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
MathSciNet review:
3291822
Full-text PDF Free Access
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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.
References
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- T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995), 1226–1229.
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References
- Shin Mi Ahn, Heesun Choi, Seung-Yeal Ha, and Ho Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci. 10 (2012), no. 2, 625–643. MR 2901323
- Shin Mi Ahn and Seung-Yeal Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys. 51 (2010), no. 10, 103301, 17. MR 2761313 (2011k:34124), DOI https://doi.org/10.1063/1.3496895
- Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, and Moon-Jin Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity 25 (2012), no. 4, 1155–1177. MR 2904273, DOI https://doi.org/10.1088/0951-7715/25/4/1155
- M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc. Nat. Acad. Sci. 105 (2008), 1232–1237.
- J. A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal. 42 (2010), no. 1, 218–236. MR 2596552 (2011c:35057), DOI https://doi.org/10.1137/090757290
- Felipe Cucker and Jiu-Gang Dong, On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci. 19 (2009), suppl., 1391–1404. MR 2554155 (2010i:37208), DOI https://doi.org/10.1142/S0218202509003851
- Felipe Cucker and Jiu-Gang Dong, Avoiding collisions in flocks, IEEE Trans. Automat. Control 55 (2010), no. 5, 1238–1243. MR 2642092 (2011d:93098), DOI https://doi.org/10.1109/TAC.2010.2042355
- Felipe Cucker and Cristián Huepe, Flocking with informed agents, MathS in Action 1 (2008), no. 1, 1–25. MR 2519063 (2010j:91188), DOI https://doi.org/10.5802/msia.1
- Felipe Cucker and Ernesto Mordecki, Flocking in noisy environments, (English, with English and French summaries), J. Math. Pures Appl. (9) 89 (2008), no. 3, 278–296. MR 2401690 (2009e:91193), DOI https://doi.org/10.1016/j.matpur.2007.12.002
- Felipe Cucker and Steve Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007), no. 5, 852–862. MR 2324245 (2008h:91132), DOI https://doi.org/10.1109/TAC.2007.895842
- Renjun Duan, Massimo Fornasier, and Giuseppe Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys. 300 (2010), no. 1, 95–145. MR 2725184 (2011i:82061), DOI https://doi.org/10.1007/s00220-010-1110-z
- Massimo Fornasier, Jan Haskovec, and Giuseppe Toscani, Fluid dynamic description of flocking via the Povzner-Boltzmann equation, Phys. D 240 (2011), no. 1, 21–31. MR 2740099 (2011j:92072), DOI https://doi.org/10.1016/j.physd.2010.08.003
- S.-Y. Ha, T. Ha, and J. Kim, Asymptotic flocking dynamics for the Cucker-Smale-type model with the Rayleigh friction, J. Phys. A: Math. Theor. 43 (2010), 315201. MR 2665672 (2011f:92073)
- Seung-Yeal Ha, Kiseop Lee, and Doron Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci. 7 (2009), no. 2, 453–469. MR 2536447 (2010f:82071)
- Seung-Yeal Ha and Jian-Guo Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 7 (2009), no. 2, 297–325. MR 2536440 (2011c:82053)
- Seung-Yeal Ha and Marshall Slemrod, Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system, J. Dynam. Differential Equations 22 (2010), no. 2, 325–330. MR 2665438 (2011g:34121), DOI https://doi.org/10.1007/s10884-009-9142-9
- Seung-Yeal Ha and Eitan Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models 1 (2008), no. 3, 415–435. MR 2425606 (2009m:76086), DOI https://doi.org/10.3934/krm.2008.1.415
- John A. Jacquez and Carl P. Simon, Qualitative theory of compartmental systems, SIAM Rev. 35 (1993), no. 1, 43–79. MR 1207797 (94c:34067), DOI https://doi.org/10.1137/1035003
- N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni, and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE 95 (2007), 48–74.
- Zhuchun Li and Xiaoping Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math. 70 (2010), no. 8, 3156–3174. MR 2763499 (2012b:92117), DOI https://doi.org/10.1137/100791774
- Sebastien Motsch and Eitan Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 144 (2011), no. 5, 923–947. MR 2836613 (2012j:92140), DOI https://doi.org/10.1007/s10955-011-0285-9
- Jaemann Park, H. Jin Kim, and Seung-Yeal Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Trans. Automat. Control 55 (2010), no. 11, 2617–2623. MR 2721906 (2011g:92138), DOI https://doi.org/10.1109/TAC.2010.2061070
- L. Perea, P. Elosegui, and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guidance, Control and Dynamics 32 (2009), 526–536.
- Jackie Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math. 68 (2007/08), no. 3, 694–719. MR 2375291 (2008k:92066), DOI https://doi.org/10.1137/060673254
- A. N. Tihonov, On systems of differential equations containing parameters, Mat. Sbornik N.S. 27(69) (1950), 147–156 (Russian). MR 0036902 (12,181d)
- John Toner and Yuhai Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. E (3) 58 (1998), no. 4, 4828–4858. MR 1651324 (99m:92051), DOI https://doi.org/10.1103/PhysRevE.58.4828
- Chad M. Topaz and Andrea L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math. 65 (2004), no. 1, 152–174. MR 2111591 (2005h:92031), DOI https://doi.org/10.1137/S0036139903437424
- T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Schochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995), 1226–1229.
- Xiaoping Xue and Liang Guo, A kind of nonnegative matrices and its application on the stability of discrete dynamical systems, J. Math. Anal. Appl. 331 (2007), no. 2, 1113–1121. MR 2313703 (2008a:39026), DOI https://doi.org/10.1016/j.jmaa.2006.09.053
- Xiaoping Xue and Zhuchun Li, Asymptotic stability analysis of a kind of switched positive linear discrete systems, IEEE Trans. Automat. Control 55 (2010), no. 9, 2198–2203. MR 2722497 (2011i:93112), DOI https://doi.org/10.1109/TAC.2010.2052144
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
MR Author ID:
684438
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
MR Author ID:
163635
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
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
