June  2018, 38(6): 2763-2793. doi: 10.3934/dcds.2018116

Remarks on the critical coupling strength for the Cucker-Smale model with unit speed

1. 

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

2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

3. 

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

4. 

Department of Mathematical Sciences, Industrial and Mathematical Data Analytics Research Center, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Yinglong Zhang

Received  April 2017 Revised  January 2018 Published  April 2018

Fund Project: The work of S.-Y. Ha is supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03. The work of D. Ko is supported by the fellowship of POSCO 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.

We present a non-trivial lower bound for the critical coupling strength to the Cucker-Smale model with unit speed constraint and short-range communication weight from the viewpoint of a mono-cluster(global) flocking. For a long-range communication weight, the critical coupling strength is zero in the sense that the mono-cluster flocking emerges from any initial configurations for any positive coupling strengths, whereas for a short-range communication weight, a mono-cluster flocking can emerge from an initial configuration only for a sufficiently large coupling strength. Our main interest lies on the condition of non-flocking. We provide a positive lower bound for the critical coupling strength. We also present numerical simulations for the upper and lower bounds for the critical coupling strength depending on initial configurations and compare them with analytical results.

Citation: Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang. Remarks on the critical coupling strength for the Cucker-Smale model with unit speed. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2763-2793. doi: 10.3934/dcds.2018116
References:
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S. AhnH. ChoiS.-Y. Ha and H. Lee, On the collision avoiding initial-configurations to the Cucker-Smale type flocking models, Comm. 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

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F. BolleyJ. 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

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J. A. CanizoJ. 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

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M. CaponigroM. FornasierB. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Math. Models Methods Appl. Sci., 25 (2015), 521-564.  doi: 10.1142/S0218202515400059.  Google Scholar

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J. A. CarrilloM. 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

[9]

J. A. CarrilloM. FornasierJ. 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

[10]

J. A. CarrilloA. KlarS. 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

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H. ChatF. GinelliG. GregoireF. Peruani and F. Raynaud, Modeling collective motion: Variations on the Vicsek model, The European Physical Journal B, 64 (2008), 451-456.  doi: 10.1140/epjb/e2008-00275-9.  Google Scholar

[12]

J. ChoS.-Y. HaF. HuangC. 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

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J. ChoS.-Y. HaF. HuangC. 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

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S.-H. Choi and S.-Y. Ha, Emergence of flocking for a multi-agent system moving with constant speed, Commun. Math. Sci., 14 (2016), 953-972.  doi: 10.4310/CMS.2016.v14.n4.a4.  Google Scholar

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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

[17]

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

[18]

P. DegondJ.-G. LiuS. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Meth. Appl. Anal., 20 (2013), 89-114.  doi: 10.4310/MAA.2013.v20.n2.a1.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

R. DuanM. 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

[23]

U. Erdmann, W. Ebeling and A. Mikhailov, Noise-induced transition from translational to rotational motion of swarms, Phys. Rev. E, 71 (2005), 051904. doi: 10.1103/PhysRevE.71.051904.  Google Scholar

[24]

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

[25]

A. Frouvelle and J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math. Anal., 44 (2012), 791-826.  doi: 10.1137/110823912.  Google Scholar

[26]

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.  Google Scholar

[27]

S.-Y. HaE. Jeong and M.-K. Kang, Emergent behavior of a generalized Viscek-type flocking model, Nonlinearity, 23 (2010), 3139-3156.  doi: 10.1088/0951-7715/23/12/008.  Google Scholar

[28]

S.-Y. HaD. Ko and Y. Zhang, Critical coupling strength of the Cucker-Smale model for flocking, Math. Models Methods Appl. Sci., 27 (2017), 1051-1087.  doi: 10.1142/S0218202517400097.  Google Scholar

[29]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic CuckerSmale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[30]

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

[31]

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

[32]

E. Justh and P. A. Krishnaprasad, Simple Control Law for UAV Formation Flying, Technical Report, 2002. Google Scholar

[33]

