Clustering phenomenon of the singular Cucker–Smale model with finite communication weight and variable coupling strength

Abstract

The clustering phenomenon of the Cucker–Smale type model has received lots of attention from many researchers. However, due to the difficulty of the study, to the best of our knowledge, there have been a few discussions about clustering only in the initial configuration or one dimension that induces very simple movements of agents.In this study, we deal with the initial configuration in which the agents of the Cucker–Smale type system in multidimensional form bi-clustering while exhibiting more complex movements without collision. For this, we design a singular interacting system of Cucker–Smale type with finite communication range and variable coupling strength, and provide a mathematical proof that the system leads to a situation in which two disjoint groups of agents pass without any collision between the agents and without changing the composition of the agents in each group. To show the proof, we first briefly discuss the system’s collision-avoidance mechanism and present sufficient conditions along with an interaction topology for an initial configuration of agents and an estimate of variable coupling strength leading to the emergence of flocking. Finally, we propose sufficient conditions to induce the desired phenomenon. We also provide the results of several numerical simulations performed to establish the significance of the conditions in our analytical results.