若干复杂系统的同步化研究
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摘要
本文以同步方法的理论构建为基础，深入讨论了各种同步方法在现实中的应用潜力。同步作为一个非常普遍的自然现象，在其进化和发展的过程中，被广泛应用到许多领域，对我们的生活起着至关重要的作用。因此如何改进和优化同步方法、如何选择同步对象、如何把更先进的同步法应用到更复杂混沌系统或者新兴的复杂网络中，已经成为众多学者广泛关注的热点话题之一。与此同时，同步化的发展让我们更深层次地认识到复杂网络的内部特性，帮我们解释许多自然界中、现实社会中遇见的复杂现象，所以对复杂网络的同步化研究具有一定的实际意义和广泛的应用前景。本文首先通过一些具体例子介绍了在不同类型的实例中几种主要同步方法的各自优势，然后通过严格的理论推导和数值模拟，把这些同步法分别应用到时空混沌和复杂网络中去。全文的主要内容可以概括为如下几点：
第一章为绪论，首先给出了混沌同步的概念及意义、复杂网络概述及研究现状，然后交代了论文的主要研究内容及创新点，这些基础概念的介绍是展开后续讨论的基石。
第二章，通过主动滑模的控制理论，探讨了被干扰的时空混沌的修正函数时滞投影同步问题。控制律被应用到时空混沌系统中去，研究两个受干扰的混沌系统，发现了渐近延迟同步到指定的缩放函数矩阵。数值模拟实验结果证明主动滑模控制器的有效性。
第三章研究不确定参数时空混沌的自适应修正函数投影时滞同步。通过Lyapunov稳定性理论设计控制方案，当加入自适应控制器式，即可实现时空的驱动和响应系统的修正函数投影时滞同步，自适应控制律和参数识别法被应用到两个相同的时空Gray-Scott系统。数值结果表明所提出的方法的可行性和有效性。
第四章给出了关于复杂网络的时空混沌同步的三个控制方案。时间延迟、投影和随机干扰在同步过程中均被考虑。通过主动滑模，Backstepping和MasterStability Function控制律设计同步方案。数值结果验证了在时空混沌同步过程中三种方法的各自优点。
第五章研究了一类由不同时滞且参数未知节点构成的社团网络的同步和辨识问题。在LaSalle不变原理的基础上，通过设计有效的控制识别方案和耦合强度自适应率，提出了相应的同步准则。最终实现了复杂社团网络的周期同步。数值模拟的结果证明了所提出的方法的可行性。
第六章研究了一类由不同的时空节点构成的社团网络的同步问题。在时空LaSalle不变原理的基础上，通过自动调节自适应耦合强度，提出了相应的同步准则。Fitzhugh–Nagumo和Panfilov作为时空节点被应用到一个复杂社团网络的同步研究中去。数值模拟的结果证明了所提出的方法的可行性。
第七章研究了一类受扰的时变混沌和时空混沌的交叉投影同步问题。在主动滑模的基础上，设计了时变和时空混沌的交叉同步控制器，提出了相应的同步准则。两个受扰的混沌系统的状态变量渐近投影同步到所需的缩放矩阵，实现了时空混沌与时变混沌系统的交叉投影同步。数值模拟的结果证明了所提出的方法的可行性。
第八章为论文的总结与展望。
This paper is based on theoretical construction of synchronization methods.Applications potential of various synchronization methods in reality were discussed indepth. Synchronization is a very common natural phenomenon, in the long course of itsevolution and development, which has been applied to many fields extensively.Synchronization plays a key role in our life. Therefore, how to improve and optimizesynchronization methods, how to build and select synchronization objects, how to apply themore advanced methods of synchronization to even more complex chaotic systems oremerging complex networks, have become a hot topic and have attracted wide attentionfrom many scholars. At the same time, with the development of synchronization, innercharacters of complex networks are widely recognized, which can help us can explain manycomplex phenomena in nature and realistic society. Therefore, synchronization study ofcomplex network has some practical significance and wider application prospects. Thispaper contrasted the respective merits of different synchronization methods in theillustration of different classificatory by some specific examples. Through rigoroustheoretical derivations and numerical simulations, those synchronization methods wereapplied to the spatiotemporal chaos and complex networks. The main content of this papercan be summarized as follows:
Chapter one is an introduction. Fistly, the concept of chaotic synchronization andsignificance, a summary of complex networks, research status of complex networks werepresented. Then, main contents and innovative points of this paper were given. Theseintroduced basic concepts are the cornerstone of subsequent discussions.
