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
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[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]

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

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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.

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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.

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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.

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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.

[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.

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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.

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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.

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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.

[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.

[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.

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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.

[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.

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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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[34]

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

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[42]

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

[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.

[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.

[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.

[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.

[47]

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

[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.

[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.

[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.

[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.

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.

[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]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[34]

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

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[42]

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

[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.

[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.

[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.

[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.

[47]

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

[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.

[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.

[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.

[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.

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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