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September  2017, 10(3): 689-723. doi: 10.3934/krm.2017028

Emergent dynamics in the interactions of Cucker-Smale ensembles

Seung-Yeal Ha 1, , Dongnam Ko 2, , Yinglong Zhang 2,, and  Xiongtao Zhang 2,

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
2. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea

* Corresponding author: Yinglong Zhang

Received  January 2016 Revised  June 2016 Published  December 2016

Fund Project: The work of S.-Y. Ha and X. Zhang is supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03. This work has been completed while the first author was visiting NCTS, National Taiwan University. He would like to thank NCTS for their hospitality during the stay. The work of D. Ko is supported by the fellowship of TJ Park Foundation. The work of Y. Zhang is partially supported by a National Research Foundation of Korea grant (2014R1A2A2A05002096) funded by the Korean government.

Merging and separation of flocking groups are often observed in our natural complex systems. In this paper, we employ the Cucker-Smale particle model to model such merging and separation phenomena. For definiteness, we consider the interaction of two homogeneous Cucker-Smale ensembles and present several sufficient frameworks for mono-cluster flocking, bi-cluster flocking and partial flocking in terms of coupling strength, communication weight, and initial configurations.

Keywords: Bi-cluster flocking, coupling strength, Cucker-Smale model, interactions, local flocking.
Mathematics Subject Classification: Primary: 34D05, 92D50; Secondary: 93D20.
Citation: Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic & Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028
References:
[1]

S. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

[2]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp.  doi: 10.1063/1.3496895.  Google Scholar

[3]

F. Bolley, J. A. Canizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Mod. Meth. Appl. Sci., 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702.  Google Scholar

[4]

J. A. Canizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod., Meth. Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

[5]

J. A. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, ESIAM Proceedings and Surveys, 47 (2014), 17-35.  doi: 10.1051/proc/201447002.  Google Scholar

[6]

J. A. Carrillo, M. R. D' Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinetic Relat. Models, 2 (2009), 363-378.  doi: 10.3934/krm.2009.2.363.  Google Scholar

[7]

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), 218-236.  doi: 10.1137/090757290.  Google Scholar

[8]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Mod. Meth. Appl. Sci., 20 (2010), 1533-1552.  doi: 10.1142/S0218202510004684.  Google Scholar

[9]

J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[10]

J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for agent-based models with unit speed constraint, Anal. Appl., 14 (2016), 39-73.  doi: 10.1142/S0219530515400023.  Google Scholar

[11]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Autom. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[12]

F. Cucker and F. C. Huepe, Flocking with informed agents, MathS in Action, 1 (2008), 1-25.  doi: 10.5802/msia.1.  Google Scholar

[13]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[14]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[15]

P. Degond and T. Yang, Diffusion in a continuum model of self-propelled particles with alignment interaction, Math. Mod. Meth. Appl. Sci., 20 (2010), 1459-1490.  doi: 10.1142/S0218202510004659.  Google Scholar

[16]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Math. Acad. Sci. Paris, 345 (2007), 555-560.  doi: 10.1016/j.crma.2007.10.024.  Google Scholar

[17]

P. Degond and S. Motsch, Large-scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[18]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[19]

R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.  Google Scholar

[20]

M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Phys. D, 240 (2011), 21-31.  doi: 10.1016/j.physd.2010.08.003.  Google Scholar

[21]

S.-Y. Ha, T. Ha and J. Kim, Asymptotic flocking dynamics for the Cucker-Smale model with the Rayleigh friction, J. Phys. A: Math. Theor., 43 (2010), 315201, 19pp.  doi: 10.1088/1751-8113/43/31/315201.  Google Scholar

[22]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.   Google Scholar

[23]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.   Google Scholar

[24]

S.-Y. Ha and M. Slemrod, Flocking dynamics of a singularly perturbed oscillator chain and the Cucker-Smale system, J. Dyn. Diff. Equat., 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.  Google Scholar

[25]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[26]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 39 (1975), 420-422.   Google Scholar

[27]

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.   Google Scholar

[28]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.  doi: 10.1137/100791774.  Google Scholar

[29]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[30]

D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Sys., 27 (2007), 89-105.   Google Scholar

[31]

J. Park, H. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Automat. Control, 55 (2010), 2617-2623.  doi: 10.1109/TAC.2010.2061070.  Google Scholar

[32]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guidance Control Dynamics, 32 (2009), 526-536.   Google Scholar

[33]

J. Peszek, Existence of piecewise weak solutions of discrete Cucker-Smale flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925.  doi: 10.1016/j.jde.2014.06.003.  Google Scholar

[34]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719.  doi: 10.1137/060673254.  Google Scholar

[35]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[36]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.  Google Scholar

