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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Emergence of a periodically rotating one-point cluster in a thermodynamic Cucker-Smale ensemble

Authors: Hangjun Cho, Linglong Du and Seung-Yeal Ha
Journal: Quart. Appl. Math. 80 (2022), 1-22
MSC (2020): Primary 34D05, 70F99, 74A15
Published electronically: November 16, 2021
MathSciNet review: 4360547
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Abstract: We study emergent behaviors of thermomechanical Cucker-Smale (TCS) ensemble confined in a harmonic potential field. In the absence of external force field, emergent dynamics of TCS particles has been extensively studied recently under various frameworks formulated in terms of initial configuration, system parameters and network topologies. Moreover, the TCS model does not exhibit rotating motions in the absence of an external force field. In this paper, we show the emergence of periodically rotating one-point cluster for the TCS model in a harmonic potential field using elementary energy estimates and continuity argument. We also provide several numerical simulations and compare them with analytical results.

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Additional Information

Hangjun Cho
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
ORCID: 0000-0001-9920-094X

Linglong Du
Affiliation: Department of Applied Mathematics, Donghua University, Shanghai, People’s Republic of China

Seung-Yeal Ha
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826; and Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Republic of Korea
MR Author ID: 684438

Keywords: Flocking, harmonic oscillator, thermodynamic Cucker-Smale model, rotating one-point cluster
Received by editor(s): June 19, 2021
Received by editor(s) in revised form: October 4, 2021
Published electronically: November 16, 2021
Additional Notes: The second author, Du Linglong, is the corresponding author
The work of the second author was supported by Natural Science Foundation of China (No.12001097 and 12171082) and China Scholarship Council. The work of the third author was supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881)
Dedicated: This paper is dedicated to Professor Constantine Dafermos
Article copyright: © Copyright 2021 Brown University