高维和时滞混沌系统的理论研究及电路实现
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摘要
在过去的二十多年中,混沌科学逐渐从简单的科学好奇过渡到重要的实际应用阶段。自从Lorenz系统作为研究混沌的典范以来,人们认为混沌提供了一个丰富的信号设计和发生机制,它们在数据加密、保密通信和信息处理等方面有潜在的应用前景。值得注意的是,有目的的产生和强化混沌是许多工程技术领域的一个研究热点。
沿着这个研究趋势,本论文重点研究了三个方面的问题:非线性电路中混沌的产生机制,混沌系统的电路设计和混沌系统的自适应同步。本论文主要贡献如下:
首先回顾了与论文研究相关的知识,包括混沌的定义、混沌运动的特征、混沌的判据和规则以及非线性电路中的混沌现象。人们知道,混沌现象不可能在一阶、二阶自治连续时间系统中产生,也不可能在一阶非自治连续时间系统中产生。但是人们在三阶自治连续时间系统中发现了混沌现象。本文对几个典型的三维混沌系统和电路实现进行了详细地分析。
针对低维混沌系统的安全性较低、容易受到威胁的缺陷,我们考虑增强非线性系统的混沌行为,提出了一种四维混沌电路的设计方案,该电路可以作为一个高维混沌发生器,在保密通信和图像加密中有潜在的应用。
但是,随着维数的增加,混沌系统的物理实现变得更加困难。幸运的是,如果加入时延,简单神经元就能够展示出混沌行为。人们发现:由两个或三个非线性神经元组成,每个神经元都有一个sigmoid传递函数的时延神经网络能产生混沌行为。我们分析了时滞非线性系统的稳定性、Hopf分岔、极限环的失稳和混沌现象,深入探讨了时滞非线性系统中的混沌产生机制。
在上述研究基础上,本文提出了带有非单调激活函数的时滞混沌神经元模型电路设计方法。我们设计的廖氏时滞混沌神经元电路既可以作为独立的时滞混沌发生器,也可以作为一个混沌神经元电路单元,甚至可以看成是细胞神经网络中的一个细胞,在保密通信和信息处理中有潜在的应用价值。值得一提的是:我们发现了一个特定的非线性电路系统,并对其稳定性、Hopf分岔的存在性进行了理论分析,构建了相应的时滞混沌电路系统,其输出波形和相图具有明显的混沌特征。
最后,我们研究了混沌同步控制。混沌同步吸引了越来越多的关注,人们在研究中发现,混沌同步在许多领域,特别是生理学、非线性光学和保密通信中起着重要的作用。利用现代控制技术,论文中研究了高维混沌系统和时滞混沌系统
Over the last two decades, chaos has gradually moved from simply being a scientific curiosity to a promising subject with practical significance and applications. Since the first paradigm of the well-known Lorenz system,chaotic systems have been considered as a rich mechanism for signal design and generation and they have potential applications to data encryption, secure communications and signal processing. It has been noticed that, purposefully generating or enhancing chaos can be a key issue in many technological and engineering applications.
This thesis addresses three important issues involved in chaotic systems: mechanism of generating chaos in nonlinear circuits, construction of chaos generators and adaptive chaos synchronization. Specifically, the main contributions of this thesis are presented as follows.
The preliminary knowledge involved in this thesis, including the definitions of chaos, the characteristics, the criterions and rules of chaos, and chaos phenomena in nonlinear circuits, are introduced.
It is well known that chaos can neither occur in first- or second-order autonomous continuous-time system, nor in a first-order non-autonomous continuous-time system. Chaos has been found in three-order autonomous continuous-time system. Several typical chaos systems and circuitry implementation are analyzed in detail.
Aiming at low-dimensional systems having low security to be easily captured, enhancement of chaos in nonlinear system should be considered. An electronic analog of a new four-dimensional continuous autonomous chaotic system is described in detail. The proposed circuit may be used as chaos generator and has potential applications to communications and signal processing.
However, with the increase of dimension, the system become more complex and hard to implement with electronic circuitry. Fortunately, it is found that simple neural networks with time delays can exhibit chaotic dynamical behavior. Chaotic phenomena for delayed neural networks consisting of two or three non-linear neurons have also been found. In this thesis, the local stability and existence of Hopf bifurcation, the loss of stability of a limit circle and the numerical transition to chaos of delayed differential equations are studied.