[37]

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.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[38]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.   Google Scholar

show all references

References:
[1]

S. Ahn, H. Choi, S.-Y. Ha and H. Lee, On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.  doi: 10.4310/CMS.2012.v10.n2.a10.  Google Scholar

[2]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301, 17pp.  doi: 10.1063/1.3496895.  Google Scholar

[3]

F. Bolley, J. A. Canizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Mod. Meth. Appl. Sci., 21 (2011), 2179-2210.  doi: 10.1142/S0218202511005702.  Google Scholar

[4]

J. A. Canizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Mod., Meth. Appl. Sci., 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.  Google Scholar

[5]

J. A. Carrillo, Y.-P. Choi and M. Hauray, Local well-posedness of the generalized Cucker-Smale model with singular kernels, ESIAM Proceedings and Surveys, 47 (2014), 17-35.  doi: 10.1051/proc/201447002.  Google Scholar

[6]

J. A. Carrillo, M. R. D' Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinetic Relat. Models, 2 (2009), 363-378.  doi: 10.3934/krm.2009.2.363.  Google Scholar

[7]

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), 218-236.  doi: 10.1137/090757290.  Google Scholar

[8]

J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Mod. Meth. Appl. Sci., 20 (2010), 1533-1552.  doi: 10.1142/S0218202510004684.  Google Scholar

[9]

J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[10]

J. Cho, S.-Y. Ha, F. Huang, C. Jin and D. Ko, Emergence of bi-cluster flocking for agent-based models with unit speed constraint, Anal. Appl., 14 (2016), 39-73.  doi: 10.1142/S0219530515400023.  Google Scholar

[11]

F. Cucker and J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Autom. Control, 55 (2010), 1238-1243.  doi: 10.1109/TAC.2010.2042355.  Google Scholar

[12]

F. Cucker and F. C. Huepe, Flocking with informed agents, MathS in Action, 1 (2008), 1-25.  doi: 10.5802/msia.1.  Google Scholar

[13]

F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002.  Google Scholar

[14]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[15]

P. Degond and T. Yang, Diffusion in a continuum model of self-propelled particles with alignment interaction, Math. Mod. Meth. Appl. Sci., 20 (2010), 1459-1490.  doi: 10.1142/S0218202510004659.  Google Scholar

[16]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Math. Acad. Sci. Paris, 345 (2007), 555-560.  doi: 10.1016/j.crma.2007.10.024.  Google Scholar

[17]

P. Degond and S. Motsch, Large-scale dynamics of the Persistent Turing Walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[18]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Mod. Meth. Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.  Google Scholar

[19]

R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Commun. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.  Google Scholar

[20]

M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Phys. D, 240 (2011), 21-31.  doi: 10.1016/j.physd.2010.08.003.  Google Scholar

[21]

S.-Y. Ha, T. Ha and J. Kim, Asymptotic flocking dynamics for the Cucker-Smale model with the Rayleigh friction, J. Phys. A: Math. Theor., 43 (2010), 315201, 19pp.  doi: 10.1088/1751-8113/43/31/315201.  Google Scholar

[22]

S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.   Google Scholar

[23]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.   Google Scholar

[24]

S.-Y. Ha and M. Slemrod, Flocking dynamics of a singularly perturbed oscillator chain and the Cucker-Smale system, J. Dyn. Diff. Equat., 22 (2010), 325-330.  doi: 10.1007/s10884-009-9142-9.  Google Scholar

[25]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinetic Relat. Models, 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[26]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 39 (1975), 420-422.   Google Scholar

[27]

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.   Google Scholar

[28]

Z. Li and X. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.  doi: 10.1137/100791774.  Google Scholar

[29]

S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9.  Google Scholar

[30]

D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Sys., 27 (2007), 89-105.   Google Scholar

[31]

J. Park, H. Kim and S.-Y. Ha, Cucker-Smale flocking with inter-particle bonding forces, IEEE Tran. Automat. Control, 55 (2010), 2617-2623.  doi: 10.1109/TAC.2010.2061070.  Google Scholar

[32]

L. Perea, P. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guidance Control Dynamics, 32 (2009), 526-536.   Google Scholar

[33]

J. Peszek, Existence of piecewise weak solutions of discrete Cucker-Smale flocking model with a singular communication weight, J. Differential Equations, 257 (2014), 2900-2925.  doi: 10.1016/j.jde.2014.06.003.  Google Scholar

[34]

J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719.  doi: 10.1137/060673254.  Google Scholar

[35]

J. Toner and Y. Tu, Flocks, herds, and Schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1998), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.  Google Scholar

[36]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.  Google Scholar

[37]

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.  doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[38]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.   Google Scholar

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