Multi-cluster flocking behavior of the hierarchical Cucker-Smale model

In this paper, we investigate the multi-cluster flocking behavior of the hierarchical Cucker-Smale model parameterized by a constant β measuring the strength of the interaction between agents. The previous studies showed that when 0≤β<1/2 unconditional flocking would occur, while for β≥1/2 conditional flocking would occur provided the initial data satisfy some given conditions. We will show in this paper that, when β=1/2, unconditional flocking would occur for the model, while for β>1/2, we give a complete understanding for the asymptotic multi-cluster formation when the initial positions and velocities have the ordering relations: by computing the initial data we can specify how many clusters emerge and which agents are in the same cluster.