?)ssler超混沌系统、Saito超混沌电路、MCK超混沌电路,利用拓扑马蹄理论,分析和验证了这些系统的超混沌行为。由于本项研究更为困难,文献中并未见到类似的工作。
3)为了实现混沌电路设计,首先本文在第五章研究了混沌系统设计问题,提出了基于反馈切换控制、拓扑马蹄构造、子系统耦合等三种设计方法,利用这些方法,设计出了多个混沌系统;然后,利用所得系统的鲁棒混沌性,适当调节参数,使系统状态方程得到简化,并易于通过(非)线性单元电路的搭接,将这些方程转化为实际电路,最终实现了多个混沌电路的设计。
4)为了使非混沌电路产生混沌,得到更简单的混沌电路,本文依据上述三种设计方法,通过分析Wien桥振荡电路的特点,提出了该电路的多种混沌化途径,使电路产生了混沌和超混沌行为,并借助拓扑马蹄理论这些行为进行了分析和判定。
本文工作使得对系统混沌的研究更加容易可靠,为研究更多的混沌系统提供方法和工具。此外,本文对混沌电路设计提供了基于切换反馈控制的若干新方法。
The complexity of behaviors in dynamics systems is not only an important problem in circuit systems, but also a core problem in many other systems with paramount scientific significance and engineering background. Therefore, how to analyze and determine these complicated behaviors, demonstrate the dynamics mechanisms of the system, and then achieve system control has become a series of principal problems of the scientific research in recently years. Among these problems, the chaoticity verification is the hardest one, since it needs the abstruse theory of dynamical systems and topological horseshoe, a great deal of numerical computation and many works on algorithm design. In this dissertation, we study two major problems on the chaotic system, i.e. chaotic system analysis and chaotic circuit design. The main study and contributions are given as follows:
1) In order to study the chaotic dynamics with one dimensional expansion, at first we provide a simple result on topological horseshoes which is very applicable for finding horseshoes with computers; then propose a method to find topological horseshoe with one dimensional expansion with this result, and the method is realized with a MATLAB GUI program; at last, we study the low dimensional Glass network and the RCLSJ(the resistive-capacitive-inductive shunted junction) model, the chaotic behaviors of the two systems are analyzed and proved by mean of topological horseshoe.
2) In order to study more complicated dynamics of chaotic phenomena, i.e. chaotic behavior with more than one directional expansion, at first we propose a technique to determine topological horseshoe with multidimensional expansion by providing a simple result on topological horseshoe; then propose a method to find this kind of horseshoe with this technique, and realize it with a MATLAB GUI program; at last, we study several famous hyperchaotic systems such as the hyperchaotic Hénon map, the Rossler hyperchaotic system, the Saito Hysteresis Chaos Generator (SHCG), and the Matsumoto-Chua-Kobayashi (MCK) circuit, the hyperchaotic behaviors are analyzed and verified by mean of topological horseshoe with multidimensional expansion. Because this research is very difficult, there is no similar work in literature according the author's knowledge.
3) In order to design chaotic circuits, at first we propose three methods to design a chaotic system, such as feedback switching control method, constructing topological horseshoe method, coupling subsystems method; we present many chaotic systems as examples to show these powerful method; since these system are robust in structure, we can obtain a simpler system by modifying their parameters, then we realize the simplified system with the chaotic circuits; finally, we get several chaotic circuits.
In order to make a nonchaotic circuit generate chaos or get a simpler chaotic circuit, we propose three ways to chaotify the Wien bridge oscillator with the three methods mentioned above. We obtain four simple chaotic circuits. Topological horseshoes are used here to analyze and verify the chaotic and hyperchaotic dynamics.
The contribution of this dissertation can make the research on chaos more reliable, and provide methods and tools for researchers to study many other chaotic systems. In addition, three new methods based on feedback switching control are also proposed to design chaotic circuits for the engineering purposes.
