October  2018, 38(10): 4875-4913. doi: 10.3934/dcds.2018213

Emergent dynamics of the Kuramoto ensemble under the effect of inertia

1. 

Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, Korea

2. 

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

3. 

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

4. 

Department of Mathematics, University of Maryland, College Park MD 20742-3289, USA

* Corresponding author

Received  August 2017 Revised  January 2018 Published  July 2018

We study the emergent collective behaviors for an ensemble of identical Kuramoto oscillators under the effect of inertia. In the absence of inertial effects, it is well known that the generic initial Kuramoto ensemble relaxes to the phase-locked states asymptotically (emergence of complete synchronization) in a large coupling regime. Similarly, even for the presence of inertial effects, similar collective behaviors are observed numerically for generic initial configurations in a large coupling strength regime. However, this phenomenon has not been verified analytically in full generality yet, although there are several partial results in some restricted set of initial configurations. In this paper, we present several improved complete synchronization estimates for the Kuramoto ensemble with inertia in two frameworks for a finite system. Our improved frameworks describe the emergence of phase-locked states and its structure. Additionally, we show that as the number of oscillators tends to infinity, the Kuramoto ensemble with infinite size can be approximated by the corresponding kinetic mean-field model uniformly in time. Moreover, we also establish the global existence of measure-valued solutions for the Kuramoto equation and its large-time asymptotics.

Citation: Young-Pil Choi, Seung-Yeal Ha, Javier Morales. Emergent dynamics of the Kuramoto ensemble under the effect of inertia. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4875-4913. doi: 10.3934/dcds.2018213
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]

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

[3]

N. J. Balmforth and R. Sassi, A shocking display of synchrony, Physica D, 143 (2000), 21-55. doi: 10.1016/S0167-2789(00)00095-6. Google Scholar

[4]

I. Barbǎlat, Systèmes d'équations différentielles d'oscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 262-270. Google Scholar

[5]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization of the Kuramoto model in the mean-field limit, Comm. Math. Sci., 13 (2015), 1775-1786. doi: 10.4310/CMS.2015.v13.n7.a6. Google Scholar

[6]

F. BolleyA. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, ESAIM: M2AN, 44 (2010), 867-884. doi: 10.1051/m2an/2010045. Google Scholar

[7]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Commun. Math. Phys., 56 (1977), 101-113. doi: 10.1007/BF01611497. Google Scholar

[8]

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

[9]

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

[10]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218. doi: 10.1142/S0218202516500287. Google Scholar

[11]

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

[12]

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

[13]

Y.-P. ChoiS.-Y. HaS. Jung and M. Slemrod, Kuramoto oscillators with inertia: A fast-slow dynamical systems approach, Quart. Appl. Math., 73 (2015), 467-482. doi: 10.1090/qam/1380. Google Scholar

[14]

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

[15]

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. Heter. Media, 8 (2013), 943-968. doi: 10.3934/nhm.2013.8.943. Google Scholar

[16]

Y.-P. ChoiS.-Y. HaZ. LiX. Xue and S.-B. Yun, Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow, J. Differ. Equat., 257 (2014), 2591-2621. doi: 10.1016/j.jde.2014.05.054. Google Scholar

[17]

Y.-P. Choi and Z. Li, On the region of attraction of phase-locked states for swing equations on connected graphs with inhomogeneous damping, submitted.Google Scholar

[18]

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

[19]

R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 48-58. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

G. B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585. doi: 10.1007/BF00164052. Google Scholar

[24]

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

[25]

S.-Y. Ha, Y.-H. Kim, J. Morales and J. Park, Emergence of phase concentration for the Kuramoto-Sakaguchi equation, submitted.Google Scholar

[26]

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

[27]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Comm. Math. Sci., 14 (2016), 1073-1091. doi: 10.4310/CMS.2016.v14.n4.a10. Google Scholar

[28]

S.-Y. HaJ. Kim and X. Zhang, Uniform $\ell_p$-stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181. doi: 10.3934/krm.2018045. Google Scholar

[29]

A. Haraux and M. A. Jendoubi, Convergence of solutions of second order gradient-like systems with analytic nonlinearities, J. Differ. Equat., 114 (1998), 313-320. doi: 10.1006/jdeq.1997.3393. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equation, Lecture Notes in Mathematics, 1048, Springer, Berlin, Heidelberg, 1984, 60-110. doi: 10.1007/BFb0071878. Google Scholar

[36]

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

[37]

H. Spohn, On the Vlasov hierarchy, Math. Methods Appl. Sci., 3 (1981), 445-455. doi: 10.1002/mma.1670030131. Google Scholar

[38]

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

[39]

