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doi: 10.3934/dcdsb.2021308
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Emergent behaviors of discrete Lohe aggregation flows

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

* Corresponding author: Hyungjun Choi

Received  July 2021 Revised  November 2021 Early access January 2022

Fund Project: The work of S.-Y. Ha was supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881)

The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.

Citation: Hyungjun Choi, Seung-Yeal Ha, Hansol Park. Emergent behaviors of discrete Lohe aggregation flows. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021308
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]

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.

[3]

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.

[4]

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.

[5]

A. Bielecki, Estimation of the Euler method error on a Riemannian manifold, Comm. Numer. Methods Engrg., 18 (2002), 757-763.  doi: 10.1002/cnm.516.

[6]

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.

[7]

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

[8]

E. CelledoniH. Marthinsen and B. Owren, An introduction to lie group integrators-basics, new developments and applications, J. Comput. Phys., 257 (2014), 1040-1061.  doi: 10.1016/j.jcp.2012.12.031.

[9]

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, 18 pp. doi: 10.1063/1.4878117.

[10]

S.-H. Choi and S.-Y. Ha, Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM J. Appl. Dyn. Syst., 13 (2014), 1417-1441. doi: 10.1137/140961699.

[11]

Y.-P. Choi and S.-Y. Ha, A simple proof of the complete consensus of discrete-time dynamical networks with time-varying couplings, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 58-69. 

[12]

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.

[13]

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.

[14]

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

[15]

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.

[16]

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

[17]

M. P. do Carmo, Riemannian Geometry, Mathematics: Theory and Applications, Birkhäuser. Boston, Boston, MA, 1992.

[18]

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.

[19]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[20]

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.

[21]

S.-Y. Ha, D. Kim, J. Kim and X. Zhang, Uniform-in-time transition from discrete to continuous dynamics in the Kuramoto synchronization, J. of Math. Phys., 60 (2019), 051508, 21 pp. doi: 10.1063/1.5051788.

[22]

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, Phys. D, 416 (2021), Paper No. 132781, 26 pp. doi: 10.1016/j.physd.2020.132781.

[23]

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.

[24]

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.

[25]

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.

[26]

S.-Y. HaD. Ko and S. W. 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.

[27]

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.

[28]

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

[29]

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.

[30]

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.

[31]

N. H. Ibragimov, Elementary Lie group analysis and ordinary differential equations, Wiley New York, 197, 1999.

[32]

A. IserlesH. Z. Munthe-KaasS. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154.

[33]

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

[34]

T. Jahnke and C. Lubich, Error bounds for exponential operator splittings, BIT Numerical Mathematics, 40 (2000), 735-744.  doi: 10.1023/A:1022396519656.

[35]

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

[36]

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

[37]

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

[38]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A: Math. Theor., 43 (2010), 465301, 20 pp. doi: 10.1088/1751-8113/43/46/465301.

[39]

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.

[40]

J. MarkdahlJ. 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.

[41]

H. Munthe-Kaas, Runge-kutta methods on lie groups, BIT Numerical Mathematics, 38 (1998), 92-111.  doi: 10.1007/BF02510919.

[42]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, IEEE 45th Conference on Decision and Control (CDC), (2006), 5060–5066.

[43]

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

[44] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[45]

W. Shim, On the generic complete synchronization of the discrete Kuramoto model, Kinetic and Related Models, 13 (2020), 979-1005.  doi: 10.3934/krm.2020034.

[46]

G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.  doi: 10.1137/0705041.

[47]

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.

[48]

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.

[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]

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.

[51]

H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10 (1959), 545-551.  doi: 10.1090/S0002-9939-1959-0108732-6.

[52]

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

[53]

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

[54]

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

[55]

X. Zhang and T. Zhu, Emergent behaviors of the discrete-time Kuramoto model for generic initial configuration, Commun. Math. Sci., 18 (2020), 535-570.  doi: 10.4310/CMS.2020.v18.n2.a11.

