October  2020, 13(5): 979-1005. doi: 10.3934/krm.2020034

On the generic complete synchronization of the discrete Kuramoto model

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

* Corresponding author: Woojoo Shim

Received  January 2020 Revised  May 2020 Published  August 2020

Fund Project: The work of Woojoo Shim is supported by the National Research Foundation of Korea (NRF-2017R1A5A1015-620)

We study the emergent behavior of discrete-time approximation of the finite-dimensional Kuramoto model. Compared to Zhang and Zhu's recent work in [38], we do not rely on the consistency of one-step foward Euler scheme but analyze the discrete model directly to obtain sharper and more explicit result. More precisely, we present the optimal condition for the convergence and order preserving for identical oscilators with generic initial data. Then, we give the exact convergence rate of the identical oscillators to their limit under the reasonable assuption on time step. Finally, we provide an alternative proof of the asymptotic phase-locking of nonidentical oscillators which can be applied whenever the given Lyapunov functional is continuous and all zeros are isolated.

Citation: Woojoo Shim. On the generic complete synchronization of the discrete Kuramoto model. Kinetic and Related Models, 2020, 13 (5) : 979-1005. doi: 10.3934/krm.2020034
References:
[1]

J. A. AcebrónL. L. BonillaC. J. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys, 77 (2005), 137-185. 

[2]

J. A. AcebrónM. M. LavrentievJr . and and R. Spigler, Spectral analysis and computation for the Kuramoto-Sakaguchi integroparabolic equation, IMA J. Numer. Anal, 21 (2001), 239-263.  doi: 10.1093/imanum/21.1.239.

[3]

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

[4]

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.

[5]

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. doi: 10.1063/1.4745197.

[6]

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. 

[7]

Y.-P. ChoiS.-Y. Ha and J. Morales, Emergent dynamics of the Kuramoto ensemble under the effect of inertia, Discrete Contin. Dyn. Syst, 38 (2018), 4875-4913.  doi: 10.3934/dcds.2018213.

[8]

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

[9]

Y.-P. ChoiS.-Y. Ha and S. Noh, Remarks on the nonlinear stability of the Kuramoto model with inertia, Quart. Appl. Math, 73 (2015), 391-399.  doi: 10.1090/qam/1383.

[10]

Y.-P. ChoiS.-Y. Ha and S.-B. Yun, Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto-Daido model with inertia, Netw. Heterog. Media, 8 (2013), 943-968.  doi: 10.3934/nhm.2013.8.943.

[11]

Y.-P. ChoiZ. LiS.-Y. HaX. Xue and S.-B. Yun, Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow, J. Differential Equations, 257 (2014), 2591-2621.  doi: 10.1016/j.jde.2014.05.054.

[12]

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

[13]

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.

[14]

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.

[15]

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.

[16]

S.-Y. HaT. Ha and J. H. Kim, On the complete synchronization of the Kuramoto phase model, Phys. D, 239 (2010), 1692-1700.  doi: 10.1016/j.physd.2010.05.003.

[17]

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

[18]

S.-Y. Ha, D. Kim, J. Lee and Y. Zhang, Remarks on the stability properties of the Kuramoto-Sakaguchi-Fokker-Planck equation with frustration, Z. Angew. Math. Phys, 69 (2018), 25 pp. doi: 10.1007/s00033-018-0984-z.

[19]

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

[20]

S.-Y. HaH. K. Kim and S. 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.

[21]

S.-Y. HaH. K. Kim and S. Ryoo, On the Finiteness of Collisions and phase-locked States for the Kuramoto Model, J. Stat. Phys., 163 (2016), 1394-1424.  doi: 10.1007/s10955-016-1528-6.

[22]

S.-Y. Ha and Z. Li, Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia, Math. Models Methods Appl. Sci., 26 (2016), 357-382.  doi: 10.1142/S0218202516400054.

[23]

J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166.  doi: 10.1007/BF01048044.

[24]

S.-Y. Ha and Q. Xiao, Nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Plank equation, J. Stat. Phys., 160 (2015), 477-496.  doi: 10.1007/s10955-015-1270-5.

[25]

S.-Y. Ha and Q. Xiao, Remarks on the nonlinear stability of the Kuramoto-Sakaguchi equation, J. Differential Equations, 259 (2015), 2430-2457.  doi: 10.1016/j.jde.2015.03.038.

[26]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Phys., 30 (1975), 420-422. 

[27]

M. M. LavrentievJr . and and R. Spigler, Existence and uniqueness of solutions to the Kuramoto-Sakaguchi nonlinear parabolic integrodifferential equation, Differential Integral Equations, 13 (2000), 649-667. 

[28]

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

[29]

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

[30]

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

[31] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[32]

H. Sakaguchi, Cooperative phenomena in coupled oscillator system under external fields, Progr. Theoret. Phys., 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.

