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September  2020, 19(9): 4621-4654. doi: 10.3934/cpaa.2020209

Fast and slow velocity alignments in a Cucker-Smale ensemble with adaptive couplings

1. 

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

2. 

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

3. 

National Institute for Mathematical Sciences, 70, Yuseong-daero 1689 beon-gil, Yuseong-gu, Daejeon 34047, Republic of Korea

4. 

Department of Mathematics and Research Institute of Natural Sciences, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Republic of Korea

* Corresponding author

Received  February 2020 Revised  February 2020 Published  June 2020

Fund Project: The work of S.-Y. Ha is partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No. 2017R1A5A1015626), the work of D. Kim was supported by National Institute for Mathematical Sciences (NIMS) grant funded by the Korea government (MSIT) (No.B20900000) and the work of J. Park has been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2018R1C1B5043861)

We study the emergent dynamics of the Cucker-Smale (C-S for brevity) ensemble under adaptive couplings. For the adaptive couplings, we basically consider two types of couplings: Hebbian vs. anti-Hebbian. When the Hebbian rule is employed, we present sufficient conditions leading to the mono-cluster flocking using the Lyapunov functional approach. On the other hand, for the anti-Hebbian rule, the possibility of mono-cluster flocking mainly depends on the integrability of the communication weight function and the regularity of the adaptive law at the origin. In addition, we perform numerical experiments and compare them with our analytic results.

Citation: Seung-Yeal Ha, Dohyun Kim, Jinyeong Park. Fast and slow velocity alignments in a Cucker-Smale ensemble with adaptive couplings. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4621-4654. doi: 10.3934/cpaa.2020209
References:
[1]

J. A. AcebronL. L. BonillaC. J. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.   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), Art. 103301. doi: 10.1063/1.3496895.  Google Scholar

[3]

J. Bronski, Y. He, X. Li, Y. Liu, R. D. Sponseller and S. Wolbert, The stability of fixed points for a Kuramoto model with Hebbian interactions, Chaos, 27 (2017), Art. 053110. doi: 10.1063/1.4983524.  Google Scholar

[4]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.   Google Scholar

[5]

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

[6]

P. CattiauxF. Delebecque and L. Pedeches, Stochastic Cucker-Smale models: old and new, Ann. Appl. Probab., 28 (2018), 3239-3286.  doi: 10.1214/18-AAP1400.  Google Scholar

[7]

Y. P. Choi, S. Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles, Volume 1. Modeling and Simulation in Science, Engineering and Technology (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser, Springer, (2017), 299–331.  Google Scholar

[8]

Y. P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033.  doi: 10.3934/krm.2017040.  Google Scholar

[9]

J. ChoS. Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Meth. Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[10]

Y. P. ChoiD. KalsieJ. Peszek and A. Peters, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.  doi: 10.1137/19M1241799.  Google Scholar

[11]

Y. P. Choi and Z. Li, Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays, Appl. Math. Lett., 86 (2018), 49-56.  doi: 10.1016/j.aml.2018.06.018.  Google Scholar

[12]

A. Crnkic and V. Jacimovic, Swarms on the 3-sphere with adaptive synapses: Hebbian and anti-Hebbian learning, Syst. Control Lett., 122 (2018), 32-38.  doi: 10.1016/j.sysconle.2018.10.004.  Google Scholar

[13]

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

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D. Cumin and C. P. Unsworth, Generalizing the Kuramoto model for the study of neuronal synchronization in the brain, Physica D, 226 (2007), 181-196.  doi: 10.1016/j.physd.2006.12.004.  Google Scholar

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P. Degond and S. Motsch, Large-scale dynamics of the persistent turning walker model of fish behavior, J. Statist. Phys., 131 (2008), 989-1022.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

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[17]

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

[18]

B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.  doi: 10.1007/BF00164052.  Google Scholar

[19]

A. GushchinE. Mallada and A. Tang, Phase-coupled oscillators with plastic coupling: synchronization and stability, IEEE. Trans. Netw. Sci. Eng., 3 (2016), 240-256.  doi: 10.1109/TNSE.2016.2605096.  Google Scholar

[20]

S. Y. HaT. Ha and J. H. Kim, Emergent behavior of a Cucker-Smale type particle model with a nonlinear velocity couplings, IEEE Trans. Automat. Control., 55 (2010), 1679-1683.  doi: 10.1109/TAC.2010.2046113.  Google Scholar

[21]

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

[22]

