August  2021, 26(8): 4227-4261. doi: 10.3934/dcdsb.2020286

Emergent behaviors of the generalized Lohe matrix model

1. 

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

2. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Hansol Park

Received  March 2020 Revised  July 2020 Published  August 2021 Early access  October 2020

Fund Project: The work of S.-Y. Ha is supported by NRF-2020R1A2C3A01003881. The work of H. Park is supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)

We present a first-order aggregation model on the space of complex matrices which can be derived from the Lohe tensor model on the space of tensors with the same rank and size. We call such matrix-valued aggregation model as "the generalized Lohe matrix model". For the proposed matrix model with two cubic coupling terms, we study several structural properties such as the conservation laws, solution splitting property. In particular, for the case of only one coupling, we reformulate the reduced Lohe matrix model into the Lohe matrix model with a diagonal frustration, and provide several sufficient frameworks leading to the complete and practical aggregations. For the estimates of collective dynamics, we use a nonlinear functional approach using an ensemble diameter which measures the degree of aggregation.

Citation: Seung-Yeal Ha, Hansol Park. Emergent behaviors of the generalized Lohe matrix model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4227-4261. doi: 10.3934/dcdsb.2020286
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. 

[2]

D. Aeyels and J. Rogge, Stability of phase locking and existence of frequency in networks of globally coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941. 

[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]

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.

[5]

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.

[6]

A. J. Bernoff and C. M. Topaz, Nonlocal aggregation models: a primer of swarm equilibria, SIAM Rev., 55 (2013), 709-747.  doi: 10.1137/130925669.

[7]

A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250.  doi: 10.1137/100804504.

[8]

J. C. Bronski, T. E. Carty and S. E. Simpson, A matrix-valued Kuramoto model, J. Stat. Phys., 178 (2020), 595-624, Archived as arXiv: 1903.09223. doi: 10.1007/s10955-019-02442-w.

[9]

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

[10]

D. Chi, S.-H. Choi and S.-Y. Ha, Emergent behaviors of a holonomic particle system on a sphere, J. Math. Phys., 55 (2014), 052703, 18pp. doi: 10.1063/1.4878117.

[11]

S.-H. Choi and S.-Y. Ha, Emergent behaviors of quantum Lohe oscillators with all-to-all couplings, J. Nonlinear Sci., 25 (2015), 1257-1283.  doi: 10.1007/s00332-015-9255-8.

[12]

S.-H. Choi and S.-Y. Ha, Time-delayed interactions and synchronization of identical Lohe oscillators, Quart. Appl. Math., 74 (2016), 297-319.  doi: 10.1090/qam/1417.

[13]

S.-H. Choi and S.-Y. Ha, Large-time dynamics of the asymptotic Lohe model with a small-time delay, J. Phys. A: Mathematical and Theoretical., 48 (2015), 425101, 34pp. doi: 10.1088/1751-8113/48/42/425101.

[14]

S.-H. Choi and S.-Y. Ha, Quantum synchronization of the Schödinger-Lohe model, J. Phys. A: Mathematical and Theoretical, 47 (2014), 355104, 16pp. doi: 10.1088/1751-8113/47/35/355104.

[15]

S.-H. Choi and S.-Y. Ha, Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM. J. App. Dyn., 13 (2013), 1417-1441.  doi: 10.1137/140961699.

[16]

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.

[17]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automatic Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884.

[18]

P. DegondA. FrouvelleS. Merino-Aceituno and A. Trescases, Quaternions in collective dynamics, Multiscale Model. Simul., 16 (2018), 28-77.  doi: 10.1137/17M1135207.

[19]

P. DegondA. 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.

[20]

L. DeVille, Aggregation and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160-187.  doi: 10.1007/s10955-018-2168-9.

[21]

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.

[22]

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.

[23]

F. Dörfler and F. Bullo, Exploring synchronization in complex oscillator networks, in IEEE 51st Annual Conference on Decision and Control (CDC), (2012), 7157-7170.

[24]

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.

[25]

S.-Y. Ha, D. Kim, H. Park and S. W. Ryoo, Constants of motions for the finite-dimensional Lohe type models with frustration and applications to emergent dynamics, Submitted.

[26]

S.-Y. Ha, M. Kang and D. Kim, Emergent behaviors of high-dimensional Kuramoto models on Stiefel manifolds, Submitted.

[27]

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.

[28]

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.

[29]

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.

[30]

S.-Y. HaD. Ko and S. W. 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.

[31]

S.-Y. HaZ. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070.  doi: 10.1016/j.jde.2013.07.013.

[32]

S.-Y. Ha and H. Park, From the Lohe tensor model to the Lohe Hermitian sphere model and emergent dynamics, SIAM J. Appl. Dyn. Syst., 19 (2020), 1312-1342.  doi: 10.1137/19M1288553.

