March  2017, 12(1): 1-24. doi: 10.3934/nhm.2017001

Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Korea

2. 

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea

3. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received  January 2016 Revised  November 2016 Published  February 2017

Fund Project: The work of S.-Y. Ha was supported by the National Research Foundation of Korea (NRF2014R1A2A205002096), and the work of Z. Li was supported by the National Natural Sci ence Foundation of China Grant 11401135 and the Fundamental Research Funds for the Central Universities of China (HIT.BRETIII.201501 and HIT.PIRS.201610)

We study the emergence of local exponential synchronization in an ensemble of generalized heterogeneous Kuramoto oscillators with different intrinsic dynamics. In the classic Kuramoto model, intrinsic dynamics are given by the Kronecker flow with constant natural frequencies. We generalize the constant natural frequencies to smooth functions that depend on the state and time so that it can describe a more realistic situation arising from neuroscience. In this setting, the ensemble of generalized Kuramoto oscillators loses its synchronization even when the coupling strength is large. This leads to the study of a concept of "relaxed" synchronization, which is called "practical synchronization" in literature. In this new concept of "weak" synchronization, the phase diameter of the entire ensemble is uniformly bounded by some constant inversely proportional to the coupling strength. We focus on the complete synchronizability of a subensemble consisting of generalized Kuramoto oscillators with the same intrinsic dynamics; moreover, we provide several sufficient frameworks leading to local exponential synchronization of each homogeneous subensemble, although the whole ensemble is not fully synchronized. This is a generalization of an earlier analytical result regarding practical synchronization. We also provide several numerical simulations and compare them with analytical results.

Citation: Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li. Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks & Heterogeneous Media, 2017, 12 (1) : 1-24. doi: 10.3934/nhm.2017001
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.  doi: 10.1103/RevModPhys.77.137.  Google Scholar

[2]

T. M. AntonsenR. T. FaghihM. GirvanE. Ott and J. Platig, External periodic driving of large systems of globally coupled phase oscillators, Chaos, 18 (2008), 037112, 10pp.  doi: 10.1063/1.2952447.  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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S. Bowong and J. Tewa, Practical adaptive synchronization of a class of uncertain chaotic systems, Nonlinear Dynam., 56 (2009), 57-68.  doi: 10.1007/s11071-008-9379-6.  Google Scholar

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

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L. M. Childs and S. H. Strogatz, Stability diagram for the forced Kuramoto model, Chaos: An Interdisciplinary Journal of Nonlinear Science, 18 (2008), arXiv: 0807.4717v2. doi: 10.1063/1.3049136.  Google Scholar

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

[10]

Y.-P. ChoiS.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D, 240 (2011), 32-44.  doi: 10.1016/j.physd.2010.08.004.  Google Scholar

[11]

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

[12]

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

[13]

F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillator, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X.  Google Scholar

[14]

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

[15]

S.-Y. Ha and M.-J. Kang, Fast and slow relaxations to bi-cluster configurations for the ensemble of Kuramoto oscillators,, Quarterly of Applied Mathematics, 71 (2013), 707-728.  doi: 10.1090/S0033-569X-2013-01302-0.  Google Scholar

[16]

S.-Y. HaH. K. Kim and J.-Y. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015), 1441-1462.  doi: 10.1088/0951-7715/28/5/1441.  Google Scholar

[17]

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

[18]

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

[19]

S.-Y. Ha and Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Commun. Math. Sciences, 12 (2014), 485-508.  doi: 10.4310/CMS.2014.v12.n3.a5.  Google Scholar

[20]

S.-Y. HaS. E. Noh and J. Park, Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators, Analysis and Applications, (2016), p1.  doi: 10.1142/S0219530516500111.  Google Scholar

[21]

S.-Y. HaS. E. Noh and J. Park, Practical synchronization of Kuramoto system with an intrinsic dynamics, Networks and Heterogeneous Media, 10 (2015), 787-807.  doi: 10.3934/nhm.2015.10.787.  Google Scholar

[22]