S. Y. HaJ. LeeZ. Li and J. Park, Emergent dynamics of Kuramoto oscillators with adaptive couplings: conservation law and fast learning, SIAM J. Appl. Dyn. Syst., 17 (2018), 1560-1588.  doi: 10.1137/17M1124048.  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. HaD. KimJ. Lee and S. E. Noh, Emergence of aggregation in the swarm sphere model with adaptive coupling laws, Kinet. Relat. Models, 12 (2019), 411-444.  doi: 10.3934/krm.2019018.  Google Scholar

[25]

S. Y. HaJ. KimJ. Park and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.  Google Scholar

[26]

S. Y. HaJ. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.  Google Scholar

[27]

S. Y. HaS. E. Noh and J. Park, Synchronization of Kuramoto oscillators with adaptive couplings, SIAM J. Appl. Dyn. Syst., 15 (2016), 162-194.  doi: 10.1137/15M101484X.  Google Scholar

[28]

S. Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.  Google Scholar

[29]

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

[30]

D. O. Hebb, The Organization of Behavior, Wiley, New York, 1949. Google Scholar

[31]

R. W. Hölzel and K. Krischer, Stability and long term behavior of a Hebbian network of Kuramoto oscillators, SIAM J. Appl. Dyn. Syst., 14 (2015), 188-201.  doi: 10.1137/140965168.  Google Scholar

[32]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar

[33]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Mathematical Physics (eds. H. Araki), Springer, Berlin, Heidelberg, (1975), 420–422.  Google Scholar

[34]

M. MachidaT. KanoS. YamadaM. OkumuraT. Imamura and T. Koyama, Quantum synchronization effects in intrinsic Josephson junctions, Physica C, 468 (2008), 689-694.   Google Scholar

[35]

Y. L. Maistrenko, B. Lysyansky, C. Hauptmann, O. Burylko and P. A. Tass, Multistability in the Kuramoto model with synaptic plasticity, Phys. Rev. E, 75 (2007), Art. 066207. doi: 10.1103/PhysRevE.75.066207.  Google Scholar

[36]

H. MarkramJ. LübkeM. Frotscher and B. Sakmann, Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs, Science, 275 (1997), 213-215.   Google Scholar

[37]

R. K. Niyogi and L. Q. English, Learning-rate-dependent clustering and self-development in a network of coupled phase oscillators, Phys. Rev. E., 80 (2009), Art. 066213. Google Scholar

[38]

W. F. Osgood, Beweis der existenz einer lösung der differentialgleichung $dy/dx = f(x, y)$ ohne hinzunahme der Cauchy-Lipschitz'schen bedingung, Monatsh. f. Mathematik und Physik, 9 (1898), 331-345.  doi: 10.1007/BF01707876.  Google Scholar

[39]

J. Park, D. Poyato and J. Soler, Filippov trajectories and clustering in the Kuramoto model with singular couplings, preprint, arXiv: 1809.04307. Google Scholar

[40]

C. J. Pérez VicenteA. Arenas and L. L. Bonilla, On the short-time dynamics of networks of Hebbian coupled oscillators, J. Phys. A, 29 (1996), 9-16.  doi: 10.1088/0305-4470/29/1/002.  Google Scholar

[41]

L. PereaP. Elosegui and G. Gomez, Extension of the Cucker-Smale control law to space flight formations, J. Guid. Control, 32 (2009), 527-537.   Google Scholar

[42] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept In Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[43]

Q. Ren and J. Zhao, Adaptive coupling and enhanced synchronization in coupled phase oscillators, Phys. Rev. E, 76 (2007), Art. 016207. Google Scholar

[44]

P. Seliger, S. C. Young and L. S. Tsimring, Plasticity and learning in a network of coupled phase oscillators, Phys. Rev. E, 65 (2002), Art. 041906. doi: 10.1103/PhysRevE.65.041906.  Google Scholar

[45]

L. Timms and L. Q. English, Synchronization in phase-coupled Kuramoto oscillator networks with axonal delay and synaptic plasticity, Phys. Rev. E, 89 (2014), Art. 032906. Google Scholar

[46]

G. M. Wittenberg and S. H. Wang, Malleability of spike-timing-dependent plasticity at the CA3-CA1 synapse, J. Neurosci., 26 (2006), 6610-6617.   Google Scholar

show all references

References:
[1]

J. A. AcebronL. L. BonillaC. J. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.   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), Art. 103301. doi: 10.1063/1.3496895.  Google Scholar

[3]

J. Bronski, Y. He, X. Li, Y. Liu, R. D. Sponseller and S. Wolbert, The stability of fixed points for a Kuramoto model with Hebbian interactions, Chaos, 27 (2017), Art. 053110. doi: 10.1063/1.4983524.  Google Scholar

[4]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.   Google Scholar

[5]

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

[6]