[33]

S.-Y. Ha and H. Park, Emergent behaviors of Lohe tensor flocks, J. Stat. Phys., 178 (2020), 1268-1292.  doi: 10.1007/s10955-020-02505-3.

[34]

S.-Y. Ha and S. W. Ryoo, On the emergence and orbital Stability of phase-locked states for the Lohe model, J. Stat. Phys, 163 (2016), 411-439.  doi: 10.1007/s10955-016-1481-4.

[35]

V. Jaćimović and A. Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos, 28 (2018), 083105, 1-8. doi: 10.1063/1.5029485.

[36]

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

[37]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 30 (1975), 420.

[38]

M. A. Lohe, Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization, J. Math. Phys., 60 (2019), 072701, 25pp. doi: 10.1063/1.5085248.

[39]

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.

[40]

M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101.

[41]

J. MarkdahlJ. Thunberg and J. Goncalves, Almost global consensus on the n-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.  doi: 10.1109/TAC.2017.2752799.

[42]

J. Markdahl, J. Thunberg and J. Goncalves, High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally, Automatica J. IFAC, 133 (2020), 108736, 9pp. doi: 10.1016/j.automatica.2019.108736.

[43]

J. MarkdahlJ. Thunberg and J. Goncalves, Towards almost global synchronization on the stiefel manifold, 2018 IEEE Conference on Decision and Control (CDC), 2018 (2018), 1664-1675. 

[44]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Science, 17 (2007), 309-347.  doi: 10.1007/s00332-006-0806-x.

[45]

R. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillators, Physica D, 205 (2005), 249-266.  doi: 10.1016/j.physd.2005.01.017.

[46]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975.

[47] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[48]

S. H. Strogatz and R. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635.  doi: 10.1007/BF01029202.

[49]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[50]

J. ThunbergJ. MarkdahlF. Bernard and J. Goncalves, A lifting method for analyzing distributed synchronization on the unit sphere, Automatica J. IFAC, 96 (2018), 253-258.  doi: 10.1016/j.automatica.2018.07.007.

[51]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[52]

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.

[53]

M. Verwoerd and O. Mason, On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph, SIAM J. Appl. Dyn. Syst., 8 (2009), 417-453.  doi: 10.1137/080725726.

[54]

M. Verwoerd and O. Mason, Global phase-locking in finite populations of phase-coupled oscillators, SIAM J. Appl. Dyn. Syst., 7 (2008), 134-160.  doi: 10.1137/070686858.

[55]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

[56]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

[57]

A. T. Winfree, The Geometry of Biological Time, Springer-Verlag, Berlin-New York, 1980.

[58]

J. Zhu, Synchronization of Kuramoto model in a high-dimensional linear space, Physics Letters A, 377 (2013), 2939-2943.  doi: 10.1016/j.physleta.2013.09.010.

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. 

[2]

D. Aeyels and J. Rogge, Stability of phase locking and existence of frequency in networks of globally coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941. 

[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]

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.

[5]

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.

[6]

A. J. Bernoff and C. M. Topaz, Nonlocal aggregation models: a primer of swarm equilibria, SIAM Rev., 55 (2013), 709-747.  doi: 10.1137/130925669.

[7]

A. J. Bernoff and C. M. Topaz, A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250.  doi: 10.1137/100804504.

[8]

J. C. Bronski, T. E. Carty and S. E. Simpson, A matrix-valued Kuramoto model, J. Stat. Phys., 178 (2020), 595-624, Archived as arXiv: 1903.09223. doi: 10.1007/s10955-019-02442-w.

[9]

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

[10]

D. Chi, S.-H. Choi and S.-Y. Ha, Emergent behaviors of a holonomic particle system on a sphere, J. Math. Phys., 55 (2014), 052703, 18pp. doi: 10.1063/1.4878117.

[11]

S.-H. Choi and S.-Y. Ha, Emergent behaviors of quantum Lohe oscillators with all-to-all couplings, J. Nonlinear Sci., 25 (2015), 1257-1283.  doi: 10.1007/s00332-015-9255-8.

[12]

S.-H. Choi and S.-Y. Ha, Time-delayed interactions and synchronization of identical Lohe oscillators, Quart. Appl. Math., 74 (2016), 297-319.  doi: 10.1090/qam/1417.

[13]

S.-H. Choi and S.-Y. Ha, Large-time dynamics of the asymptotic Lohe model with a small-time delay, J. Phys. A: Mathematical and Theoretical., 48 (2015), 425101, 34pp. doi: 10.1088/1751-8113/48/42/425101.

[14]

S.-H. Choi and S.-Y. Ha, Quantum synchronization of the Schödinger-Lohe model, J. Phys. A: Mathematical and Theoretical, 47 (2014), 355104, 16pp. doi: 10.1088/1751-8113/47/35/355104.

[15]

S.-H. Choi and S.-Y. Ha, Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM. J. App. Dyn., 13 (2013), 1417-1441.  doi: 10.1137/140961699.

[16]

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.

