具有关键节点的复杂动态相似网络聚类广义同步控制研究

Cluster synchronization of networks is ubiquitous in the fields of engineering, brain science, biological science and so on, thus, cluster synchronization is very worth of researching. There exists a class of real-world networks that possess some special features. For example, there are a few crucial nodes in each cluster. The nodes in the same cluster have different dynamics and even have different state dimensions. Besides, the networks are not always satisfy the dissipatively coupled condition. To realize the cluster synchronization for this kind of networks, the existing schemes are invalid. For this kind of networks and considering that it is the function of each node’s state vector that can be easily obtained in the realist networks, this project investigates the cluster generalized synchronization schemes for the complex dynamical networks with crucial nodes and dynamic similarities. Firstly, we will discuss the characters of the crucial nodes to obtain the significant parameters that represent their importance. And based on the theories of differential vector fields, we will discover the dynamic similarities between nodes in the same cluster when they are not controlled or only a few of them are controlled. Then, some kinds of multi-cluster complex dynamical network models with crucial nodes and dynamic similarities are to be established, which embodies those features mentioned above in realist networks. After that, the cluster generalized synchronization between the functions of the nodes’ state vectors will be defined. Finally, associated with the definition of cluster generalized synchronization, the information of crucial nodes and the dynamic similarities of nodes, many cluster generalized synchronization schemes will be proposed for our network models. We will try our best to reduce the number of controlled nodes and the control costs as far as possible in this project. The investigation of this project will enrich the control theory on cluster synchronization for complex dynamical networks and will bring great significance.

网络聚类同步广泛存在于工程、脑科学和生物科学等领域，具有重要研究价值。有一类现实网络，其分成多个聚类且每个聚类内有少量关键节点、同聚类节点不同甚至其状态维数也不同但有相似性、网络并非耗散耦合，现有成果已不能实现该类网络的聚类同步。针对这类网络并关注到实际网络能测得的经常是节点状态向量的函数，本项目研究具有关键节点的复杂动态相似网络聚类广义同步控制。首先，探讨关键节点的特征以获取表征其重要性的关键参数，并基于微分向量场理论给出同聚类节点在没有控制输入或少量节点有控制输入时动态相似性的定量描述。其次，建立多种具有关键节点的多聚类复杂动态相似网络模型以准确体现上述现实网络的特点。最后，提出网络节点间基于其状态向量函数的聚类广义同步概念，并充分运用关键节点和节点间的动态相似信息设计聚类广义同步控制策略以尽量减少受控节点数并降低控制成本。本项目的研究将丰富复杂动态网络聚类同步控制理论，具有重要意义。

网络聚类同步广泛存在于工程、脑科学和生物科学等领域，具有重要研究价值。现实网络中有一类是具有关键节点、节点维数不同但具有相似性、网络拓扑结构时变非耗散的聚类网络，该类网络的聚类广义同步值得深入研究。本课题针对该类网络，所做的主要研究内容和重要结果总结如下：(1)探讨了关键节点的特征以及同聚类节点与关键节点的相似性，这里的相似性，既包括在控制输入下的定量描述，也包括无控制输入时的定量表述，既有整个节点状态间的定量描述，也有节点部分动力学间的定量描述；(2)基于关键节点和相似性信息，根据实际网络的特征，建立了多种聚类网络模型，并给出了指数或渐近的聚类广义同步、矩阵投影聚类同步等概念；(3)针对建立的每类聚类网络模型，借助微分流形理论、Lyapunov稳定性理论、LaSalle不变集理论、Lyapunov- Krasovskii稳定性理论、控制理论等主要的理论分析工具，通过严谨的理论推导，分别设计了合适的聚类同步控制策略，并分别通过数值仿真验证了理论结果的有效性和正确性。本课题所得理论成果，已整理成论文发表，其中SCI已检索论文12篇。本课题研究表明：关键节点和相似信息，有助于设计控制策略实现控制目标；网络模型越复杂，或者对网络限制条件越少，控制策略将越复杂，但所得理论结果更具有普遍性和更广的适用范围。在未来的实际应用中，需要根据实际情况，权衡选择合适的控制策略。值得指出的是，本课题建立的网络模型充分关注了关键节点及同聚类中节点与关键节点的相似信息，拓展了聚类同步的概念，减弱了对节点动力学特征与网络拓扑结构特征的诸多限制条件，使得所提出的稳定与聚类同步控制策略更具有一般性和重要的理论意义。整个研究完成了预定的研究内容和数量指标。本课题的研究，在一定程度上丰富了复杂动态网络聚类同步控制理论，具有重要的理论意义。