On the Cucker-Smale flocking with alternating leaders
Authors:
Zhuchun Li and Seung-Yeal Ha
Journal:
Quart. Appl. Math. 73 (2015), 693-709
MSC (2010):
Primary 92D25; Secondary 92C42
DOI:
https://doi.org/10.1090/qam/1401
Published electronically:
September 11, 2015
MathSciNet review:
3432279
Full-text PDF Free Access
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Additional Information
Abstract: We discuss the emergent flocking behavior in a group of Cucker-Smale flocking agents under rooted leadership with alternating leaders. It is well known that the network topology regulates the emergent behaviors of flocks. All existing results on the Cucker-Smale model with leader-follower topologies assume a fixed leader during the temporal evolution process. The rooted leadership is the most general topology taking a leadership. Motivated by collective behaviors observed in the flocks of birds, swarming fishes and potential engineering applications, we consider the rooted leadership with alternating leaders; that is, at each time slice there is a leader, but it can be switched among the agents from time to time. We provide several sufficient conditions leading to the asymptotic flocking among the Cucker-Smale agents under rooted leadership with alternating leaders.
References
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- Felipe Cucker and Cristián Huepe, Flocking with informed agents, MathS in Action 1 (2008), no. 1, 1–25. MR 2519063, DOI https://doi.org/10.5802/msia.1
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- 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, DOI https://doi.org/10.1103/PhysRevE.58.4828
- 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.
- T. Vicsek and A. Zafeiris, Collective motion, Physics Reports 517 (2012), 71-140.
- 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, 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, DOI https://doi.org/10.1109/TAC.2010.2052144
References
- Shinmi Ahn, Hyeong-Ohk Bae, Seung-Yeal Ha, Yongsik Kim, and Hyuncheul Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci. 23 (2013), no. 9, 1603–1628. MR 3062922, DOI https://doi.org/10.1142/S0218202513500176
- 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, DOI https://doi.org/10.4310/CMS.2012.v10.n2.a10
- 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
- Ming Cao, A. Stephen Morse, and Brian D. O. Anderson, Reaching a consensus in a dynamically changing environment: a graphical approach, SIAM J. Control Optim. 47 (2008), no. 2, 575–600. MR 2385854 (2009e:68106), DOI https://doi.org/10.1137/060657005
- Ming Cao, A. Stephen Morse, and Brian D. O. Anderson, Reaching a consensus in a dynamically changing environment: convergence rates, measurement delays, and asynchronous events, SIAM J. Control Optim. 47 (2008), no. 2, 601–623. MR 2385855 (2009e:68107), DOI https://doi.org/10.1137/060657029
- 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, J. Math. Pures Appl. (9) 89 (2008), no. 3, 278–296 (English, with English and French summaries). 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
- Felipe Cucker and Steve Smale, On the mathematics of emergence, Jpn. J. Math. 2 (2007), no. 1, 197–227. MR 2295620 (2007m:91126), DOI https://doi.org/10.1007/s11537-007-0647-x
- Felipe Cucker, Steve Smale, and Ding-Xuan Zhou, Modeling language evolution, Found. Comput. Math. 4 (2004), no. 3, 315–343. MR 2078666 (2005b:91192), DOI https://doi.org/10.1007/s10208-003-0101-2
- Reinhard Diestel, Graph theory, 4th ed., Graduate Texts in Mathematics, vol. 173, Springer, Heidelberg, 2010. MR 2744811 (2011m:05002)
- Seung-Yeal Ha, Kyoung-Kuk Kim, and Kiseop Lee, A mathematical model for multi-name credit based on community flocking, Quant. Finance 15 (2015), no. 5, 841–851. MR 3334575, DOI https://doi.org/10.1080/14697688.2012.744085
- 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, Zhuchun Li, Marshall Slemrod, and Xiaoping Xue, Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit, Quart. Appl. Math. 72 (2014), no. 4, 689–701. MR 3291822, DOI https://doi.org/10.1090/S0033-569X-2014-01350-5
- 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 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
- Ali Jadbabaie, Jie Lin, and A. Stephen Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control 48 (2003), no. 6, 988–1001. MR 1986266 (2004m:82095), DOI https://doi.org/10.1109/TAC.2003.812781
- Jeong-Han Kang, Seung-Yeal Ha, Kyungkeun Kang, and Eunhee Jeong, How do cultural classes emerge from assimilation and distinction? An extension of the Cucker-Smale flocking model, J. Math. Sociol. 38 (2014), no. 1, 47–71. MR 3173651, DOI https://doi.org/10.1080/0022250X.2011.629063
- G. Latouche and V. Ramaswami, Introduction to matrix analytic methods in stochastic modeling, ASA-SIAM Series on Statistics and Applied Probability, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; American Statistical Association, Alexandria, VA, 1999. MR 1674122 (2000b:60224)
- 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, Seung-Yeal Ha, and Xiaoping Xue, Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci. 24 (2014), no. 7, 1389–1419. MR 3192593, DOI https://doi.org/10.1142/S0218202514500043
- 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
- M. Nagy, Z. Ákos, D. Biro and T. Vicsek, Hierarchical group dynamics in pigeon flocks, Nature 464 (2010) 890-893.
- 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, G. Gómez and P. Elosegui, 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
- 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
- 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.
- T. Vicsek and A. Zafeiris, Collective motion, Physics Reports 517 (2012), 71-140.
- 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
Zhuchun Li
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
Email:
lizhuchun@gmail.com
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Keywords:
Cucker-Smale agent,
alternating leaders,
rooted leadership,
flocking
Received by editor(s):
February 24, 2014
Received by editor(s) in revised form:
March 14, 2014
Published electronically:
September 11, 2015
Article copyright:
© Copyright 2015
Brown University