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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 and 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. 

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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. 

[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.

[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. 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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[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.

[34]

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

[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.

[36]

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

[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.

[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.

[39]

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

[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.

[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. 

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

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

[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.

[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.

[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. 

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. 

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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. 

[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.

[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. 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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

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

[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.

[34]

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

[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.

[36]

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

[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.

[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.

[39]

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

[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.

[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. 

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

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

[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.

[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.

[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. 

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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