A. Jadbabaie, N. Motee and M. Barahona, On the Stability of the Kuramoto Model of Coupled Nonlinear Oscillators, In Proceedings of the American Control Conference. Boston, Massachusetts, 2004. Google Scholar

[23]

J. Kim, J. Yang, J. Kim and H. Shim, Practical Consensus for Heterogeneous Linear Timevarying Multi-agent Systems, In Proceedings of 12th International Conference on Control, Automation and Systems. Jeju Island, Korea, 2012. Google Scholar

[24]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture notes in theoretical physics, 30 (1975), p420.   Google Scholar

[25]

Z LiX. Xue and D. Yu, On the Lojasiewicz exponent of Kuramoto model, J. Math. Phys., 56 (2015), 0227041, 20pp.  doi: 10.1063/1.4908104.  Google Scholar

[26]

Z. LiX. Xue and D. Yu, Synchronization and tansient stability in power grids based on Lojasiewicz inequalities, SIAM J. Control Optim., 52 (2014), 2482-2511.  doi: 10.1137/130950604.  Google Scholar

[27]

M. MaJ. Zhou and J. Cai, Practical synchronization of second-order nonautonomous systems with parameter mismatch and its applications, Nonlinear Dynam., 69 (2012), 1285-1292.  doi: 10.1007/s11071-012-0346-x.  Google Scholar

[28]

M. MaJ. Zhou and J. Cai, Practical synchronization of non autonomous systems with uncertain parameter mismatch via a single feedback control, Int. J. Mod Phys C, 23 (2012), 1250073.   Google Scholar

[29]

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

[30]

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

[31]

E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18 (2008), 37113, 6pp.  doi: 10.1063/1.2930766.  Google Scholar

[32] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Pres, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[33]

J. G. RestrepoE. Ott and B. R. Hunt, Synchronization in large directed networks of coupled phase oscillators, Chaos, 16 (2006), 015107, 10pp.  doi: 10.1063/1.2148388.  Google Scholar

[34]

H. Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Prog. Theor. Phys., 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.  Google Scholar

[35]

P. S SkardaE. Ott and J. G. Restrepo, Cluster synchrony in systems of coupled phase oscillators with higher-order coupling, Phys. Rev. E, 143 (2000), 036208.   Google Scholar

[36]

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

[37]

S. H. Strogatz, Human sleep and circadian rhythms: A simple model based on two coupled oscillators, J. Math. Biol., 25 (1987), 327-347.  doi: 10.1007/BF00276440.  Google Scholar

[38]

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

[39]

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

[40]

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

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.  doi: 10.1103/RevModPhys.77.137.  Google Scholar

[2]

T. M. AntonsenR. T. FaghihM. GirvanE. Ott and J. Platig, External periodic driving of large systems of globally coupled phase oscillators, Chaos, 18 (2008), 037112, 10pp.  doi: 10.1063/1.2952447.  Google Scholar

[3]

A. ArenasA. Díaz-GuileraJ. KurthsY. Moreno and C. Zhou, Synchronization in complex networks, Physics Reports, 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002.  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]

S. Bowong and J. Tewa, Practical adaptive synchronization of a class of uncertain chaotic systems, Nonlinear Dynam., 56 (2009), 57-68.  doi: 10.1007/s11071-008-9379-6.  Google Scholar

[6]

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

[7]

L. M. Childs and S. H. Strogatz, Stability diagram for the forced Kuramoto model, Chaos: An Interdisciplinary Journal of Nonlinear Science, 18 (2008), arXiv: 0807.4717v2. doi: 10.1063/1.3049136.  Google Scholar

[8]

J. ChoS.-Y HaF. HuangJ. Jin and D. Ko, Emergence of bi-cluster flocking for agent-based models with unit speed constraint, Analysis and Applications, 14 (2016), p39.  doi: 10.1142/S0219530515400023.  Google Scholar

[9]

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

[10]

Y.-P. ChoiS.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D, 240 (2011), 32-44.  doi: 10.1016/j.physd.2010.08.004.  Google Scholar

[11]

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

[12]

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

[13]

F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillator, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099.  doi: 10.1137/10081530X.  Google Scholar

