June  2021, 14(3): 429-468. doi: 10.3934/krm.2021011

A mean-field limit of the particle swarmalator model

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, and Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

2. 

Research Institute of Basic Sciences, Seoul National University, Seoul, 08826, Republic of Korea

3. 

Institute of New Media and Communications, Seoul National University, Seoul, 08826, 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

5. 

Center for Mathematical Sciences, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, 430074, China

* Corresponding author: Jinwook Jung

Received  June 2020 Revised  November 2020 Published  June 2021 Early access  March 2021

We present a mean-field limit of the particle swarmalator model introduced in [46] with singular communication weights. For a mean-field limit, we employ a probabilistic approach for the propagation of molecular chaos and suitable cut-offs in singular terms, which results in the validation of the mean-field limit. We also provide a local-in-time well-posedness of strong and weak solutions to the derived kinetic swarmalator equation.

Citation: Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011
References:
[1]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.

[2]

G. Ajmone MarsanN. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251.

[3]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[4]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci., 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105.

[5]

N. Bellomo and S.-Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.  doi: 10.1142/S0218202517500154.

[6]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006, 29 pp. doi: 10.1142/S0218202511400069.

[7]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.

[8]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.

[9]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.

[10]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65.  doi: 10.4310/CMS.2010.v8.n1.a4.

[11]

A. L. BertozziT. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334.

[12]

N. Boers and P. Pickl, On mean field limits for dynamical systems, J. Stat. Phys., 164 (2016), 1-16.  doi: 10.1007/s10955-015-1351-5.

[13]

L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier- Stokes equations, Differ. Integral Equ., 22 (2009), 1247-1271. 

[14]

J. C. Bronski, L. Deville and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-$N$ Kuramoto model, Chaos, 22 (2012), 033133, 17 pp. doi: 10.1063/1.4745197.

[15]

J. Buck and E. Buck, Biology of sychronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

[16]

J. A. Carrillo, M. Chipot and Y. Huang, On global minimizers of repulsive–attractive power–law interaction energies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130399, 13 pp. doi: 10.1098/rsta.2013.0399.

[17]

J. A. CarrilloY.-P. ChoiP. B. Mucha and Jan Peszek, Sharp conditions to avoid collisions in singular Cucker–Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017.

[18]

A. CavagnaL. D. CastelloI. GiardinaT. GrigeraA. JelicS. MelilloT. MoraL. ParisiE. SilvestriM. Viale and A. M. Walczak, Flocking and turning: A new model for self-organized collective motion, J. Stat. Phys., 158 (2015), 601-627.  doi: 10.1007/s10955-014-1119-3.

[19]

Y.-P. ChoiS.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.

[20]

Y.-P. ChoiS.-Y. HaS. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011.

[21]

T. ChampionL. D. Pascale and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal., 40 (2008), 1-20.  doi: 10.1137/07069938X.

[22]

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

[23]

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

[24]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.

[25]

R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.  doi: 10.1007/BF01077243.

[26]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.

[27]

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.

[28]

F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X.

[29]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.

[30]

S.-Y. HaJ. JungJ. KimJ. Park and X. Zhang, Emergent behaviors of the swarmalator model for position-phase aggregation, Math. Models Methods Appl. Sci., 29 (2019), 2225-2269.  doi: 10.1142/S0218202519500453.

[31]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[32]

S.-Y. HaJ. KimP. Pickl and X. Zhang, A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication, Kinet. Relat. Models, 12 (2019), 1045-1067.  doi: 10.3934/krm.2019039.

[33]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.

[34]

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.  doi: 10.4310/CMS.2009.v7.n2.a2.

[35]

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.

[36]

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.

[37]

P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinet. Relat. Models, 7 (2014), 661-711.  doi: 10.3934/krm.2014.7.661.

[38]

P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $W^{-1, \infty}$ kernels, Invent. Math., 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.

[39]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Theoretical Physics, 30 (1975), 420–422. doi: 10.1007/BFb0013365.

[40]

D. Lazarovici and P. Pickl, A mean field limit for the Vlasov-Poisson system, Arch. Ration. Mech. Anal., 225 (2017), 1201-1231.  doi: 10.1007/s00205-017-1125-0.

[41]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.  doi: 10.1016/j.matpur.2006.01.005.

[42]

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

[43]

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.

[44]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation (ed. C. Cercignani), Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1048 (1984), 60–110. doi: 10.1007/BFb0071878.

[45]

K. P. O'Keeffe, J. H. Evers and T. Kolokolnikov, Ring states in swarmalator systems, Phys. Rev. E, 98 (2018), 022203. doi: 10.1103/PhysRevE.98.022203.

[46]

K. P. O'Keeffe, H. Hong and S. H. Strogatz, Oscillators that sync and swarm, Nature Communications, 8 (2017), 1504. doi: 10.1038/s41467-017-01190-3.

[47]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Syst. Mag., 27 (2007), 89-105. 

[48]

J. Park, D. Poyato and J. Soler, Filippov trajectories and clustering in the Kuramoto model with singular couplings, J. Eur. Math. Soc., to appear, arXiv: 1809.04307.

[49] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[50]

S. Serfaty, Mean field limit for Coulomb-type flows, Duke Math. J., 169 (2020), 2887-2935.  doi: 10.1215/00127094-2020-0019.

[51]

J. Toner and Y. Tu, Flocks, herds and Schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1988), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

[52]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.

[53]

T. VicsekCzirókE. 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.

[54]

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]

J. A. AcebronL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137.

[2]

G. Ajmone MarsanN. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251.

[3]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. Poyato and J. Soler, Vehicular traffic, crowds and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374.