[56]

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]

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.

[3]

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.

[4]

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.

[5]

A. Bielecki, Estimation of the Euler method error on a Riemannian manifold, Comm. Numer. Methods Engrg., 18 (2002), 757-763.  doi: 10.1002/cnm.516.

[6]

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.

[7]

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

[8]

E. CelledoniH. Marthinsen and B. Owren, An introduction to lie group integrators-basics, new developments and applications, J. Comput. Phys., 257 (2014), 1040-1061.  doi: 10.1016/j.jcp.2012.12.031.

[9]

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, 18 pp. doi: 10.1063/1.4878117.

[10]

S.-H. Choi and S.-Y. Ha, Complete entrainment of Lohe oscillators under attractive and repulsive couplings, SIAM J. Appl. Dyn. Syst., 13 (2014), 1417-1441. doi: 10.1137/140961699.

[11]

Y.-P. Choi and S.-Y. Ha, A simple proof of the complete consensus of discrete-time dynamical networks with time-varying couplings, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 58-69. 

[12]

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.

[13]

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.

[14]

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

[15]

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.

[16]

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

[17]

M. P. do Carmo, Riemannian Geometry, Mathematics: Theory and Applications, Birkhäuser. Boston, Boston, MA, 1992.

[18]

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.

[19]

F. Dörfler and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica J. IFAC, 50 (2014), 1539-1564.  doi: 10.1016/j.automatica.2014.04.012.

[20]

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.

[21]

S.-Y. Ha, D. Kim, J. Kim and X. Zhang, Uniform-in-time transition from discrete to continuous dynamics in the Kuramoto synchronization, J. of Math. Phys., 60 (2019), 051508, 21 pp. doi: 10.1063/1.5051788.

[22]

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, Phys. D, 416 (2021), Paper No. 132781, 26 pp. doi: 10.1016/j.physd.2020.132781.

[23]

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.

[24]

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.

[25]

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.

[26]

S.-Y. HaD. Ko and S. W. 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.

[27]

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.

[28]

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

[29]

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.

[30]

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.

[31]

N. H. Ibragimov, Elementary Lie group analysis and ordinary differential equations, Wiley New York, 197, 1999.

[32]

A. IserlesH. Z. Munthe-KaasS. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365.  doi: 10.1017/S0962492900002154.

[33]

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

[34]

T. Jahnke and C. Lubich, Error bounds for exponential operator splittings, BIT Numerical Mathematics, 40 (2000), 735-744.  doi: 10.1023/A:1022396519656.

[35]

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

[36]

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

[37]

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

[38]

M. A. Lohe, Quantum synchronization over quantum networks, J. Phys. A: Math. Theor., 43 (2010), 465301, 20 pp. doi: 10.1088/1751-8113/43/46/465301.

[39]

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.

[40]

J. MarkdahlJ. 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.

[41]

H. Munthe-Kaas, Runge-kutta methods on lie groups, BIT Numerical Mathematics, 38 (1998), 92-111.  doi: 10.1007/BF02510919.

[42]

R. Olfati-Saber, Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks, IEEE 45th Conference on Decision and Control (CDC), (2006), 5060–5066.

[43]

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

[44] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[45]

W. Shim, On the generic complete synchronization of the discrete Kuramoto model, Kinetic and Related Models, 13 (2020), 979-1005.  doi: 10.3934/krm.2020034.

[46]

G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.  doi: 10.1137/0705041.

[47]

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.

[48]

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.

[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]

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.

[51]

H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc., 10 (1959), 545-551.  doi: 10.1090/S0002-9939-1959-0108732-6.

[52]

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

[53]

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

[54]

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

[55]

X. Zhang and T. Zhu, Emergent behaviors of the discrete-time Kuramoto model for generic initial configuration, Commun. Math. Sci., 18 (2020), 535-570.  doi: 10.4310/CMS.2020.v18.n2.a11.

[56]

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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