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Emergent dynamics in the interactions of Cucker-Smale ensembles
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
F. Cucker and F. C. Huepe,
Flocking with informed agents, MathS in Action, 1 (2008), 1-25.
doi: 10.5802/msia.1. |
[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. |
[14] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[26] |
Y. Kuramoto,
International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 39 (1975), 420-422.
|
[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.
|
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
[34] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719.
doi: 10.1137/060673254. |
[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. |
[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. |
[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. |
[38] |
A. T. Winfree,
Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.
|
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
F. Cucker and F. C. Huepe,
Flocking with informed agents, MathS in Action, 1 (2008), 1-25.
doi: 10.5802/msia.1. |
[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. |
[14] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[26] |
Y. Kuramoto,
International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 39 (1975), 420-422.
|
[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.
|
[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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
[34] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007), 694-719.
doi: 10.1137/060673254. |
[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. |
[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. |
[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. |
[38] |
A. T. Winfree,
Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.
|
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