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

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。

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.