1)为了更严格地讨论具有一维拉伸的混沌现象,首先给出一个简单的拓扑马蹄结果,该结果对于计算机寻找拓扑马蹄而言非常适用;然后,在此基础上,提出了关于一维拉伸拓扑马蹄的寻找方法,并为其设计了MATLAB软件;作为应用实例,本文研究了低维Glass网络模型和超导Josephson结电路模型,利用找到的拓扑马蹄分析和证明了这些系统的混沌行为。
2)为了进一步探讨更复杂的混沌现象,即,存在多个方向拉伸的混沌行为,首先提出了若干适用于多维拉伸的拓扑马蹄判定方法;然后,在此基础上,提出了三维空间中二维拉伸马蹄的寻找方法,并为其设计了MATLAB软件;作为应用实例,依次研究了著名的Hénon超混沌映射、R(?)ssler超混沌系统、Saito超混沌电路、MCK超混沌电路,利用拓扑马蹄理论,分析和验证了这些系统的超混沌行为。由于本项研究更为困难,文献中并未见到类似的工作。
3)为了实现混沌电路设计,首先本文在第五章研究了混沌系统设计问题,提出了基于反馈切换控制、拓扑马蹄构造、子系统耦合等三种设计方法,利用这些方法,设计出了多个混沌系统;然后,利用所得系统的鲁棒混沌性,适当调节参数,使系统状态方程得到简化,并易于通过(非)线性单元电路的搭接,将这些方程转化为实际电路,最终实现了多个混沌电路的设计。
4)为了使非混沌电路产生混沌,得到更简单的混沌电路,本文依据上述三种设计方法,通过分析Wien桥振荡电路的特点,提出了该电路的多种混沌化途径,使电路产生了混沌和超混沌行为,并借助拓扑马蹄理论这些行为进行了分析和判定。
本文工作使得对系统混沌的研究更加容易可靠,为研究更多的混沌系统提供方法和工具。此外,本文对混沌电路设计提供了基于切换反馈控制的若干新方法。
The complexity of behaviors in dynamics systems is not only an important problem in circuit systems, but also a core problem in many other systems with paramount scientific significance and engineering background. Therefore, how to analyze and determine these complicated behaviors, demonstrate the dynamics mechanisms of the system, and then achieve system control has become a series of principal problems of the scientific research in recently years. Among these problems, the chaoticity verification is the hardest one, since it needs the abstruse theory of dynamical systems and topological horseshoe, a great deal of numerical computation and many works on algorithm design. In this dissertation, we study two major problems on the chaotic system, i.e. chaotic system analysis and chaotic circuit design. The main study and contributions are given as follows:
1) In order to study the chaotic dynamics with one dimensional expansion, at first we provide a simple result on topological horseshoes which is very applicable for finding horseshoes with computers; then propose a method to find topological horseshoe with one dimensional expansion with this result, and the method is realized with a MATLAB GUI program; at last, we study the low dimensional Glass network and the RCLSJ(the resistive-capacitive-inductive shunted junction) model, the chaotic behaviors of the two systems are analyzed and proved by mean of topological horseshoe.
2) In order to study more complicated dynamics of chaotic phenomena, i.e. chaotic behavior with more than one directional expansion, at first we propose a technique to determine topological horseshoe with multidimensional expansion by providing a simple result on topological horseshoe; then propose a method to find this kind of horseshoe with this technique, and realize it with a MATLAB GUI program; at last, we study several famous hyperchaotic systems such as the hyperchaotic Hénon map, the Rossler hyperchaotic system, the Saito Hysteresis Chaos Generator (SHCG), and the Matsumoto-Chua-Kobayashi (MCK) circuit, the hyperchaotic behaviors are analyzed and verified by mean of topological horseshoe with multidimensional expansion. Because this research is very difficult, there is no similar work in literature according the author's knowledge.
3) In order to design chaotic circuits, at first we propose three methods to design a chaotic system, such as feedback switching control method, constructing topological horseshoe method, coupling subsystems method; we present many chaotic systems as examples to show these powerful method; since these system are robust in structure, we can obtain a simpler system by modifying their parameters, then we realize the simplified system with the chaotic circuits; finally, we get several chaotic circuits.
In order to make a nonchaotic circuit generate chaos or get a simpler chaotic circuit, we propose three ways to chaotify the Wien bridge oscillator with the three methods mentioned above. We obtain four simple chaotic circuits. Topological horseshoes are used here to analyze and verify the chaotic and hyperchaotic dynamics.
The contribution of this dissertation can make the research on chaos more reliable, and provide methods and tools for researchers to study many other chaotic systems. In addition, three new methods based on feedback switching control are also proposed to design chaotic circuits for the engineering purposes.
引文
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