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

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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.

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Acknowledgements

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

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Republic of Korea

    Seung-Yeal Ha

  2. Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

    Seung-Yeal Ha

  3. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Republic of Korea

    Jeongho Kim

  4. Department of Mathematics and Research Institute of Natural Sciences, Hanyang University, Wangsimni-ro 222, Seongdong-gu, Seoul, 04763, Republic of Korea

    Jinyeong Park

  5. Center for Mathematical Sciences, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan, 430074, China

    Xiongtao Zhang

Authors
  1. Seung-Yeal Ha
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  2. Jeongho Kim
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  4. Xiongtao Zhang
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Corresponding author

Correspondence to Jinyeong Park.

Additional information

Communicated by T.-P. Liu

The work of S.-Y. Ha is partially supported by the Samsung Science and Technology Foundation under Project (Number SSTF-BA1401-03) and the work of J. Kim was supported by the German Research Foundation (DFG) under project number IRTG 2235 and 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.

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Ha, SY., Kim, J., Park, J. et al. Complete Cluster Predictability of the Cucker–Smale Flocking Model on the Real Line. Arch Rational Mech Anal 231, 319–365 (2019). https://doi.org/10.1007/s00205-018-1281-x

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  • DOI: https://doi.org/10.1007/s00205-018-1281-x

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