Communications in Mathematical Sciences

Volume 16 (2018)

Number 7

A first-order reduction of the Cucker–Smale model on the real line and its clustering dynamics

Pages: 1907 – 1931



Seung-Yeal Ha (Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, South Korea; and Korea Institute for Advanced Study, Hoegiro, Seoul, South Korea)

Jinyeong Park (Department of Mathematics and Research Institute of Natural Sciences, Hanyang University, Seoul, South Korea)

Xiongtao Zhang (Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, China)


We present a first-order reduction for the Cucker–Smale (C–S) model on the real line, and discuss its clustering dynamics in terms of spatial configurations and system parameters. In previous literature, flocking estimates for the C–S model were mainly focused on the relaxation dynamics of the particle’s velocities toward the common velocity. In contrast, the relaxation dynamics of spatial configurations was treated as a secondary issue except for the uniform boundedness of the spatial diameter. In this paper, we first derive a first-order system for the spatial coordinate that can be rewritten as a gradient flow, and then use this first-order formulation to derive several sufficient conditions on the clustering dynamics based on the spatial positions depending on the natural velocities characterized by initial position-velocity configurations.


clustering, collective dynamics, Cucker–Smale model, flocking, synchronization

2010 Mathematics Subject Classification

70F99, 82C22

The work of S.-Y. Ha is partially supported by the Samsung Science and Technology Foundation under project number SSTF-BA1401-03. The work of J. Kim was supported by the German Research Foundation (DFG) under project number IRTG 2235. The work of J. Park was supported by the research fund of Hanyang University (HY-2018). The work of X. Zhang is supported by Scientific Research Foundation of Huazhong University of Science and Technology.

Received 29 March 2017

Accepted 22 July 2018

Published 7 March 2019