[14]

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

[15]

S.-Y. Ha and M.-J. Kang, Fast and slow relaxations to bi-cluster configurations for the ensemble of Kuramoto oscillators,, Quarterly of Applied Mathematics, 71 (2013), 707-728.  doi: 10.1090/S0033-569X-2013-01302-0.  Google Scholar

[16]

S.-Y. HaH. K. Kim and J.-Y. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015), 1441-1462.  doi: 10.1088/0951-7715/28/5/1441.  Google Scholar

[17]

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

[18]

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

[19]

S.-Y. Ha and Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Commun. Math. Sciences, 12 (2014), 485-508.  doi: 10.4310/CMS.2014.v12.n3.a5.  Google Scholar

[20]

S.-Y. HaS. E. Noh and J. Park, Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators, Analysis and Applications, (2016), p1.  doi: 10.1142/S0219530516500111.  Google Scholar

[21]

S.-Y. HaS. E. Noh and J. Park, Practical synchronization of Kuramoto system with an intrinsic dynamics, Networks and Heterogeneous Media, 10 (2015), 787-807.  doi: 10.3934/nhm.2015.10.787.  Google Scholar

[22]

A. Jadbabaie, N. Motee and M. Barahona, On the Stability of the Kuramoto Model of Coupled Nonlinear Oscillators, In Proceedings of the American Control Conference. Boston, Massachusetts, 2004. Google Scholar

[23]

J. Kim, J. Yang, J. Kim and H. Shim, Practical Consensus for Heterogeneous Linear Timevarying Multi-agent Systems, In Proceedings of 12th International Conference on Control, Automation and Systems. Jeju Island, Korea, 2012. Google Scholar

[24]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture notes in theoretical physics, 30 (1975), p420.   Google Scholar

[25]

Z LiX. Xue and D. Yu, On the Lojasiewicz exponent of Kuramoto model, J. Math. Phys., 56 (2015), 0227041, 20pp.  doi: 10.1063/1.4908104.  Google Scholar

[26]

Z. LiX. Xue and D. Yu, Synchronization and tansient stability in power grids based on Lojasiewicz inequalities, SIAM J. Control Optim., 52 (2014), 2482-2511.  doi: 10.1137/130950604.  Google Scholar

[27]

M. MaJ. Zhou and J. Cai, Practical synchronization of second-order nonautonomous systems with parameter mismatch and its applications, Nonlinear Dynam., 69 (2012), 1285-1292.  doi: 10.1007/s11071-012-0346-x.  Google Scholar

[28]

M. MaJ. Zhou and J. Cai, Practical synchronization of non autonomous systems with uncertain parameter mismatch via a single feedback control, Int. J. Mod Phys C, 23 (2012), 1250073.   Google Scholar

[29]

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

[30]

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

[31]

E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18 (2008), 37113, 6pp.  doi: 10.1063/1.2930766.  Google Scholar

[32] A. PikovskyM. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Pres, Cambridge, 2001.  doi: 10.1017/CBO9780511755743.  Google Scholar
[33]

J. G. RestrepoE. Ott and B. R. Hunt, Synchronization in large directed networks of coupled phase oscillators, Chaos, 16 (2006), 015107, 10pp.  doi: 10.1063/1.2148388.  Google Scholar

[34]

H. Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Prog. Theor. Phys., 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.  Google Scholar

[35]

P. S SkardaE. Ott and J. G. Restrepo, Cluster synchrony in systems of coupled phase oscillators with higher-order coupling, Phys. Rev. E, 143 (2000), 036208.   Google Scholar

[36]

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

[37]

S. H. Strogatz, Human sleep and circadian rhythms: A simple model based on two coupled oscillators, J. Math. Biol., 25 (1987), 327-347.  doi: 10.1007/BF00276440.  Google Scholar

[38]

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

[39]

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

[40]

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

Figure 1.  Plot of phases versus time
Figure 2.  Bipartite network
Figure 3.  All-to-all network
Figure 4.  Mixture of three homogeneous ensembles on an all-toall network
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