Flocking behaviours of a delayed collective model with local rule and critical neighbourhood situation

摘要

It is of particular significance in both theories and applications to understand how self-organized particles use limited environment and simple rules to organize into ordered emergence. In this paper, we study a modified Cucker-Smale-type system with a simple and local cut-off weight. Also, a communication delay is introduced into both the velocity adjoint terms and the cut-off weight. We extend the previous criteria for flocking to the delayed model, using an approach based on the invariant subspace decomposition in the infinite-dimensional space. For the noncritical neighbourhood situation, a criterion of flocking emergence with an exponential convergent rate is established by the standard arguments on the functional differential system. For the general neighbourhood situation, when all the switching intervals have a uniform minimum gap, another criterion of flocking emergence is also found.