Chapter two investigates modified function projective lag synchronization ofspatiotemporal chaos with disturbances. A control scheme is designed via active slidingmode control. The states of two chaotic systems with disturbances are asymptotically lagsynchronized up to a desired scaling function matrix. The control law is applied tospatiotemporal Gray-Scott systems. Numerical simulations are presented to demonstrate theeffectiveness of the proposed active sliding mode controllers.
Chapter three investigates modified function projective lag synchronization ofspatiotemporal chaos with uncertain parameters using adaptive method. A control scheme isdesigned via Lyapunov stability theory. The synchronization of spatiotemporal chaosbetween a drive system and a response system with time-delay is implemented by addingthe adaptive controllers. The adaptive control law and the parameter update law are appliedto two identical spatiotemporal Gray-Scott systems. Numerical results demonstrate thefeasibility and the effectiveness of the proposed approach.
In chapter four, three control schemes are presented in synchronization ofspatiotemporal chaos in complex networks. The time-delay, the projection and the randomdisturbances are accounted in the synchronization. The control laws are designed via the active sliding mode control, the backstepping control and the master stability function. Themerits of each scheme are numerically demonstrated in synchronization of spatiotemporalGray-Scott systems.
Chapter five initiates a novel approach for simultaneously identifying unknownparameters and synchronizing time-delayed complex community networks withnonidentical nodes. Based on the LaSalle’s invariance principle, a criterion is established byconstructing an effective control identification scheme and adjusting automatically theadaptive coupling strength. The proposed control law is applied to a complex communitynetwork which is periodically synchronized to a different chaotic trajectory. Numericalsimulations are presented to demonstrate the feasibility of the proposed methods.
Chapter six initiates an approach for simultaneously synchronizing time-delayedcomplex community networks with nonidentical spatiotemporal nodes. Based on theLaSalle’s invariance principle, a criterion is established by adjusting automatically theadaptive coupling strength. The Fitzhugh–Nagumo and Panfilov as spatiotemporal nodesare applied to the synchronization of a complex community network. Numerical simulationsare presented to demonstrate the feasibility of the proposed methods.
Chapter seven investigates cross projective synchronization of spatiotemporal andtime-varying chaos with disturbances. A control scheme is designed via active sliding modecontrol. The states of two chaotic systems with disturbances are asymptotically projectivesynchronized up to a desired scaling matrix. The control law is applied to spatiotemporaland time-varying systems. Numerical simulations are presented to demonstrate theeffectiveness of the cross projective synchronization between spatiotemporal andtime-varying chaos.
Chapter8is the summary and forecasting.
第一章为绪论，首先给出了混沌同步的概念及意义、复杂网络概述及研究现状，然后交代了论文的主要研究内容及创新点，这些基础概念的介绍是展开后续讨论的基石。
第二章，通过主动滑模的控制理论，探讨了被干扰的时空混沌的修正函数时滞投影同步问题。控制律被应用到时空混沌系统中去，研究两个受干扰的混沌系统，发现了渐近延迟同步到指定的缩放函数矩阵。数值模拟实验结果证明主动滑模控制器的有效性。
第三章研究不确定参数时空混沌的自适应修正函数投影时滞同步。通过Lyapunov稳定性理论设计控制方案，当加入自适应控制器式，即可实现时空的驱动和响应系统的修正函数投影时滞同步，自适应控制律和参数识别法被应用到两个相同的时空Gray-Scott系统。数值结果表明所提出的方法的可行性和有效性。
第四章给出了关于复杂网络的时空混沌同步的三个控制方案。时间延迟、投影和随机干扰在同步过程中均被考虑。通过主动滑模，Backstepping和MasterStability Function控制律设计同步方案。数值结果验证了在时空混沌同步过程中三种方法的各自优点。
第五章研究了一类由不同时滞且参数未知节点构成的社团网络的同步和辨识问题。在LaSalle不变原理的基础上，通过设计有效的控制识别方案和耦合强度自适应率，提出了相应的同步准则。最终实现了复杂社团网络的周期同步。数值模拟的结果证明了所提出的方法的可行性。
第六章研究了一类由不同的时空节点构成的社团网络的同步问题。在时空LaSalle不变原理的基础上，通过自动调节自适应耦合强度，提出了相应的同步准则。Fitzhugh–Nagumo和Panfilov作为时空节点被应用到一个复杂社团网络的同步研究中去。数值模拟的结果证明了所提出的方法的可行性。
第七章研究了一类受扰的时变混沌和时空混沌的交叉投影同步问题。在主动滑模的基础上，设计了时变和时空混沌的交叉同步控制器，提出了相应的同步准则。两个受扰的混沌系统的状态变量渐近投影同步到所需的缩放矩阵，实现了时空混沌与时变混沌系统的交叉投影同步。数值模拟的结果证明了所提出的方法的可行性。
第八章为论文的总结与展望。
