复杂网络上的同步及趋同性研究
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摘要
在本文中,我们探讨了复杂网络模型的两种动力学行为：同步和趋同性。
第一类是同步问题,我们在前人研究的基础上,主要研究了线性耦合的复杂网络上的几类同步问题：首先,我们研究由右端非Lipschitz甚至不连续的常微分方程所描述的动力系统通过线性耦合构成的复杂网络上的同步问题。结合了Fillipov关于右端不连续的微分方程理论和传统的同步分析的手段,我们给出了在此类复杂网络上实现完全同步的一个充分条件。如果把连续函数看成是不连续函数的特例,则我们的结果推广了前人关于满足Lipschitz条件或者QUAD条件的复杂网络上的同步分析的结果,使之可以应用到更多的网络上去。同时,以前的关于Lipschitz或者QUAD的条件也作为特例包含于我们的条件之中。其次,我们研究了线性耦合的复杂网络上的分群同步问题。在我们所研究的网络里,同一群内的节点具有相同的动力学性质,而不同的群之间的动力学方程则不同。在这一框架下,我们分别研究了连续时间和离散时间的网络模型,并给出了网络实现分群同步的充要条件。同时,我们还分别给出了通过自适应控制在这两类网络中实现分群同步的方案。最后,我们研究了动态结构网络上的完全同步问题。也就是,网络的耦合结构是随时间而变化的,我们首先给出了在一般情形下实现完全同步的一个充分条件。然后,我们又讨论了两种具体的随机切换的耦合结构：独立同分布过程和Markov过程。
第二类是趋同性问题。这是一类与同步问题密切关联的问题。首先,我们研究了具有随机切换的拓扑结构的加权网络上的趋同性问题。在这一框架下,我们也主要做了两方面的工作：一是在任意权重的复杂网络上的趋同性。这意味着描述网络结构的拓扑图上可以出现负权重的边,这可以用来刻画网络中两种相互竞争的倾向。我们给出了在这一情形下具有一般随机切换结构的网络实现趋同的充分条件。二是在网络的权重为非负的情况下,我们仍然假定网络的结构切换为一般的随机过程,不同的是这次我们用了一个适应过程来刻画它。在这一框架下,我们以条件期望的形式给出了网络实现趋同的充分条件。在这两种情形下,我们的研究都包含了离散时间网络和连续时间网络。并且我们都给出了一般理论结果在切换过程为独立同分布和Markov链的情形下的具体推论。其次,我们还研究了网络在不连续的趋同性协议下的趋同性问题。我们也是同时研究了固定网络结构和切换网络结构。在固定网络结构的情形下,我们给出了网络实现趋同性的一个充要条件。而在网络结构随机切换的情形下,我们则给出了网络实现几乎必然趋同性的一个充分条件。
In this dissertation, we mainly discuss two aspects of problems concerning the dynamics in complex networks. One is synchronization and the other is consensus.
In the part of synchronization analysis, we investigate the following problems:
·Complete synchronization. We investigate the complete synchronization in complex networks of linearly coupled dynamical systems described by dif-ferential equations with non-Lipschitz or even discontinuous righthand sides; By integrating the Fillipov theory of differential equations with discontinuous righthand sides and the classical synchronization analysis as the basic tool, we give some sufficient conditions for complete synchronization of such net-works. These conditions can be seen as a generalization of previous results on synchronization of complex networks of linearly coupled dynamical systems described by differential equations with continuous righthand sides.
·Cluster synchronization. We investigate cluster synchronization of lin-early coupled nonidentical dynamical systems. In such networks, the nodes of the complex networks are divided into serval clusters. The nodes in the same cluster have the same dynamical properties, while different clusters have distinguishable nodes. Under this framework, we investigate the cluster syn-chronization in both discrete-time and continuous-time network models. In both cases, we provide the sufficient and necessary conditions for the cluster synchronization of the networks. Furthermore, we propose an adaptive control scheme to realize cluster synchronization in these networks.
·Synchronization under time-varying coupling We investigate complete synchronization in networks with time-varying coupling. First, we give a suf-ficient condition for a class of such networks to achieve complete synchroniza-tion. Then, we discuss two special stochastic switching processes:independent and identically distributed process and Markov process. In both cases, we have give sufficient conditions for the network to achieve complete synchronization almost surely.
In the part of consensus analysis, we investigate the following consensus prob-lems in networks of multiagents:
·Consensus in networks with cooperation and competition. It means that the underlying graph topologies can have both positive and negative weights, which can be used to describe cooperative and competitive trends in the networks. We give sufficient conditions for almost sure consensus in both discrete-time and continuous-time network models with general stochastically switching topologies. And the results are applied to the switching topologies of independent and identically distributed process and Markov chains.
·Consensus in networks with adapted switching processes. When the weights are nonnegative, we use an adapted process to describe the topology switching process. Under this framework, we provide sufficient conditions for almost sure consensus and moment consensus, called Lp consensus, which is the first time to be introduced and is equivalent to almost sure consensus in our case. The conditions are in terms of conditional expectations of the union of the graphs with a fixed length and include both discrete-time and continuous-time network models. And the results apply to the switching topologies of independent and identically distributed process and Markov chains.
