复杂动力网络的同步与拓扑识别
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摘要
同步在各种自然界或人造的系统中是一个广泛存在的现象,比如,动物群落的迁移、电力网格、传感器网络、细胞神经网络,等等.20世纪末,自从人们发现真实网络的小世界和无尺度的特征后,就开始关注这些复杂网络上的同步问题.将节点动力学与复杂网络的拓扑结构结合起来考虑,提出一个新的研究领域-复杂动力网络,它是刻画不同网络拓扑结构和动力学性质的一种有力工具.
本文在得到几类复杂动力学网络同步准则的基础上,主要研究了如何通过复杂动力学网络的动力学性质来识别复杂网络的未知拓扑结构,复杂网络的拓扑结构特征如何影响其动力学网络的广义同步行为,以及脉冲耦合方式如何影响离散动力学网络的同步能力.
本论文主要内容分为六章,第一章简要介绍了与后文相关的一些非线性动力学和复杂网络的预备知识,并指出目前复杂动力学网络的研究现状.二至五章为文章的正文,详细介绍了本文的工作及创新点.第六章是我们对今后工作的展望.全文的主要内容和思想概括为如下几个方面：
1)拓扑识别和参数识别在复杂网络的研究中是具有挑战性的问题.利用同步理论,我们研究含有未知系统参数的复杂动力学网络的拓扑识别问题,提出了同时识别系统不确定参数和未知网络拓扑结构的自适应法则.通过严格的理论分析,指出了两簇函数组在同步流形上的线性无关性是保证网络拓扑识别正确的必要条件.此外,我们利用主稳定函数的方法对相同节点动力学的网络,分析了其耦合强度大小对网络拓扑结构识别的影响.对于不同节点动力学的网络,分析了节点动力学的差异性对网络拓扑识别效率的影响.
2)在各种现实的复杂动力学网络中,时滞常常在状态变量和耦合因子中出现.基于以上因素,提出了一种新的方法同时识别含时滞的复杂动力学网络的未知拓扑结构及不确定的节点动力学参数.该方法也适用于含不同节点的未知的时滞复杂动力学网络,并能在线追踪演化的网络拓扑结构.
3)广义同步是比完全同步更弱的一种同步现象.它在真实世界中广泛地存在.基于节点的局部广义同步信息,我们提出自适应地调节各节点耦合强度的策略,从而来实现复杂动力学网络的广义同步.采用辅助系统方法和Lyapunov函数的方法,证明了对任意给定的初始耦合强度,由不同节点构成的复杂动力学网络上都能发生广义同步.我们对三类典型网络的广义同步行为进行了比较分析：无尺度网络模型、小世界网络模型、以及一簇介于无尺度网络与ER随机网络之间的网络模型.研究发现,由于度分布的异质性,无标度网络在广义同步的过程中具有分层特征,而小世界网络不具此特征.并且,我们详细地讨论了网络度分布的异质性是如何影响网络广义同步行为的.
4)离散的动力学网络模型在自然界和人造的系统中也是一类典型的例子.在过去的十年里,连续动力学网络的脉冲控制及同步得到了广泛的研究.但是对于离散动力学网络的脉冲控制和同步问题的研究还很少.我们提出离散动力学网络的脉冲耦合模型,即耦合作用只在某些动力学演化时刻发生.得到了该脉冲耦合的离散动力学网络模型的全局同步准则和局部同步准则.在给定节点动力学的前提下,指出其脉冲耦合矩阵,脉冲耦合强度以及脉冲区间的长度应满足的关系.
Synchronization is related to various research topics in natural and man-made systems, such as animal groups, power grids, sensor networks, cellular neural networks, and so on. In recent years, more and more researchers have become to be interested in the synchronization behavior in complex dynamical networks, since the "scale-free" and "small-world" properties of real-world net-works were discovered in the end of 20th century. Scientists jointly consider topological structures of complex networks and their dynamical behaviors, then a new research discipline comes out-complex dynamical networks, which is an efficient tool to study different network topologies and their dynamics character-istic.
Based on several synchronization criteria of four kinds of complex dynamical networks, the thesis has studied on the following topics:how to identify uncertain topological structures of complex dynamical networks via time series of nodes' dynamics; how topological structures of complex networks affect their general-ized synchronization behaviors; how the impulsive coupling strategy affects the synchronizability of discrete dynamical networks.
The thesis is composed of six chapters. In Chapter 1, we will introduce some fundamental knowledge and key concepts of nonlinear dynamical systems and complex networks. And we will also summarize some present work on the synchronization of complex dynamical networks which are related to the topic of the thesis. Then, main results and ideas of our work will be given in Chapter 2-Chapter 5. In Chapter 6, some outlooks of our further research work are discussed. The main contents and innovation points are summarized as follows.
