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On group symmetries of the hydrodynamic equations for rarefied gas
Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds
1. | Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea |
2. | Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea |
3. | School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea |
4. | Research Institute of Basic Sciences, Seoul National University, Seoul 08826, Republic of Korea |
We study emergent collective behaviors of a thermodynamic Cucker-Smale (TCS) ensemble on complete smooth Riemannian manifolds. For this, we extend the TCS model on the Euclidean space to a complete smooth Riemannian manifold by adopting the work [
References:
[1] |
J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys, 77 (2005), 137-185. Google Scholar |
[2] |
G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. 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. |
[3] |
A. Aydo$\breve{g}$du, S. T. McQuade and N. Pouradier Duteil,
Opinion dynamics on a general compact Riemannian manifold, Netw. Heterog. Media, 12 (2017), 489-523.
doi: 10.3934/nhm.2017021. |
[4] |
I. Barbalat,
Systèmes déquations différentielles doscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.
|
[5] |
J. Buck and E. Buck,
Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.
doi: 10.1038/211562a0. |
[6] |
J. C. Bronski, T. E. Carty and S. E. Simpson,
A matrix-valued Kuramoto model, J. Stat. Phys., 178 (2020), 595-624.
doi: 10.1007/s10955-019-02442-w. |
[7] |
H. Chaté, F. Ginelli, G. Grégoire, F. Peruani and F. Raynaud, Modeling collective motion: variations on the Vicsek model, The European Physical Journal B, 64 (2008), 451-456. Google Scholar |
[8] |
Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim,
Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.
doi: 10.1088/1361-6544/aafaae. |
[9] |
Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim,
On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, J. Math. Fluid Mech., 22 (2020), 4-38.
doi: 10.1007/s00021-019-0466-x. |
[10] |
Y.-P. Choi, S.-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. |
[11] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[12] |
L. DeVille,
Synchronization and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160-187.
doi: 10.1007/s10955-018-2168-9. |
[13] |
P. Degond, A. Frouvelle and S. Merino-Aceituno,
A new flocking model through body attitude coordination, Math. Models Methods Appl. Sci., 27 (2017), 1005-1049.
doi: 10.1142/S0218202517400085. |
[14] |
J.-G. Dong, S.-Y. Ha and D. Kim, From discrete Cucker-Smale model to continuous Cucker-Smale model in a temperature field, J. Math. Phys., 60 (2019), 072705, 22 pp.
doi: 10.1063/1.5084770. |
[15] |
J.-G. Dong, S.-Y. Ha and D. Kim,
On the Cucker-Smale with q-closest neighbors in a self-consistent temperature field, SIAM J. Control and Optimization, 58 (2020), 368-392.
doi: 10.1137/18M1195462. |
[16] |
J.-G. Dong, S.-Y. Ha and D. Kim, Emergence of mono-cluster flocking in the thermomechanical Cucker-Smale model under switching topologies, Analysis and Applications, (2020), 1-38.
doi: 10.1142/S0219530520500025. |
[17] |
J.-G. Dong, S.-Y. Ha, D. Kim and J. Kim,
Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.
doi: 10.1016/j.jde.2018.08.034. |
[18] |
R. Fetecau, H. Park and F. S. Patacchini, Well-posedness and asymptotic behaviour of an aggregation model with intrinsic interactions on sphere and other manifolds, Analysis and Applications. Google Scholar |
[19] |
A. Frouvelle and J.-G. Liu, Long-time dynamics for a simple aggregation equation on the sphere, in International Workshop on Stochastic Dynamics out of Equilibrium, Springer, Cham, 282 (2017), 457-479.
doi: 10.1007/978-3-030-15096-9_16. |
[20] |
R. C. Fetecau and B. Zhang,
Self-organization on Riemannian manifolds, J. Geom. Mech., 11 (2019), 397-426.
doi: 10.3934/jgm.2019020. |
[21] |
S.-Y. Ha, S. Hwang, D. Kim, S.-C. Kim and C. Min, Emergent behaviors of a first-order particle swarm model on the hyperbolic space, J. Math. Phys., 61 (2020), 042701, 23 pp.
doi: 10.1063/1.5066255. |
[22] |
S.-Y. Ha and D. Kim,
A second-order particle swarm model on a sphere and emergent dynamics, SIAM J. Appl. Dyn. Syst., 18 (2019), 80-116.
doi: 10.1137/18M1205996. |
[23] |
S.-Y. Ha and D. Kim,
Emergent behavior of a second-order Lohe matrix model on the unitary group, J. Stat. Phys., 175 (2019), 904-931.