P. CattiauxF. Delebecque and L. Pedeches, Stochastic Cucker-Smale models: old and new, Ann. Appl. Probab., 28 (2018), 3239-3286.  doi: 10.1214/18-AAP1400.  Google Scholar

[7]

Y. P. Choi, S. Y. Ha and Z. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, in Active Particles, Volume 1. Modeling and Simulation in Science, Engineering and Technology (eds. N. Bellomo, P. Degond and E. Tadmor), Birkhäuser, Springer, (2017), 299–331.  Google Scholar

[8]

Y. P. Choi and J. Haskovec, Cucker-Smale model with normalized communication weights and time delay, Kinet. Relat. Models, 10 (2017), 1011-1033.  doi: 10.3934/krm.2017040.  Google Scholar

[9]

J. ChoS. Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Meth. Appl. Sci., 26 (2016), 1191-1218.  doi: 10.1142/S0218202516500287.  Google Scholar

[10]

Y. P. ChoiD. KalsieJ. Peszek and A. Peters, A collisionless singular Cucker-Smale model with decentralized formation control, SIAM J. Appl. Dyn. Syst., 18 (2019), 1954-1981.  doi: 10.1137/19M1241799.  Google Scholar

[11]

Y. P. Choi and Z. Li, Emergent behavior of Cucker-Smale flocking particles with heterogeneous time delays, Appl. Math. Lett., 86 (2018), 49-56.  doi: 10.1016/j.aml.2018.06.018.  Google Scholar

[12]

A. Crnkic and V. Jacimovic, Swarms on the 3-sphere with adaptive synapses: Hebbian and anti-Hebbian learning, Syst. Control Lett., 122 (2018), 32-38.  doi: 10.1016/j.sysconle.2018.10.004.  Google Scholar

[13]

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

[14]

D. Cumin and C. P. Unsworth, Generalizing the Kuramoto model for the study of neuronal synchronization in the brain, Physica D, 226 (2007), 181-196.  doi: 10.1016/j.physd.2006.12.004.  Google Scholar

[15]

P. Degond and S. Motsch, Large-scale dynamics of the persistent turning walker model of fish behavior, J. Statist. Phys., 131 (2008), 989-1022.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[16]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.  Google Scholar

[17]

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

[18]

B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.  doi: 10.1007/BF00164052.  Google Scholar

[19]

A. GushchinE. Mallada and A. Tang, Phase-coupled oscillators with plastic coupling: synchronization and stability, IEEE. Trans. Netw. Sci. Eng., 3 (2016), 240-256.  doi: 10.1109/TNSE.2016.2605096.  Google Scholar

[20]

S. Y. HaT. Ha and J. H. Kim, Emergent behavior of a Cucker-Smale type particle model with a nonlinear velocity couplings, IEEE Trans. Automat. Control., 55 (2010), 1679-1683.  doi: 10.1109/TAC.2010.2046113.  Google Scholar

[21]

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

[22]

S. Y. HaJ. LeeZ. Li and J. Park, Emergent dynamics of Kuramoto oscillators with adaptive couplings: conservation law and fast learning, SIAM J. Appl. Dyn. Syst., 17 (2018), 1560-1588.  doi: 10.1137/17M1124048.  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. HaD. KimJ. Lee and S. E. Noh, Emergence of aggregation in the swarm sphere model with adaptive coupling laws, Kinet. Relat. Models, 12 (2019), 411-444.  doi: 10.3934/krm.2019018.  Google Scholar

[25]

S. Y. HaJ. KimJ. Park and X. Zhang, Complete cluster predictability of the Cucker-Smale flocking model on the real line, Arch. Ration. Mech. Anal., 231 (2019), 319-365.  doi: 10.1007/s00205-018-1281-x.  Google Scholar

[26]

S. Y. HaJ. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181.  doi: 10.3934/krm.2018045.  Google Scholar

[27]

S. Y. HaS. E. Noh and J. Park, Synchronization of Kuramoto oscillators with adaptive couplings, SIAM J. Appl. Dyn. Syst., 15 (2016), 162-194.  doi: 10.1137/15M101484X.  Google Scholar

[28]

S. Y. Ha and T. Ruggeri, Emergent dynamics of a thermodynamically consistent particle model, Arch. Ration. Mech. Anal., 223 (2017), 1397-1425.  doi: 10.1007/s00205-016-1062-3.  Google Scholar

[29]

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

[30]

D. O. Hebb, The Organization of Behavior, Wiley, New York, 1949. Google Scholar

[31]

R. W. Hölzel and K. Krischer, Stability and long term behavior of a Hebbian network of Kuramoto oscillators, SIAM J. Appl. Dyn. Syst., 14 (2015), 188-201.  doi: 10.1137/140965168.  Google Scholar

