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  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 & Continuous Dynamical Systems - B, 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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

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