H. A. Tanaka, A. J. Lichtenberg and S. Oishi, First order phase transition resulting from finite inertia in coupled oscillator systems Phys. Rev. Lett. 78 (1997), 2104. doi: 10.1103/PhysRevLett.78.2104. Google Scholar

[40]

H. A. TanakaA. J. Lichtenberg and S. Oishi, Self-synchronization of coupled oscillators with hysteretic responses, Physica D, 100 (1997), 279-300. doi: 10.1016/S0167-2789(96)00193-5. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

C. Villani, Optimal Transport, Old and New, Springer 2008. doi: 10.1007/978-3-540-71050-9. Google Scholar

[45]

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

[46]

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]

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

[3]

N. J. Balmforth and R. Sassi, A shocking display of synchrony, Physica D, 143 (2000), 21-55. doi: 10.1016/S0167-2789(00)00095-6. Google Scholar

[4]

I. Barbǎlat, Systèmes d'équations différentielles d'oscillations non Linéaires, Rev. Math. Pures Appl., 4 (1959), 262-270. Google Scholar

[5]

D. BenedettoE. Caglioti and U. Montemagno, On the complete phase synchronization of the Kuramoto model in the mean-field limit, Comm. Math. Sci., 13 (2015), 1775-1786. doi: 10.4310/CMS.2015.v13.n7.a6. Google Scholar

[6]

F. BolleyA. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, ESAIM: M2AN, 44 (2010), 867-884. doi: 10.1051/m2an/2010045. Google Scholar

[7]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Commun. Math. Phys., 56 (1977), 101-113. doi: 10.1007/BF01611497. Google Scholar

[8]

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

[9]

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

[10]

J. ChoS.-Y. HaF. HuangC. Jin and D. Ko, Emergence of bi-cluster flocking for the Cucker-Smale model, Math. Models Methods Appl. Sci., 26 (2016), 1191-1218. doi: 10.1142/S0218202516500287. Google Scholar

[11]

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

[12]

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

[13]

Y.-P. ChoiS.-Y. HaS. Jung and M. Slemrod, Kuramoto oscillators with inertia: A fast-slow dynamical systems approach, Quart. Appl. Math., 73 (2015), 467-482. doi: 10.1090/qam/1380. Google Scholar

[14]

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

[15]

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. Heter. Media, 8 (2013), 943-968. doi: 10.3934/nhm.2013.8.943. Google Scholar

[16]

Y.-P. ChoiS.-Y. HaZ. LiX. Xue and S.-B. Yun, Complete entrainment of Kuramoto oscillators with inertia on networks via gradient-like flow, J. Differ. Equat., 257 (2014), 2591-2621. doi: 10.1016/j.jde.2014.05.054. Google Scholar

[17]

Y.-P. Choi and Z. Li, On the region of attraction of phase-locked states for swing equations on connected graphs with inhomogeneous damping, submitted.Google Scholar

[18]

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

[19]

R. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 48-58. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

G. B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585. doi: 10.1007/BF00164052. Google Scholar

[24]

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

[25]

S.-Y. Ha, Y.-H. Kim, J. Morales and J. Park, Emergence of phase concentration for the Kuramoto-Sakaguchi equation, submitted.Google Scholar

[26]

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

[27]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Comm. Math. Sci., 14 (2016), 1073-1091. doi: 10.4310/CMS.2016.v14.n4.a10. Google Scholar

[28]

S.-Y. HaJ. Kim and X. Zhang, Uniform $\ell_p$-stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11 (2018), 1157-1181. doi: 10.3934/krm.2018045. Google Scholar

[29]

A. Haraux and M. A. Jendoubi, Convergence of solutions of second order gradient-like systems with analytic nonlinearities, J. Differ. Equat., 114 (1998), 313-320. doi: 10.1006/jdeq.1997.3393. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

H. Neunzert, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equation, Lecture Notes in Mathematics, 1048, Springer, Berlin, Heidelberg, 1984, 60-110. doi: 10.1007/BFb0071878. Google Scholar

[36]

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

[37]

H. Spohn, On the Vlasov hierarchy, Math. Methods Appl. Sci., 3 (1981), 445-455. doi: 10.1002/mma.1670030131. Google Scholar

[38]

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

[39]

H. A. Tanaka, A. J. Lichtenberg and S. Oishi, First order phase transition resulting from finite inertia in coupled oscillator systems Phys. Rev. Lett. 78 (1997), 2104. doi: 10.1103/PhysRevLett.78.2104. Google Scholar

[40]

H. A. TanakaA. J. Lichtenberg and S. Oishi, Self-synchronization of coupled oscillators with hysteretic responses, Physica D, 100 (1997), 279-300. doi: 10.1016/S0167-2789(96)00193-5. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

C. Villani, Optimal Transport, Old and New, Springer 2008. doi: 10.1007/978-3-540-71050-9. Google Scholar

[45]

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

[46]

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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