E. Justh and P. A. Krishnaprasad, Steering laws and continuum models for planar formations, Proc. 42nd IEEE Conf. on Decision and Control, (2003), 3609-3614.   Google Scholar

[34]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 30 (1975), p420. Google Scholar

[35]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Berlin, Springer, 1984.  Google Scholar

[36]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, Lecture Notes in Theoretical Physics, 39 (1975), 420-422.   Google Scholar

[37]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[38]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A: Math. Theor., 43 (2010), 465301, 20pp. doi: 10.1088/1751-8113/43/46/465301.  Google Scholar

[39]

M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101, 25 pp. doi: 10.1088/1751-8113/42/39/395101.  Google Scholar

[40]

A. S. Mikhailov and D. H. Zanette, Noise-induced breakdown of coherent collective motion in swarms, Phys. Rev. E, 60 (1999), 4571-4575.  doi: 10.1103/PhysRevE.60.4571.  Google Scholar

[41]

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

[42]

S. Motsch and E. Tadmor, Heterophilious dynamics enhanced consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[43]

D. A. PaleyN. E. Leonard and R. Sepulchre, Stabilization of symmetric formations to motion around convex loops, Syst. Control Lett., 57 (2008), 209-215.  doi: 10.1016/j.sysconle.2007.08.005.  Google Scholar

[44]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Sys., 27 (2007), 89-105.  doi: 10.1109/MCS.2007.384123.  Google Scholar

[45]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guidance Control Dynamics, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[46]

R. Sepulchre, D. Paley and N. Leonard, Stabilization of Collective Motion of Self-Propelled Particles, Proc. 16th Int. Symp. Mathematical Theory of Networks and Systems (Leuven, Belgium, July 2004) Available at cdcl.umd.edu/papers/mtns04.pdf. Google Scholar

[47]

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

[48]

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

[49]

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

[50]

T. VicsekA. CzirókE. Ben-JacobI. 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

[51]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3.  Google Scholar

show all references

References:
[1]

S. AhnH. ChoiS.-Y. Ha and H. Lee, On the collision avoiding initial-configurations to the Cucker-Smale type flocking models, Comm. 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]

G. Albi and L. Pareschi, Selective model-predictive control for flocking systems, preprint, arXiv: 1603.05012, (2016). Google Scholar

[4]

F. BolleyJ. 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

[5]

M. BonginiM. Fornasier and D. Kalise, (Un)conditional consensus emergence under perturbed and decentralized feedback controls, Discrete Contin. Dyn. Syst., 35 (2015), 4071-4094.  doi: 10.3934/dcds.2015.35.4071.  Google Scholar

[6]

J. A. CanizoJ. 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

[7]

M. CaponigroM. FornasierB. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Math. Models Methods Appl. Sci., 25 (2015), 521-564.  doi: 10.1142/S0218202515400059.  Google Scholar

[8]

J. A. CarrilloM. 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

[9]

J. A. CarrilloM. FornasierJ. 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

[10]

J. A. CarrilloA. KlarS. 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

[11]

H. ChatF. GinelliG. GregoireF. Peruani and F. Raynaud, Modeling collective motion: Variations on the Vicsek model, The European Physical Journal B, 64 (2008), 451-456.  doi: 10.1140/epjb/e2008-00275-9.  Google Scholar

[12]

J. ChoS.-Y. HaF. HuangC. 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

[13]

J. ChoS.-Y. HaF. HuangC. 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

[14]

S.-H. Choi and S.-Y. Ha, Emergence of flocking for a multi-agent system moving with constant speed, Commun. Math. Sci., 14 (2016), 953-972.  doi: 10.4310/CMS.2016.v14.n4.a4.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

P. DegondJ.-G. LiuS. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Meth. Appl. Anal., 20 (2013), 89-114.  doi: 10.4310/MAA.2013.v20.n2.a1.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

R. DuanM. 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

[23]

U. Erdmann, W. Ebeling and A. Mikhailov, Noise-induced transition from translational to rotational motion of swarms, Phys. Rev. E, 71 (2005), 051904. doi: 10.1103/PhysRevE.71.051904.  Google Scholar