doi: 10.1007/s10955-019-02270-y. |
[24] |
S.-Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang,
Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.
doi: 10.1090/qam/1517. |
[25] |
S.-Y. Ha, J. Kim, J. 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. Ha, J. Kim and T. Ruggeri,
Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.
doi: 10.1137/17M111064X. |
[27] |
S.-Y. Ha, J. Kim and T. Ruggeri,
From the relativistic mixture of gases to the relativistic cucker-smale flocking, Arch. Rational Mech. Anal., 235 (2020), 1661-1706.
doi: 10.1007/s00205-019-01452-y. |
[28] |
S.-Y. Ha, D. Ko and S. Ryoo,
Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys., 168 (2017), 171-207.
doi: 10.1007/s10955-017-1797-8. |
[29] |
S.-Y. Ha, D. Ko and S. Ryoo,
On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds, J. Stat. Phys., 172 (2018), 1427-1478.
doi: 10.1007/s10955-018-2091-0. |
[30] |
S.-Y. Ha, D. Kim and F. W. Schlöder, Emergent behaviors of Cucker-Smale flocks on Riemannian manifolds, IEEE Trans. Automat. Control, (2020).
doi: 10.1109/TAC.2020.3014096. |
[31] |
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. |
[32] |
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. |
[33] |
M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical systems, and an Introduction to Chaos, Third edition, Elsevier/Academic Press, Amsterdam, 2013.
doi: 10.1016/B978-0-12-382010-5.00001-4.![]() ![]() |
[34] |
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. |
[35] |
J. J$\ddot{u}$rgen, Riemannian Geometry and Geometric Analysis, Universitext. Springer 2011.
doi: 10.1007/978-3-642-21298-7. |
[36] |
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. |
[37] |
J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Trans. Automat. Control, (2019).
doi: 10.1109/TAC.2020.3030849. |
[38] |
J. Markdahl, J. Thunberg and J. Gonçalves,
Almost global consensus on the $n$-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.
doi: 10.1109/TAC.2017.2752799. |
[39] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[40] |
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. |
[41] |
R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, Proc. of the 45th IEEE conference on Decision and Control, (2006), 5060-5066.
doi: 10.1109/CDC.2006.376811. |
[42] |
R. Olfati-Saber, J. A. Fax and R. M. Murray,
Consensus and cooperation in networked multi-agent systems, Proc. IEEE, 95 (2007), 215-233.
doi: 10.1109/JPROC.2006.887293. |
[43] |
A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511755743.![]() ![]() |
[44] |
C. W. Reynolds,
Flocks, herds, and schools: A distributed behavioral model, Comput. Graph, 21 (1987), 25-34.
doi: 10.1145/280811.281008. |
[45] |
L. M. Ritchie, M. A. Lohe and A. G. Williams, Synchronization of relativistic particles in the hyperbolic Kuramoto model, Chaos, 28 (2018), 053116, 11 pp.
doi: 10.1063/1.5021701. |
[46] |
A. Sarlette, S. Bonnabel and R. Sepulchre,
Coordinated motion design on Lie groups, IEEE Trans. Automat. Control, 55 (2010), 1047-1058.
doi: 10.1109/TAC.2010.2042003. |
[47] |
A. Sarlette and R. Sepulchre,
Consensus optimization on manifolds, SIAM J. Control Optim., 48 (2009), 56-76.
doi: 10.1137/060673400. |
[48] |
R. Tron, B. Afsari and R. Vidal,
Riemannian consensus for manifolds with bounded curvature, IEEE Trans. Automat. Contr., 58 (2013), 921-934.
doi: 10.1109/TAC.2012.2225533. |
[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] |
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. |
[51] |
T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet,
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. |
[52] |
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. |
[53] |
A. T. Winfree, The Geometry of Biological Time, Springer-Verlag, New York, 1980. |
[54] |
J. Zhu,
Synchronization of Kuramoto model in a high-dimensional linear space, Phys. Lett. A, 377 (2013), 2939-2943.
doi: 10.1016/j.physleta.2013.09.010. |
[55] |
A. A. Z$\ddot{u}$lke and H. Varela, The effect of temperature on the coupled slow and fast dynamics of an electrochemical oscillator, Rep. Sci., (2016), 24553.
doi: 10.1038/srep24553. |
[56] |
J. Zhang, J. Zhu and C. Qian,
On equilibria and consensus of the Lohe model with identical oscillators, SIAM J. Appl. Dyn. Syst., 17 (2018), 1716-1741.
