具有多拓扑结构的时滞集群模型的分组分析

Clustering Analysis for the Delayed Flocking Model with Multi-topology Structures

集群行为是自然界的常见现象.集群理论的研究在经济,军事,管理等领域具有广泛的应用价值.本文研究两类具有不同拓扑结构和时滞影响的集群模型的分组行为,获得了集群分组的充分必要条件.对于具有有向图的Cucker-Smale型分组模型,考虑时滞的影响,给出了时滞的充分必要条件.同样地对于具有二分图结构且个体之间是竞争关系的分组模型,我们也探索了时滞对分组行为的影响规律,给出了时滞的充分必要条件.最后,通过数值仿真方法验证了结论的正确性.

Flocking behavior is a common phenomenon in nature.The research of flocking theory has been widely applied in economy,military and management.This paper studies the group behavior of two models with time delays under different topology structures,and obtains necessary and sufficient conditions.For the Cucker-Smale type model with directed graphs,we get the necessary and sufficient conditions of time delay.For another model with competitive relationship between individuals under bipartite graph,we also explore the influence of time delay on group behavior,and get the necessary and sufficient conditions.Finally,we verify the correctness of the conclusion by numerical simulation.