[33]

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

[34]

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.

[35]

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.

[36]

M. Verwoerd and O. Mason, A convergence result for the Kuramoto model with all-to-all coupling, SIAM J. Appl. Dyn. Syst., 10 (2011), 906-920.  doi: 10.1137/090771946.

[37]

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.

[38]

X. Zhang and T. Zhu, Emergent behaviors of the discrete-time Kuramoto model for generic initial configuration, preprint, arXiv: 1909.03358. doi: 10.4310/CMS.2020.v18.n2.a11.

show all references

References:
[1]

J. A. AcebrónL. L. BonillaC. J. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys, 77 (2005), 137-185. 

[2]

J. A. AcebrónM. M. LavrentievJr . and and R. Spigler, Spectral analysis and computation for the Kuramoto-Sakaguchi integroparabolic equation, IMA J. Numer. Anal, 21 (2001), 239-263.  doi: 10.1093/imanum/21.1.239.

[3]

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

[4]

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.

[5]

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. doi: 10.1063/1.4745197.

[6]

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. 

[7]

Y.-P. ChoiS.-Y. Ha and J. Morales, Emergent dynamics of the Kuramoto ensemble under the effect of inertia, Discrete Contin. Dyn. Syst, 38 (2018), 4875-4913.  doi: 10.3934/dcds.2018213.

[8]

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

[9]

Y.-P. ChoiS.-Y. Ha and S. Noh, Remarks on the nonlinear stability of the Kuramoto model with inertia, Quart. Appl. Math, 73 (2015), 391-399.  doi: 10.1090/qam/1383.

[10]

Y.-P. ChoiS.-Y. Ha and S.-B. Yun, Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto-Daido model with inertia, Netw. Heterog. Media, 8 (2013), 943-968.  doi: 10.3934/nhm.2013.8.943.

[11]

Y.-P. ChoiZ. LiS.-Y. HaX. Xue and S.-B. Yun, Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow, J. Differential Equations, 257 (2014), 2591-2621.  doi: 10.1016/j.jde.2014.05.054.

[12]

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

[13]

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.

[14]

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.

[15]

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.

[16]

S.-Y. HaT. Ha and J. H. Kim, On the complete synchronization of the Kuramoto phase model, Phys. D, 239 (2010), 1692-1700.  doi: 10.1016/j.physd.2010.05.003.

[17]

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

[18]

S.-Y. Ha, D. Kim, J. Lee and Y. Zhang, Remarks on the stability properties of the Kuramoto-Sakaguchi-Fokker-Planck equation with frustration, Z. Angew. Math. Phys, 69 (2018), 25 pp. doi: 10.1007/s00033-018-0984-z.

[19]

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

[20]

S.-Y. HaH. K. Kim and S. 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.

[21]

S.-Y. HaH. K. Kim and S. Ryoo, On the Finiteness of Collisions and phase-locked States for the Kuramoto Model, J. Stat. Phys., 163 (2016), 1394-1424.  doi: 10.1007/s10955-016-1528-6.

[22]

S.-Y. Ha and Z. Li, Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia, Math. Models Methods Appl. Sci., 26 (2016), 357-382.  doi: 10.1142/S0218202516400054.

[23]

J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166.  doi: 10.1007/BF01048044.

[24]

S.-Y. Ha and Q. Xiao, Nonlinear instability of the incoherent state for the Kuramoto-Sakaguchi-Fokker-Plank equation, J. Stat. Phys., 160 (2015), 477-496.  doi: 10.1007/s10955-015-1270-5.

[25]

S.-Y. Ha and Q. Xiao, Remarks on the nonlinear stability of the Kuramoto-Sakaguchi equation, J. Differential Equations, 259 (2015), 2430-2457.  doi: 10.1016/j.jde.2015.03.038.

[26]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Mathematical Physics, Lecture Notes in Phys., 30 (1975), 420-422. 

[27]

M. M. LavrentievJr . and and R. Spigler, Existence and uniqueness of solutions to the Kuramoto-Sakaguchi nonlinear parabolic integrodifferential equation, Differential Integral Equations, 13 (2000), 649-667. 

[28]

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

[29]

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

[30]

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

[31] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.
[32]

H. Sakaguchi, Cooperative phenomena in coupled oscillator system under external fields, Progr. Theoret. Phys., 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.

[33]

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

[34]

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.

[35]

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.

[36]

M. Verwoerd and O. Mason, A convergence result for the Kuramoto model with all-to-all coupling, SIAM J. Appl. Dyn. Syst., 10 (2011), 906-920.  doi: 10.1137/090771946.

[37]

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.

[38]

X. Zhang and T. Zhu, Emergent behaviors of the discrete-time Kuramoto model for generic initial configuration, preprint, arXiv: 1909.03358. doi: 10.4310/CMS.2020.v18.n2.a11.

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