doi: 10.1137/17M112765X. |
show all references
References:
[1] |
J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys, 77 (2005), 137-185. Google Scholar |
[2] |
G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. 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. |
[3] |
A. Aydo$\breve{g}$du, S. T. McQuade and N. Pouradier Duteil,
Opinion dynamics on a general compact Riemannian manifold, Netw. Heterog. Media, 12 (2017), 489-523.
doi: 10.3934/nhm.2017021. |
[4] |
I. Barbalat,
Systèmes déquations différentielles doscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 267-270.
|
[5] |
J. Buck and E. Buck,
Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.
doi: 10.1038/211562a0. |
[6] |
J. C. Bronski, T. E. Carty and S. E. Simpson,
A matrix-valued Kuramoto model, J. Stat. Phys., 178 (2020), 595-624.
doi: 10.1007/s10955-019-02442-w. |
[7] |
H. Chaté, F. Ginelli, G. Grégoire, F. Peruani and F. Raynaud, Modeling collective motion: variations on the Vicsek model, The European Physical Journal B, 64 (2008), 451-456. Google Scholar |
[8] |
Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim,
Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids, Nonlinearity, 32 (2019), 1597-1640.
doi: 10.1088/1361-6544/aafaae. |
[9] |
Y.-P. Choi, S.-Y. Ha, J. Jung and J. Kim,
On the coupling of kinetic thermomechanical Cucker-Smale equation and compressible viscous fluid system, J. Math. Fluid Mech., 22 (2020), 4-38.
doi: 10.1007/s00021-019-0466-x. |
[10] |
Y.-P. Choi, S.-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. |
[11] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[12] |
L. DeVille,
Synchronization and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160-187.
doi: 10.1007/s10955-018-2168-9. |
[13] |
P. Degond, A. Frouvelle and S. Merino-Aceituno,
A new flocking model through body attitude coordination, Math. Models Methods Appl. Sci., 27 (2017), 1005-1049.
doi: 10.1142/S0218202517400085. |
[14] |
J.-G. Dong, S.-Y. Ha and D. Kim, From discrete Cucker-Smale model to continuous Cucker-Smale model in a temperature field, J. Math. Phys., 60 (2019), 072705, 22 pp.
doi: 10.1063/1.5084770. |
[15] |
J.-G. Dong, S.-Y. Ha and D. Kim,
On the Cucker-Smale with q-closest neighbors in a self-consistent temperature field, SIAM J. Control and Optimization, 58 (2020), 368-392.
doi: 10.1137/18M1195462. |
[16] |
J.-G. Dong, S.-Y. Ha and D. Kim, Emergence of mono-cluster flocking in the thermomechanical Cucker-Smale model under switching topologies, Analysis and Applications, (2020), 1-38.
doi: 10.1142/S0219530520500025. |
[17] |
J.-G. Dong, S.-Y. Ha, D. Kim and J. Kim,
Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles, J. Differential Equations, 266 (2019), 2373-2407.
doi: 10.1016/j.jde.2018.08.034. |
[18] |
R. Fetecau, H. Park and F. S. Patacchini, Well-posedness and asymptotic behaviour of an aggregation model with intrinsic interactions on sphere and other manifolds, Analysis and Applications. Google Scholar |
[19] |
A. Frouvelle and J.-G. Liu, Long-time dynamics for a simple aggregation equation on the sphere, in International Workshop on Stochastic Dynamics out of Equilibrium, Springer, Cham, 282 (2017), 457-479.
doi: 10.1007/978-3-030-15096-9_16. |
[20] |
R. C. Fetecau and B. Zhang,
Self-organization on Riemannian manifolds, J. Geom. Mech., 11 (2019), 397-426.
doi: 10.3934/jgm.2019020. |
[21] |
S.-Y. Ha, S. Hwang, D. Kim, S.-C. Kim and C. Min, Emergent behaviors of a first-order particle swarm model on the hyperbolic space, J. Math. Phys., 61 (2020), 042701, 23 pp.
doi: 10.1063/1.5066255. |
[22] |
S.-Y. Ha and D. Kim,
A second-order particle swarm model on a sphere and emergent dynamics, SIAM J. Appl. Dyn. Syst., 18 (2019), 80-116.
doi: 10.1137/18M1205996. |
[23] |
S.-Y. Ha and D. Kim,
Emergent behavior of a second-order Lohe matrix model on the unitary group, J. Stat. Phys., 175 (2019), 904-931.
doi: 10.1007/s10955-019-02270-y. |
[24] |
S.-Y. Ha, J. Kim, C. Min, T. Ruggeri and X. Zhang,
Uniform stability and mean-field limit of thermodynamic Cucker-Smale model, Quart. Appl. Math., 77 (2019), 131-176.
