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 & 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.   Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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