1) Topology identification and parameter identification are challenging ques-tions in complex networks. By the synchronization-based estimation theory, iden-tification of the topological structure and unknown parameters of a complex dy-namical network with nonidentical nodes/identical nodes is carefully studied. Based on rigorously theoretical analysis, it points out that the so-called linear in-dependency of drive signals is essential for an effective and correct estimation of the topological structure and unknown parameters. Moreover, how the coupling strength affects the topological identification is analyzed through the method of Master Stability Function. And one key factor-nodes' dissimilarity-that de-termine the efficiency of the proposed adaptive control approach is then further investigated.
2) Time delay often appears in the state variables or coupling coefficients of various practical complex networks. The paper initiates a novel approach for simultaneously identifying the topological structure and unknown parameters of uncertain general complex networks with time delay. In particular, this method is also effective for uncertain delayed complex dynamical networks with differ-ent node dynamics. Moreover, the proposed method can be easily extended to monitor the on-line evolution of network topological structure.
3) Generalized synchronization, which is weaker than complete synchroniza-tion, plays an important role in many networked systems. The proposed general-ized synchronization strategy is to adjust adaptively a node's coupling strength based on the node's local generalized synchronization information. By taking the auxiliary-system approach and using the Lyapunov function method, we prove that for any given initial coupling strengths, the generalized synchronization can take place in complex networks consisting of nonidentical dynamical systems. We investigates generalized synchronization in three typical classes of complex dy-namical networks:scale-free networks, small-world networks, and interpolating networks. It is demonstrated that the coupling strengths are affected by topolo-gies of the networks. Furthermore, it is found that there are hierarchical features in the processes of generalized synchronization in scale-free networks because of their highly heterogeneous distributions of connection degree. Finally, we dis-cuss in detail how a network's degree of heterogeneity affects its generalization synchronization behavior.
4) Over the last decade, impulsive control and synchronization of continu-ous dynamical networks has been extensively investigated in various disciplines. However, impulsive control and synchronization of discrete dynamical networks has only lightly been covered. In this paper, a novel model is proposed for the synchronization of a class of discrete dynamical networks through impulsive cou-plings. Moreover, the global and local stability of synchronization manifold are then further studied. As a result, several effective synchronization criteria are attained, which describe conditions for the impulsively coupling matrix, coupling strengths, and the impulsive intervals, as the nodes' dynamics is given.
本文在得到几类复杂动力学网络同步准则的基础上,主要研究了如何通过复杂动力学网络的动力学性质来识别复杂网络的未知拓扑结构,复杂网络的拓扑结构特征如何影响其动力学网络的广义同步行为,以及脉冲耦合方式如何影响离散动力学网络的同步能力.
本论文主要内容分为六章,第一章简要介绍了与后文相关的一些非线性动力学和复杂网络的预备知识,并指出目前复杂动力学网络的研究现状.二至五章为文章的正文,详细介绍了本文的工作及创新点.第六章是我们对今后工作的展望.全文的主要内容和思想概括为如下几个方面：
1)拓扑识别和参数识别在复杂网络的研究中是具有挑战性的问题.利用同步理论,我们研究含有未知系统参数的复杂动力学网络的拓扑识别问题,提出了同时识别系统不确定参数和未知网络拓扑结构的自适应法则.通过严格的理论分析,指出了两簇函数组在同步流形上的线性无关性是保证网络拓扑识别正确的必要条件.此外,我们利用主稳定函数的方法对相同节点动力学的网络,分析了其耦合强度大小对网络拓扑结构识别的影响.对于不同节点动力学的网络,分析了节点动力学的差异性对网络拓扑识别效率的影响.
2)在各种现实的复杂动力学网络中,时滞常常在状态变量和耦合因子中出现.基于以上因素,提出了一种新的方法同时识别含时滞的复杂动力学网络的未知拓扑结构及不确定的节点动力学参数.该方法也适用于含不同节点的未知的时滞复杂动力学网络,并能在线追踪演化的网络拓扑结构.
3)广义同步是比完全同步更弱的一种同步现象.它在真实世界中广泛地存在.基于节点的局部广义同步信息,我们提出自适应地调节各节点耦合强度的策略,从而来实现复杂动力学网络的广义同步.采用辅助系统方法和Lyapunov函数的方法,证明了对任意给定的初始耦合强度,由不同节点构成的复杂动力学网络上都能发生广义同步.我们对三类典型网络的广义同步行为进行了比较分析：无尺度网络模型、小世界网络模型、以及一簇介于无尺度网络与ER随机网络之间的网络模型.研究发现,由于度分布的异质性,无标度网络在广义同步的过程中具有分层特征,而小世界网络不具此特征.并且,我们详细地讨论了网络度分布的异质性是如何影响网络广义同步行为的.
