June  2018, 13(2): 297-322. doi: 10.3934/nhm.2018013

Uniform stability and mean-field limit for the augmented Kuramoto model

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea

2. 

Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea

3. 

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

4. 

Department of Mathematics and Research Institute of Natural Sciences, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Korea

5. 

Center for Mathematical sciences, Huazhong University of Science and Technology, Wuhan, China

* Corresponding author: Jinyeong Park

Received  July 2017 Revised  March 2018 Published  May 2018

Fund Project: The works of S.-Y. Ha and X. Zhang are supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03). The work of J. Kim is supported by the German Research Foundation (DFG), project number IRTG2235

We present two uniform estimates on stability and mean-field limit for the "augmented Kuramoto model (AKM)" arising from the second-order lifting of the first-order Kuramoto model (KM) for synchronization. In particular, we address three issues such as synchronization estimate, uniform stability and mean-field limit which are valid uniformly in time for the AKM. The derived mean-field equation for the AKM corresponds to the dissipative Vlasov-McKean type equation. The kinetic Kuramoto equation for distributed natural frequencies is not compatible with the frequency variance functional approach for the complete synchronization. In contrast, the kinetic equation for the AKM has a similar structural similarity with the kinetic Cucker-Smale equation which admits the Lyapunov functional approach for the variance. We present sufficient frameworks leading to the uniform stability and mean-field limit for the AKM.

Citation: Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013
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]

D. Aeyels and J. Rogge, Existence of partial entrainment and stability of phase-locking behavior of coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941.   Google Scholar

[3]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.  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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J. 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, 17pp. doi: 10.1063/1.4745197.  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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J. A. CarrilloY.-P. ChoiS.-Y. HaM.-J. Kang and Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys., 156 (2014), 395-415.  doi: 10.1007/s10955-014-1005-z.  Google Scholar

[8]

L. CasettiM. Pettini and E. G. D. Cohen, Phase transitions and topology changes in configuration space, J. Statist. Phys., 111 (2003), 1091-1123.  doi: 10.1023/A:1023044014341.  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

[10]

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

[11]

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

[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 oscillators, SIAM. J. Appl. Dyn. Syst, 10 (2011), 1070-1099.  doi: 10.1137/10081530X.  Google Scholar

[14]

G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol, 22 (1985), 1-9.  doi: 10.1007/BF00276542.  Google Scholar

[15]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[16]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. and Relat. Model., 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[17]

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

[18]

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

[19]

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, 4 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.  Google Scholar

[20]

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

[21]

S.-Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear to Kinet. and Relat. Model. Google Scholar

[22]

S.-Y. HaC. LattanzioB. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.  doi: 10.1090/S0033-569X-2010-01200-7.  Google Scholar

[23]

S.-Y. Ha and M. Slemrod, A fast-slow dynamical systems theory for the Kuramoto type phase model, J. Differential Equations, 251 (2011), 2685-2695.  doi: 10.1016/j.jde.2011.04.004.  Google Scholar

[24]

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

[25]

A. JadbabaieN. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2014), 4296-4301.   Google Scholar

[26]

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

[27]

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

[28]

C. Lancellotti, On the vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory and Statistical Physics, 34 (2005), 523-535.  doi: 10.1080/00411450508951152.  Google Scholar

[29]

D. Mehta, N. S. Daleo, F. Dörfler and J. D. Hauenstein, Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis, Chaos, 25 (2015), 053103, 7pp. doi: 10.1063/1.4919696.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

H. Neunzert, An introduction to the nonlinear boltzmann-vlasov equation, In Kinetic Theories and the Boltzmann Equation(Montecatini, 1981), 60-110, Lecture Notes in Math., 1048, Springer, Berlin, 1984. doi: 10.1007/BFb0071878.  Google Scholar

[34]

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

[35]

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

[36]

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.   Google Scholar

[37]

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

[38]

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

[39]

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

[40]

C. Villani, Optimal Transport, Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[41]

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]

D. Aeyels and J. Rogge, Existence of partial entrainment and stability of phase-locking behavior of coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941.   Google Scholar

[3]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823.  doi: 10.1007/s10955-015-1426-3.  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. 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, 17pp. doi: 10.1063/1.4745197.  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]

J. A. CarrilloY.-P. ChoiS.-Y. HaM.-J. Kang and Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys., 156 (2014), 395-415.  doi: 10.1007/s10955-014-1005-z.  Google Scholar

[8]

L. CasettiM. Pettini and E. G. D. Cohen, Phase transitions and topology changes in configuration space, J. Statist. Phys., 111 (2003), 1091-1123.  doi: 10.1023/A:1023044014341.  Google Scholar

[9]

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

[10]

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

[11]

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

[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 oscillators, SIAM. J. Appl. Dyn. Syst, 10 (2011), 1070-1099.  doi: 10.1137/10081530X.  Google Scholar

[14]

G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol, 22 (1985), 1-9.  doi: 10.1007/BF00276542.  Google Scholar

[15]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2.  Google Scholar

[16]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. and Relat. Model., 1 (2008), 415-435.  doi: 10.3934/krm.2008.1.415.  Google Scholar

[17]

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

[18]

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

[19]

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, 4 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.  Google Scholar

[20]

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

[21]

S.-Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear to Kinet. and Relat. Model. Google Scholar

[22]

S.-Y. HaC. LattanzioB. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.  doi: 10.1090/S0033-569X-2010-01200-7.  Google Scholar

[23]

S.-Y. Ha and M. Slemrod, A fast-slow dynamical systems theory for the Kuramoto type phase model, J. Differential Equations, 251 (2011), 2685-2695.  doi: 10.1016/j.jde.2011.04.004.  Google Scholar

[24]

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

[25]

A. JadbabaieN. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2014), 4296-4301.   Google Scholar

[26]

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

[27]

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

[28]

C. Lancellotti, On the vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory and Statistical Physics, 34 (2005), 523-535.  doi: 10.1080/00411450508951152.  Google Scholar

[29]

D. Mehta, N. S. Daleo, F. Dörfler and J. D. Hauenstein, Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis, Chaos, 25 (2015), 053103, 7pp. doi: 10.1063/1.4919696.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

H. Neunzert, An introduction to the nonlinear boltzmann-vlasov equation, In Kinetic Theories and the Boltzmann Equation(Montecatini, 1981), 60-110, Lecture Notes in Math., 1048, Springer, Berlin, 1984. doi: 10.1007/BFb0071878.  Google Scholar

[34]

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

[35]

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

[36]

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.   Google Scholar

[37]

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

[38]

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

[39]

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

[40]

C. Villani, Optimal Transport, Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[41]

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

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