Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit


Authors: Seung-Yeal Ha, Zhuchun Li, Marshall Slemrod and Xiaoping Xue
Journal: Quart. Appl. Math. 72 (2014), 689-701
MSC (2010): Primary 92D25, 74A25, 76N10
DOI: https://doi.org/10.1090/S0033-569X-2014-01350-5
Published electronically: November 3, 2014
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Abstract: We present an asymptotic flocking estimate for the Cucker-Smale flocking model under the rooted leadership in a large coupling limit. For this, we reformulate the Cucker-Smale model into a fast-slow dynamical system involving a small parameter which corresponds to the inverse of a coupling strength. When the coupling strength tends to infinity, the spatial configuration will be frozen instantaneously, whereas the velocity configuration shrinks to the global leader's velocity immediately. For the rigorous explanation of this phenomenon, we use Tikhonov's singular perturbation theory. We also present several numerical simulations to confirm our analytical theory.


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Additional Information

Seung-Yeal Ha
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Email: syha@snu.ac.kr

Zhuchun Li
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
Address at time of publication: Z. Li is currently with the Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Email: lizhuchun@gmail.com

Marshall Slemrod
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Wisconsin 53706
Email: slemrod@math.wisc.edu

Xiaoping Xue
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
Email: xiaopingxue@263.net

DOI: https://doi.org/10.1090/S0033-569X-2014-01350-5
Keywords: Cucker-Smale flocking, rooted leadership, fast-slow dynamics
Received by editor(s): October 15, 2012
Published electronically: November 3, 2014
Additional Notes: The work of S.-Y. Ha was partially supported by KRF-2011-0015388. The work of Z. Li was supported by KRF-2009-0093137 and NSF of China grants 11126220 and 11400135. The work of X. Xue was supported by NSF of China grant 11271099 and the Fundamental Research Funds for Central Universities grant HIT.KLOF.2010047.
Article copyright: © Copyright 2014 Brown University


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