[4]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci., 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105.

[5]

N. Bellomo and S.-Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770.  doi: 10.1142/S0218202517500154.

[6]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006, 29 pp. doi: 10.1142/S0218202511400069.

[7]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.

[8]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.

[9]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.

[10]

A. L. Bertozzi and J. Brandman, Finite-time blow-up of $L^\infty$-weak solutions of an aggregation equation, Commun. Math. Sci., 8 (2010), 45-65.  doi: 10.4310/CMS.2010.v8.n1.a4.

[11]

A. L. BertozziT. Laurent and J. Rosado, $L^p$ theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., 64 (2011), 45-83.  doi: 10.1002/cpa.20334.

[12]

N. Boers and P. Pickl, On mean field limits for dynamical systems, J. Stat. Phys., 164 (2016), 1-16.  doi: 10.1007/s10955-015-1351-5.

[13]

L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier- Stokes equations, Differ. Integral Equ., 22 (2009), 1247-1271. 

[14]

J. C. Bronski, L. Deville and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-$N$ Kuramoto model, Chaos, 22 (2012), 033133, 17 pp. doi: 10.1063/1.4745197.

[15]

J. Buck and E. Buck, Biology of sychronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

[16]

J. A. Carrillo, M. Chipot and Y. Huang, On global minimizers of repulsive–attractive power–law interaction energies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130399, 13 pp. doi: 10.1098/rsta.2013.0399.

[17]

J. A. CarrilloY.-P. ChoiP. B. Mucha and Jan Peszek, Sharp conditions to avoid collisions in singular Cucker–Smale interactions, Nonlinear Anal. Real World Appl., 37 (2017), 317-328.  doi: 10.1016/j.nonrwa.2017.02.017.

[18]

A. CavagnaL. D. CastelloI. GiardinaT. GrigeraA. JelicS. MelilloT. MoraL. ParisiE. SilvestriM. Viale and A. M. Walczak, Flocking and turning: A new model for self-organized collective motion, J. Stat. Phys., 158 (2015), 601-627.  doi: 10.1007/s10955-014-1119-3.

[19]

Y.-P. ChoiS.-Y. Ha and J. Kim, Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication, Netw. Heterog. Media, 13 (2018), 379-407.  doi: 10.3934/nhm.2018017.

[20]

Y.-P. ChoiS.-Y. HaS. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.  doi: 10.1016/j.physd.2011.11.011.

[21]

T. ChampionL. D. Pascale and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal., 40 (2008), 1-20.  doi: 10.1137/07069938X.

[22]

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

[23]

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

[24]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005.

[25]

R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.  doi: 10.1007/BF01077243.

[26]

J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.  doi: 10.4310/CMS.2013.v11.n2.a7.

[27]

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.

[28]

F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X.

[29]

R. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.

[30]

S.-Y. HaJ. JungJ. KimJ. Park and X. Zhang, Emergent behaviors of the swarmalator model for position-phase aggregation, Math. Models Methods Appl. Sci., 29 (2019), 2225-2269.  doi: 10.1142/S0218202519500453.

[31]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[32]

S.-Y. HaJ. KimP. Pickl and X. Zhang, A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication, Kinet. Relat. Models, 12 (2019), 1045-1067.  doi: 10.3934/krm.2019039.

[33]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267.  doi: 10.4171/EMSS/17.

[34]

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.  doi: 10.4310/CMS.2009.v7.n2.a2.

[35]

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.

[36]

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.

[37]

P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinet. Relat. Models, 7 (2014), 661-711.  doi: 10.3934/krm.2014.7.661.

[38]

P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $W^{-1, \infty}$ kernels, Invent. Math., 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y.

[39]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Theoretical Physics, 30 (1975), 420–422. doi: 10.1007/BFb0013365.

[40]

D. Lazarovici and P. Pickl, A mean field limit for the Vlasov-Poisson system, Arch. Ration. Mech. Anal., 225 (2017), 1201-1231.  doi: 10.1007/s00205-017-1125-0.

[41]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.  doi: 10.1016/j.matpur.2006.01.005.

[42]

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

[43]

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.

[44]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, in Kinetic Theories and the Boltzmann Equation (ed. C. Cercignani), Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1048 (1984), 60–110. doi: 10.1007/BFb0071878.

[45]

K. P. O'Keeffe, J. H. Evers and T. Kolokolnikov, Ring states in swarmalator systems, Phys. Rev. E, 98 (2018), 022203. doi: 10.1103/PhysRevE.98.022203.

[46]

K. P. O'Keeffe, H. Hong and S. H. Strogatz, Oscillators that sync and swarm, Nature Communications, 8 (2017), 1504. doi: 10.1038/s41467-017-01190-3.

[47]

D. A. PaleyN. E. LeonardR. SepulchreD. Grunbaum and J. K. Parrish, Oscillator models and collective motion, IEEE Control Syst. Mag., 27 (2007), 89-105. 

[48]

J. Park, D. Poyato and J. Soler, Filippov trajectories and clustering in the Kuramoto model with singular couplings, J. Eur. Math. Soc., to appear, arXiv: 1809.04307.

[49] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[50]

S. Serfaty, Mean field limit for Coulomb-type flows, Duke Math. J., 169 (2020), 2887-2935.  doi: 10.1215/00127094-2020-0019.

[51]

J. Toner and Y. Tu, Flocks, herds and Schools: A quantitative theory of flocking, Phys. Rev. E, 58 (1988), 4828-4858.  doi: 10.1103/PhysRevE.58.4828.

[52]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004.

[53]

T. VicsekCzirókE. 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.

[54]

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.

[1]

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