Complete Cluster Predictability of the Cucker-Smale Flocking Model on the Real Line

Abstract

We present the complete predictability of clustering for the Cucker-Smale (C-S) model on the line. Emergence of multi-cluster flocking is often observed in numerical simulations for the C-S model with short-range interactions. However, the explicit computation of the number of emergent multi-clusters a priori is a challenging problem for the Cucker-Smale flocking model. In this paper, we present an explicit criterion and algorithm to calculate the number of clusters and their bulk velocities in terms of initial configuration, coupling strength and communication weight function in a one-dimensional setting. We present a finite increasing sequence of coupling strengths in which the number of asymptotic clusters has a jump. We also provide several numerical examples and compare them with analytical results.