Based on the above work, a circuitry implementation of chaotic Liao’s delayed
沿着这个研究趋势,本论文重点研究了三个方面的问题:非线性电路中混沌的产生机制,混沌系统的电路设计和混沌系统的自适应同步。本论文主要贡献如下:
首先回顾了与论文研究相关的知识,包括混沌的定义、混沌运动的特征、混沌的判据和规则以及非线性电路中的混沌现象。人们知道,混沌现象不可能在一阶、二阶自治连续时间系统中产生,也不可能在一阶非自治连续时间系统中产生。但是人们在三阶自治连续时间系统中发现了混沌现象。本文对几个典型的三维混沌系统和电路实现进行了详细地分析。
针对低维混沌系统的安全性较低、容易受到威胁的缺陷,我们考虑增强非线性系统的混沌行为,提出了一种四维混沌电路的设计方案,该电路可以作为一个高维混沌发生器,在保密通信和图像加密中有潜在的应用。
但是,随着维数的增加,混沌系统的物理实现变得更加困难。幸运的是,如果加入时延,简单神经元就能够展示出混沌行为。人们发现:由两个或三个非线性神经元组成,每个神经元都有一个sigmoid传递函数的时延神经网络能产生混沌行为。我们分析了时滞非线性系统的稳定性、Hopf分岔、极限环的失稳和混沌现象,深入探讨了时滞非线性系统中的混沌产生机制。
在上述研究基础上,本文提出了带有非单调激活函数的时滞混沌神经元模型电路设计方法。我们设计的廖氏时滞混沌神经元电路既可以作为独立的时滞混沌发生器,也可以作为一个混沌神经元电路单元,甚至可以看成是细胞神经网络中的一个细胞,在保密通信和信息处理中有潜在的应用价值。值得一提的是:我们发现了一个特定的非线性电路系统,并对其稳定性、Hopf分岔的存在性进行了理论分析,构建了相应的时滞混沌电路系统,其输出波形和相图具有明显的混沌特征。
最后,我们研究了混沌同步控制。混沌同步吸引了越来越多的关注,人们在研究中发现,混沌同步在许多领域,特别是生理学、非线性光学和保密通信中起着重要的作用。利用现代控制技术,论文中研究了高维混沌系统和时滞混沌系统
Over the last two decades, chaos has gradually moved from simply being a scientific curiosity to a promising subject with practical significance and applications. Since the first paradigm of the well-known Lorenz system,chaotic systems have been considered as a rich mechanism for signal design and generation and they have potential applications to data encryption, secure communications and signal processing. It has been noticed that, purposefully generating or enhancing chaos can be a key issue in many technological and engineering applications.
This thesis addresses three important issues involved in chaotic systems: mechanism of generating chaos in nonlinear circuits, construction of chaos generators and adaptive chaos synchronization. Specifically, the main contributions of this thesis are presented as follows.
The preliminary knowledge involved in this thesis, including the definitions of chaos, the characteristics, the criterions and rules of chaos, and chaos phenomena in nonlinear circuits, are introduced.
It is well known that chaos can neither occur in first- or second-order autonomous continuous-time system, nor in a first-order non-autonomous continuous-time system. Chaos has been found in three-order autonomous continuous-time system. Several typical chaos systems and circuitry implementation are analyzed in detail.
Aiming at low-dimensional systems having low security to be easily captured, enhancement of chaos in nonlinear system should be considered. An electronic analog of a new four-dimensional continuous autonomous chaotic system is described in detail. The proposed circuit may be used as chaos generator and has potential applications to communications and signal processing.
However, with the increase of dimension, the system become more complex and hard to implement with electronic circuitry. Fortunately, it is found that simple neural networks with time delays can exhibit chaotic dynamical behavior. Chaotic phenomena for delayed neural networks consisting of two or three non-linear neurons have also been found. In this thesis, the local stability and existence of Hopf bifurcation, the loss of stability of a limit circle and the numerical transition to chaos of delayed differential equations are studied.
Based on the above work, a circuitry implementation of chaotic Liao’s delayed
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