4)离散的动力学网络模型在自然界和人造的系统中也是一类典型的例子.在过去的十年里,连续动力学网络的脉冲控制及同步得到了广泛的研究.但是对于离散动力学网络的脉冲控制和同步问题的研究还很少.我们提出离散动力学网络的脉冲耦合模型,即耦合作用只在某些动力学演化时刻发生.得到了该脉冲耦合的离散动力学网络模型的全局同步准则和局部同步准则.在给定节点动力学的前提下,指出其脉冲耦合矩阵,脉冲耦合强度以及脉冲区间的长度应满足的关系.
Synchronization is related to various research topics in natural and man-made systems, such as animal groups, power grids, sensor networks, cellular neural networks, and so on. In recent years, more and more researchers have become to be interested in the synchronization behavior in complex dynamical networks, since the "scale-free" and "small-world" properties of real-world net-works were discovered in the end of 20th century. Scientists jointly consider topological structures of complex networks and their dynamical behaviors, then a new research discipline comes out-complex dynamical networks, which is an efficient tool to study different network topologies and their dynamics character-istic.
Based on several synchronization criteria of four kinds of complex dynamical networks, the thesis has studied on the following topics:how to identify uncertain topological structures of complex dynamical networks via time series of nodes' dynamics; how topological structures of complex networks affect their general-ized synchronization behaviors; how the impulsive coupling strategy affects the synchronizability of discrete dynamical networks.
The thesis is composed of six chapters. In Chapter 1, we will introduce some fundamental knowledge and key concepts of nonlinear dynamical systems and complex networks. And we will also summarize some present work on the synchronization of complex dynamical networks which are related to the topic of the thesis. Then, main results and ideas of our work will be given in Chapter 2-Chapter 5. In Chapter 6, some outlooks of our further research work are discussed. The main contents and innovation points are summarized as follows.
1) Topology identification and parameter identification are challenging ques-tions in complex networks. By the synchronization-based estimation theory, iden-tification of the topological structure and unknown parameters of a complex dy-namical network with nonidentical nodes/identical nodes is carefully studied. Based on rigorously theoretical analysis, it points out that the so-called linear in-dependency of drive signals is essential for an effective and correct estimation of the topological structure and unknown parameters. Moreover, how the coupling strength affects the topological identification is analyzed through the method of Master Stability Function. And one key factor-nodes' dissimilarity-that de-termine the efficiency of the proposed adaptive control approach is then further investigated.
2) Time delay often appears in the state variables or coupling coefficients of various practical complex networks. The paper initiates a novel approach for simultaneously identifying the topological structure and unknown parameters of uncertain general complex networks with time delay. In particular, this method is also effective for uncertain delayed complex dynamical networks with differ-ent node dynamics. Moreover, the proposed method can be easily extended to monitor the on-line evolution of network topological structure.
3) Generalized synchronization, which is weaker than complete synchroniza-tion, plays an important role in many networked systems. The proposed general-ized synchronization strategy is to adjust adaptively a node's coupling strength based on the node's local generalized synchronization information. By taking the auxiliary-system approach and using the Lyapunov function method, we prove that for any given initial coupling strengths, the generalized synchronization can take place in complex networks consisting of nonidentical dynamical systems. We investigates generalized synchronization in three typical classes of complex dy-namical networks:scale-free networks, small-world networks, and interpolating networks. It is demonstrated that the coupling strengths are affected by topolo-gies of the networks. Furthermore, it is found that there are hierarchical features in the processes of generalized synchronization in scale-free networks because of their highly heterogeneous distributions of connection degree. Finally, we dis-cuss in detail how a network's degree of heterogeneity affects its generalization synchronization behavior.
4) Over the last decade, impulsive control and synchronization of continu-ous dynamical networks has been extensively investigated in various disciplines. However, impulsive control and synchronization of discrete dynamical networks has only lightly been covered. In this paper, a novel model is proposed for the synchronization of a class of discrete dynamical networks through impulsive cou-plings. Moreover, the global and local stability of synchronization manifold are then further studied. As a result, several effective synchronization criteria are attained, which describe conditions for the impulsively coupling matrix, coupling strengths, and the impulsive intervals, as the nodes' dynamics is given.
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