[17]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automatic Control, 54 (2009), 353-357.  doi: 10.1109/TAC.2008.2007884.

[18]

P. DegondA. FrouvelleS. Merino-Aceituno and A. Trescases, Quaternions in collective dynamics, Multiscale Model. Simul., 16 (2018), 28-77.  doi: 10.1137/17M1135207.

[19]

P. DegondA. 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.

[20]

L. DeVille, Aggregation and stability for quantum Kuramoto, J. Stat. Phys., 174 (2019), 160-187.  doi: 10.1007/s10955-018-2168-9.

[21]

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.

[22]

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.

[23]

F. Dörfler and F. Bullo, Exploring synchronization in complex oscillator networks, in IEEE 51st Annual Conference on Decision and Control (CDC), (2012), 7157-7170.

[24]

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.

[25]

S.-Y. Ha, D. Kim, H. Park and S. W. Ryoo, Constants of motions for the finite-dimensional Lohe type models with frustration and applications to emergent dynamics, Submitted.

[26]

S.-Y. Ha, M. Kang and D. Kim, Emergent behaviors of high-dimensional Kuramoto models on Stiefel manifolds, Submitted.

[27]

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.

[28]

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.

[29]

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.

[30]

S.-Y. HaD. Ko and S. W. 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.

[31]

S.-Y. HaZ. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070.  doi: 10.1016/j.jde.2013.07.013.

[32]

S.-Y. Ha and H. Park, From the Lohe tensor model to the Lohe Hermitian sphere model and emergent dynamics, SIAM J. Appl. Dyn. Syst., 19 (2020), 1312-1342.  doi: 10.1137/19M1288553.

[33]

S.-Y. Ha and H. Park, Emergent behaviors of Lohe tensor flocks, J. Stat. Phys., 178 (2020), 1268-1292.  doi: 10.1007/s10955-020-02505-3.

[34]

S.-Y. Ha and S. W. Ryoo, On the emergence and orbital Stability of phase-locked states for the Lohe model, J. Stat. Phys, 163 (2016), 411-439.  doi: 10.1007/s10955-016-1481-4.

[35]

V. Jaćimović and A. Crnkić, Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere, Chaos, 28 (2018), 083105, 1-8. doi: 10.1063/1.5029485.

[36]

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

[37]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes Theor. Phys., 30 (1975), 420.

[38]

M. A. Lohe, Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization, J. Math. Phys., 60 (2019), 072701, 25pp. doi: 10.1063/1.5085248.

[39]

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.

[40]

M. A. Lohe, Non-abelian Kuramoto model and synchronization, J. Phys. A: Math. Theor., 42 (2009), 395101.

[41]

J. MarkdahlJ. Thunberg and J. Goncalves, Almost global consensus on the n-sphere, IEEE Trans. Automat. Control, 63 (2018), 1664-1675.  doi: 10.1109/TAC.2017.2752799.

[42]

J. Markdahl, J. Thunberg and J. Goncalves, High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally, Automatica J. IFAC, 133 (2020), 108736, 9pp. doi: 10.1016/j.automatica.2019.108736.

[43]

J. MarkdahlJ. Thunberg and J. Goncalves, Towards almost global synchronization on the stiefel manifold, 2018 IEEE Conference on Decision and Control (CDC), 2018 (2018), 1664-1675. 

[44]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Science, 17 (2007), 309-347.  doi: 10.1007/s00332-006-0806-x.

[45]

R. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillators, Physica D, 205 (2005), 249-266.  doi: 10.1016/j.physd.2005.01.017.

[46]

C. S. Peskin, Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York, 1975.

[47] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[48]

S. H. Strogatz and R. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635.  doi: 10.1007/BF01029202.

[49]

S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.  doi: 10.1016/S0167-2789(00)00094-4.

[50]

J. ThunbergJ. MarkdahlF. Bernard and J. Goncalves, A lifting method for analyzing distributed synchronization on the unit sphere, Automatica J. IFAC, 96 (2018), 253-258.  doi: 10.1016/j.automatica.2018.07.007.

[51]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[52]

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.

[53]

M. Verwoerd and O. Mason, On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph, SIAM J. Appl. Dyn. Syst., 8 (2009), 417-453.  doi: 10.1137/080725726.

[54]

M. Verwoerd and O. Mason, Global phase-locking in finite populations of phase-coupled oscillators, SIAM J. Appl. Dyn. Syst., 7 (2008), 134-160.  doi: 10.1137/070686858.

[55]

T. Vicsek and A. Zefeiris, Collective motion, Phys. Rep., 517 (2012), 71-140. 

[56]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. 

[57]

A. T. Winfree, The Geometry of Biological Time, Springer-Verlag, Berlin-New York, 1980.

[58]

J. Zhu, Synchronization of Kuramoto model in a high-dimensional linear space, Physics Letters A, 377 (2013), 2939-2943.  doi: 10.1016/j.physleta.2013.09.010.

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