[24]

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

[25]

A. Frouvelle and J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math. Anal., 44 (2012), 791-826.  doi: 10.1137/110823912.  Google Scholar

[26]

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.  Google Scholar

[27]

S.-Y. HaE. Jeong and M.-K. Kang, Emergent behavior of a generalized Viscek-type flocking model, Nonlinearity, 23 (2010), 3139-3156.  doi: 10.1088/0951-7715/23/12/008.  Google Scholar

[28]

S.-Y. HaD. Ko and Y. Zhang, Critical coupling strength of the Cucker-Smale model for flocking, Math. Models Methods Appl. Sci., 27 (2017), 1051-1087.  doi: 10.1142/S0218202517400097.  Google Scholar

[29]

S.-Y. HaK. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic CuckerSmale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9.  Google Scholar

[30]

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

[31]

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

[32]

E. Justh and P. A. Krishnaprasad, Simple Control Law for UAV Formation Flying, Technical Report, 2002. Google Scholar

[33]

E. Justh and P. A. Krishnaprasad, Steering laws and continuum models for planar formations, Proc. 42nd IEEE Conf. on Decision and Control, (2003), 3609-3614.   Google Scholar

[34]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 30 (1975), p420. Google Scholar

[35]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Berlin, Springer, 1984.  Google Scholar

[36]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, Lecture Notes in Theoretical Physics, 39 (1975), 420-422.   Google Scholar

[37]

N. E. LeonardD. A. PaleyF. LekienR. SepulchreD. M. Fratantoni and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE, 95 (2007), 48-74.  doi: 10.1109/JPROC.2006.887295.  Google Scholar

[38]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A: Math. Theor., 43 (2010), 465301, 20pp. doi: 10.1088/1751-8113/43/46/465301.  Google Scholar

[39]

M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101, 25 pp. doi: 10.1088/1751-8113/42/39/395101.  Google Scholar

[40]

A. S. Mikhailov and D. H. Zanette, Noise-induced breakdown of coherent collective motion in swarms, Phys. Rev. E, 60 (1999), 4571-4575.  doi: 10.1103/PhysRevE.60.4571.  Google Scholar

[41]

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

[42]

S. Motsch and E. Tadmor, Heterophilious dynamics enhanced consensus, SIAM Review, 56 (2014), 577-621.  doi: 10.1137/120901866.  Google Scholar

[43]

D. A. PaleyN. E. Leonard and R. Sepulchre, Stabilization of symmetric formations to motion around convex loops, Syst. Control Lett., 57 (2008), 209-215.  doi: 10.1016/j.sysconle.2007.08.005.  Google Scholar

[44]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Sys., 27 (2007), 89-105.  doi: 10.1109/MCS.2007.384123.  Google Scholar

[45]

L. PereaP. Elosegui and G. Gómez, Extension of the Cucker-Smale control law to space flight formation, J. Guidance Control Dynamics, 32 (2009), 527-537.  doi: 10.2514/1.36269.  Google Scholar

[46]

R. Sepulchre, D. Paley and N. Leonard, Stabilization of Collective Motion of Self-Propelled Particles, Proc. 16th Int. Symp. Mathematical Theory of Networks and Systems (Leuven, Belgium, July 2004) Available at cdcl.umd.edu/papers/mtns04.pdf. Google Scholar

[47]

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

[48]

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

[49]

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

[50]

T. VicsekA. CzirókE. Ben-JacobI. 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

[51]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.  doi: 10.1016/0022-5193(67)90051-3.  Google Scholar

Figure 1.  Position-velocity configurations
Figure 2.  Diameter of positions $D({\bf{x}})$ along time axis
Figure 3.  Temporal evolution of $\theta({\bf{x}}(t), {\bf{v}}(t))$
Figure 4.  Randomly chosen Position-velocity distribution
Figure 5.  Initial configuration and its evidence of non-flocking
Figure 6.  Emergence of local flocking
Figure 7.  Emergence of local flocking for smaller $\kappa$
Figure 8.  Emergence of larger cluster flocking for much smaller $\kappa$
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