This paper is based on theoretical construction of synchronization methods.Applications potential of various synchronization methods in reality were discussed indepth. Synchronization is a very common natural phenomenon, in the long course of itsevolution and development, which has been applied to many fields extensively.Synchronization plays a key role in our life. Therefore, how to improve and optimizesynchronization methods, how to build and select synchronization objects, how to apply themore advanced methods of synchronization to even more complex chaotic systems oremerging complex networks, have become a hot topic and have attracted wide attentionfrom many scholars. At the same time, with the development of synchronization, innercharacters of complex networks are widely recognized, which can help us can explain manycomplex phenomena in nature and realistic society. Therefore, synchronization study ofcomplex network has some practical significance and wider application prospects. Thispaper contrasted the respective merits of different synchronization methods in theillustration of different classificatory by some specific examples. Through rigoroustheoretical derivations and numerical simulations, those synchronization methods wereapplied to the spatiotemporal chaos and complex networks. The main content of this papercan be summarized as follows:
Chapter one is an introduction. Fistly, the concept of chaotic synchronization andsignificance, a summary of complex networks, research status of complex networks werepresented. Then, main contents and innovative points of this paper were given. Theseintroduced basic concepts are the cornerstone of subsequent discussions.
Chapter two investigates modified function projective lag synchronization ofspatiotemporal chaos with disturbances. A control scheme is designed via active slidingmode control. The states of two chaotic systems with disturbances are asymptotically lagsynchronized up to a desired scaling function matrix. The control law is applied tospatiotemporal Gray-Scott systems. Numerical simulations are presented to demonstrate theeffectiveness of the proposed active sliding mode controllers.
Chapter three investigates modified function projective lag synchronization ofspatiotemporal chaos with uncertain parameters using adaptive method. A control scheme isdesigned via Lyapunov stability theory. The synchronization of spatiotemporal chaosbetween a drive system and a response system with time-delay is implemented by addingthe adaptive controllers. The adaptive control law and the parameter update law are appliedto two identical spatiotemporal Gray-Scott systems. Numerical results demonstrate thefeasibility and the effectiveness of the proposed approach.
In chapter four, three control schemes are presented in synchronization ofspatiotemporal chaos in complex networks. The time-delay, the projection and the randomdisturbances are accounted in the synchronization. The control laws are designed via the active sliding mode control, the backstepping control and the master stability function. Themerits of each scheme are numerically demonstrated in synchronization of spatiotemporalGray-Scott systems.
Chapter five initiates a novel approach for simultaneously identifying unknownparameters and synchronizing time-delayed complex community networks withnonidentical nodes. Based on the LaSalle’s invariance principle, a criterion is established byconstructing an effective control identification scheme and adjusting automatically theadaptive coupling strength. The proposed control law is applied to a complex communitynetwork which is periodically synchronized to a different chaotic trajectory. Numericalsimulations are presented to demonstrate the feasibility of the proposed methods.
Chapter six initiates an approach for simultaneously synchronizing time-delayedcomplex community networks with nonidentical spatiotemporal nodes. Based on theLaSalle’s invariance principle, a criterion is established by adjusting automatically theadaptive coupling strength. The Fitzhugh–Nagumo and Panfilov as spatiotemporal nodesare applied to the synchronization of a complex community network. Numerical simulationsare presented to demonstrate the feasibility of the proposed methods.
Chapter seven investigates cross projective synchronization of spatiotemporal andtime-varying chaos with disturbances. A control scheme is designed via active sliding modecontrol. The states of two chaotic systems with disturbances are asymptotically projectivesynchronized up to a desired scaling matrix. The control law is applied to spatiotemporaland time-varying systems. Numerical simulations are presented to demonstrate theeffectiveness of the cross projective synchronization between spatiotemporal andtime-varying chaos.
Chapter8is the summary and forecasting.
引文
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