doi: 10.1090/qam/1517. |
[25] |
S.-Y. Ha, J. Kim, J. 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. Ha, J. Kim and T. Ruggeri,
Emergent behaviors of thermodynamic Cucker-Smale particles, SIAM J. Math. Anal., 50 (2018), 3092-3121.
doi: 10.1137/17M111064X. |
[27] |
S.-Y. Ha, J. Kim and T. Ruggeri,
From the relativistic mixture of gases to the relativistic cucker-smale flocking, Arch. Rational Mech. Anal., 235 (2020), 1661-1706.
doi: 10.1007/s00205-019-01452-y. |
[28] |
S.-Y. Ha, D. Ko and S. Ryoo,
Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys., 168 (2017), 171-207.
doi: 10.1007/s10955-017-1797-8. |
[29] |
S.-Y. Ha, D. Ko and S. Ryoo,
On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds, J. Stat. Phys., 172 (2018), 1427-1478.
doi: 10.1007/s10955-018-2091-0. |
[30] |
S.-Y. Ha, D. Kim and F. W. Schlöder, Emergent behaviors of Cucker-Smale flocks on Riemannian manifolds, IEEE Trans. Automat. Control, (2020).
doi: 10.1109/TAC.2020.3014096. |
[31] |
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. |
[32] |
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. |
[33] |
M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical systems, and an Introduction to Chaos, Third edition, Elsevier/Academic Press, Amsterdam, 2013.
doi: 10.1016/B978-0-12-382010-5.00001-4.![]() ![]() |
[34] |
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. |
[35] |
J. J$\ddot{u}$rgen, Riemannian Geometry and Geometric Analysis, Universitext. Springer 2011.
doi: 10.1007/978-3-642-21298-7. |
[36] |
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. |
[37] |
J. Markdahl, Synchronization on Riemannian manifolds: Multiply connected implies multistable, IEEE Trans. Automat. Control, (2019).
doi: 10.1109/TAC.2020.3030849. |
[38] |
J. Markdahl, J. Thunberg and J. Gonçalves,
Almost global consensus on the $n$-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.
doi: 10.1109/TAC.2017.2752799. |
[39] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[40] |
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. |
[41] |
R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, Proc. of the 45th IEEE conference on Decision and Control, (2006), 5060-5066.
doi: 10.1109/CDC.2006.376811. |
[42] |
R. Olfati-Saber, J. A. Fax and R. M. Murray,
Consensus and cooperation in networked multi-agent systems, Proc. IEEE, 95 (2007), 215-233.
doi: 10.1109/JPROC.2006.887293. |
[43] |
A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511755743.![]() ![]() |
[44] |
C. W. Reynolds,
Flocks, herds, and schools: A distributed behavioral model, Comput. Graph, 21 (1987), 25-34.
doi: 10.1145/280811.281008. |
[45] |
L. M. Ritchie, M. A. Lohe and A. G. Williams, Synchronization of relativistic particles in the hyperbolic Kuramoto model, Chaos, 28 (2018), 053116, 11 pp.
doi: 10.1063/1.5021701. |
[46] |
A. Sarlette, S. Bonnabel and R. Sepulchre,
Coordinated motion design on Lie groups, IEEE Trans. Automat. Control, 55 (2010), 1047-1058.
doi: 10.1109/TAC.2010.2042003. |
[47] |
A. Sarlette and R. Sepulchre,
Consensus optimization on manifolds, SIAM J. Control Optim., 48 (2009), 56-76.
doi: 10.1137/060673400. |
[48] |
R. Tron, B. Afsari and R. Vidal,
Riemannian consensus for manifolds with bounded curvature, IEEE Trans. Automat. Contr., 58 (2013), 921-934.
doi: 10.1109/TAC.2012.2225533. |
[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] |
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. |
[51] |
T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet,
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. |
[52] |
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. |
[53] |
A. T. Winfree, The Geometry of Biological Time, Springer-Verlag, New York, 1980. |
[54] |
J. Zhu,
Synchronization of Kuramoto model in a high-dimensional linear space, Phys. Lett. A, 377 (2013), 2939-2943.
doi: 10.1016/j.physleta.2013.09.010. |
[55] |
A. A. Z$\ddot{u}$lke and H. Varela, The effect of temperature on the coupled slow and fast dynamics of an electrochemical oscillator, Rep. Sci., (2016), 24553.
doi: 10.1038/srep24553. |
[56] |
J. Zhang, J. Zhu and C. Qian,
On equilibria and consensus of the Lohe model with identical oscillators, SIAM J. Appl. Dyn. Syst., 17 (2018), 1716-1741.
doi: 10.1137/17M112765X. |
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