[32]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar

[33]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Mathematical Physics (eds. H. Araki), Springer, Berlin, Heidelberg, (1975), 420–422.  Google Scholar

[34]

M. MachidaT. KanoS. YamadaM. OkumuraT. Imamura and T. Koyama, Quantum synchronization effects in intrinsic Josephson junctions, Physica C, 468 (2008), 689-694.   Google Scholar

[35]

Y. L. Maistrenko, B. Lysyansky, C. Hauptmann, O. Burylko and P. A. Tass, Multistability in the Kuramoto model with synaptic plasticity, Phys. Rev. E, 75 (2007), Art. 066207. doi: 10.1103/PhysRevE.75.066207.  Google Scholar

[36]

H. MarkramJ. LübkeM. Frotscher and B. Sakmann, Regulation of synaptic efficacy by coincidence of postsynaptic APs and EPSPs, Science, 275 (1997), 213-215.   Google Scholar

[37]

R. K. Niyogi and L. Q. English, Learning-rate-dependent clustering and self-development in a network of coupled phase oscillators, Phys. Rev. E., 80 (2009), Art. 066213. Google Scholar

[38]

W. F. Osgood, Beweis der existenz einer lösung der differentialgleichung $dy/dx = f(x, y)$ ohne hinzunahme der Cauchy-Lipschitz'schen bedingung, Monatsh. f. Mathematik und Physik, 9 (1898), 331-345.  doi: 10.1007/BF01707876.  Google Scholar

[39]

J. Park, D. Poyato and J. Soler, Filippov trajectories and clustering in the Kuramoto model with singular couplings, preprint, arXiv: 1809.04307. Google Scholar

[40]

C. J. Pérez VicenteA. Arenas and L. L. Bonilla, On the short-time dynamics of networks of Hebbian coupled oscillators, J. Phys. A, 29 (1996), 9-16.  doi: 10.1088/0305-4470/29/1/002.  Google Scholar

[41]

L. PereaP. Elosegui and G. Gomez, Extension of the Cucker-Smale control law to space flight formations, J. Guid. Control, 32 (2009), 527-537.   Google Scholar

[42] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept In Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[43]

Q. Ren and J. Zhao, Adaptive coupling and enhanced synchronization in coupled phase oscillators, Phys. Rev. E, 76 (2007), Art. 016207. Google Scholar

[44]

P. Seliger, S. C. Young and L. S. Tsimring, Plasticity and learning in a network of coupled phase oscillators, Phys. Rev. E, 65 (2002), Art. 041906. doi: 10.1103/PhysRevE.65.041906.  Google Scholar

[45]

L. Timms and L. Q. English, Synchronization in phase-coupled Kuramoto oscillator networks with axonal delay and synaptic plasticity, Phys. Rev. E, 89 (2014), Art. 032906. Google Scholar

[46]

G. M. Wittenberg and S. H. Wang, Malleability of spike-timing-dependent plasticity at the CA3-CA1 synapse, J. Neurosci., 26 (2006), 6610-6617.   Google Scholar

Figure 1.  Hebbian rule
Figure 2.  Anti-Hebbian rule and short-ranged interaction
Figure 3.  Anti-Hebbian rule with $ \eta = 2 $ and long-ranged interaction
Figure 4.  Anti-Hebbian rule with $ \eta = 0.5 $ and long-ranged interaction
Table 1.  Main results
$ \Gamma $ $ \psi $ Asymptotic behavior Corresponding result
Hebbian $ \Gamma(0) >0 $ Short-ranged Conditional flocking Theorem 3.1
Figure 1
Long-ranged Unconditional flocking
Anti-Hebbian $ \Gamma(s) = s^\eta $ Short-ranged No alignment Theorem 3.3
Figure 2
Long-ranged $ \eta\geq 1 $ Slow velocity alignment Theorem 3.5
Figure 3
$ 0<\eta<1 $ Unconditional flocking Theorems 3.7 and 3.8
Figure 4
$ \Gamma $ $ \psi $ Asymptotic behavior Corresponding result
Hebbian $ \Gamma(0) >0 $ Short-ranged Conditional flocking Theorem 3.1
Figure 1
Long-ranged Unconditional flocking
Anti-Hebbian $ \Gamma(s) = s^\eta $ Short-ranged No alignment Theorem 3.3
Figure 2
Long-ranged $ \eta\geq 1 $ Slow velocity alignment Theorem 3.5
Figure 3
$ 0<\eta<1 $ Unconditional flocking Theorems 3.7 and 3.8
Figure 4
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