Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 319–365 | Cite as

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

  • Seung-Yeal Ha
  • Jeongho Kim
  • Jinyeong ParkEmail author
  • Xiongtao Zhang


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.


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The authors would like to thank the anonymous referee for his/her careful reading and constructive comments on the original manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulRepublic of Korea
  2. 2.Korea Institute for Advanced StudySeoulRepublic of Korea
  3. 3.Department of Mathematical SciencesSeoul National UniversitySeoulRepublic of Korea
  4. 4.Department of Mathematics and Research Institute of Natural SciencesHanyang UniversitySeoulRepublic of Korea
  5. 5.Center for Mathematical SciencesHuazhong University of Science and TechnologyWuhanChina

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