第一类是同步问题,我们在前人研究的基础上,主要研究了线性耦合的复杂网络上的几类同步问题：首先,我们研究由右端非Lipschitz甚至不连续的常微分方程所描述的动力系统通过线性耦合构成的复杂网络上的同步问题。结合了Fillipov关于右端不连续的微分方程理论和传统的同步分析的手段,我们给出了在此类复杂网络上实现完全同步的一个充分条件。如果把连续函数看成是不连续函数的特例,则我们的结果推广了前人关于满足Lipschitz条件或者QUAD条件的复杂网络上的同步分析的结果,使之可以应用到更多的网络上去。同时,以前的关于Lipschitz或者QUAD的条件也作为特例包含于我们的条件之中。其次,我们研究了线性耦合的复杂网络上的分群同步问题。在我们所研究的网络里,同一群内的节点具有相同的动力学性质,而不同的群之间的动力学方程则不同。在这一框架下,我们分别研究了连续时间和离散时间的网络模型,并给出了网络实现分群同步的充要条件。同时,我们还分别给出了通过自适应控制在这两类网络中实现分群同步的方案。最后,我们研究了动态结构网络上的完全同步问题。也就是,网络的耦合结构是随时间而变化的,我们首先给出了在一般情形下实现完全同步的一个充分条件。然后,我们又讨论了两种具体的随机切换的耦合结构：独立同分布过程和Markov过程。
第二类是趋同性问题。这是一类与同步问题密切关联的问题。首先,我们研究了具有随机切换的拓扑结构的加权网络上的趋同性问题。在这一框架下,我们也主要做了两方面的工作：一是在任意权重的复杂网络上的趋同性。这意味着描述网络结构的拓扑图上可以出现负权重的边,这可以用来刻画网络中两种相互竞争的倾向。我们给出了在这一情形下具有一般随机切换结构的网络实现趋同的充分条件。二是在网络的权重为非负的情况下,我们仍然假定网络的结构切换为一般的随机过程,不同的是这次我们用了一个适应过程来刻画它。在这一框架下,我们以条件期望的形式给出了网络实现趋同的充分条件。在这两种情形下,我们的研究都包含了离散时间网络和连续时间网络。并且我们都给出了一般理论结果在切换过程为独立同分布和Markov链的情形下的具体推论。其次,我们还研究了网络在不连续的趋同性协议下的趋同性问题。我们也是同时研究了固定网络结构和切换网络结构。在固定网络结构的情形下,我们给出了网络实现趋同性的一个充要条件。而在网络结构随机切换的情形下,我们则给出了网络实现几乎必然趋同性的一个充分条件。
In this dissertation, we mainly discuss two aspects of problems concerning the dynamics in complex networks. One is synchronization and the other is consensus.
In the part of synchronization analysis, we investigate the following problems:
·Complete synchronization. We investigate the complete synchronization in complex networks of linearly coupled dynamical systems described by dif-ferential equations with non-Lipschitz or even discontinuous righthand sides; By integrating the Fillipov theory of differential equations with discontinuous righthand sides and the classical synchronization analysis as the basic tool, we give some sufficient conditions for complete synchronization of such net-works. These conditions can be seen as a generalization of previous results on synchronization of complex networks of linearly coupled dynamical systems described by differential equations with continuous righthand sides.
·Cluster synchronization. We investigate cluster synchronization of lin-early coupled nonidentical dynamical systems. In such networks, the nodes of the complex networks are divided into serval clusters. The nodes in the same cluster have the same dynamical properties, while different clusters have distinguishable nodes. Under this framework, we investigate the cluster syn-chronization in both discrete-time and continuous-time network models. In both cases, we provide the sufficient and necessary conditions for the cluster synchronization of the networks. Furthermore, we propose an adaptive control scheme to realize cluster synchronization in these networks.
·Synchronization under time-varying coupling We investigate complete synchronization in networks with time-varying coupling. First, we give a suf-ficient condition for a class of such networks to achieve complete synchroniza-tion. Then, we discuss two special stochastic switching processes:independent and identically distributed process and Markov process. In both cases, we have give sufficient conditions for the network to achieve complete synchronization almost surely.
In the part of consensus analysis, we investigate the following consensus prob-lems in networks of multiagents:
·Consensus in networks with cooperation and competition. It means that the underlying graph topologies can have both positive and negative weights, which can be used to describe cooperative and competitive trends in the networks. We give sufficient conditions for almost sure consensus in both discrete-time and continuous-time network models with general stochastically switching topologies. And the results are applied to the switching topologies of independent and identically distributed process and Markov chains.
·Consensus in networks with adapted switching processes. When the weights are nonnegative, we use an adapted process to describe the topology switching process. Under this framework, we provide sufficient conditions for almost sure consensus and moment consensus, called Lp consensus, which is the first time to be introduced and is equivalent to almost sure consensus in our case. The conditions are in terms of conditional expectations of the union of the graphs with a fixed length and include both discrete-time and continuous-time network models. And the results apply to the switching topologies of independent and identically distributed process and Markov chains.
引文
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