# American Institute of Mathematical Sciences

March  2014, 9(1): 33-64. doi: 10.3934/nhm.2014.9.33

## Asymptotic synchronous behavior of Kuramoto type models with frustrations

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747 2 Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea 3 Department of Mathematics, Harbin Institute of Technology, Harbin 150001

Received  September 2013 Revised  February 2014 Published  April 2014

We present a quantitative asymptotic behavior of coupled Kuramoto oscillators with frustrations and give some sufficient conditions for the parameters and initial condition leading to phase or frequency synchronization. We consider three Kuramoto-type models with frustrations. First, we study a general case with nonidentical oscillators; i.e., the natural frequencies are distributed. Second, as a special case, we study an ensemble of two groups of identical oscillators. For these mixture of two identical Kuramoto oscillator groups, we study the relaxation dynamics from the mixed stage to the phase-locked states via the segregation stage. Finally, we consider a Kuramoto-type model that was recently derived from the Van der Pol equations for two coupled oscillator systems in the work of Lück and Pikovsky [27]. In this case, we provide a framework in which the phase synchronization of each group is attained. Moreover, the constant frustration causes the two groups to segregate from each other, although they have the same natural frequency. We also provide several numerical simulations to confirm our analytical results.
Citation: Seung-Yeal Ha, Yongduck Kim, Zhuchun Li. Asymptotic synchronous behavior of Kuramoto type models with frustrations. Networks & Heterogeneous Media, 2014, 9 (1) : 33-64. doi: 10.3934/nhm.2014.9.33
##### References:
 [1] J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. 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. A. Rogge, Stability of phase locking and existence of entrainment in networks of globally coupled oscillators, in Proc. 6th IFAC Symposium on Nonlinear Control Systems, 3 (2004), 1031-1036. Google Scholar [3] P. Ashwin and J. W. Swift, The dynamics of $n$ weakly coupled identical oscillators, J. Nonlinear Sci., 2 (1992), 69-108. doi: 10.1007/BF02429852.  Google Scholar [4] L. L. Bonilla, J. C. Neu and R. Spigler, Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators, J. Stat. Phys., 67 (1992), 313-330. doi: 10.1007/BF01049037.  Google Scholar [5] H. Chiba, A proof of the Kuramoto's conjecture for a bifurcation structure of the infinite dimensional Kuramoto model,, preprint, ().   Google Scholar [6] Y. Choi, S.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D, 240 (2010), 32-44. doi: 10.1016/j.physd.2010.08.004.  Google Scholar [7] Y. Choi, S.-Y. Ha, S.-E. 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 [8] N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Autom. Control, 54 (2009), 353-357. doi: 10.1109/TAC.2008.2007884.  Google Scholar [9] H. Daido, Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett., 68 (1992), 1073-1076. doi: 10.1103/PhysRevLett.68.1073.  Google Scholar [10] F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Physica D, 234 (2007), 81-89. doi: 10.1016/j.physd.2007.06.025.  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 oscillator networks: A survey, submitted, (2013). 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, M. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010. doi: 10.1073/pnas.1212134110.  Google Scholar [15] 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 [16] S.-Y. Ha and M.-J. Kang, Fast and slow relaxations to bi-cluster configurations for the ensemble of Kuramoto oscillators, Quart. Appl. Math., 71 (2013), 707-728. doi: 10.1090/S0033-569X-2013-01302-0.  Google Scholar [17] S.-Y. Ha, T. 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. Ha, C. Lattanzio, B. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.  Google Scholar [19] S.-Y. Ha, and Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Commun. Math. Sci., 12 (2014), 485-508. doi: 10.4310/CMS.2014.v12.n3.a5.  Google Scholar [20] S.-Y. Ha, Z. 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 [21] S.-Y. Ha and M. Slemrod, A fast-slow dynamical systems theory for the Kuramoto phase model, J. Differential Equations, 251 (2011), 2685-2695. doi: 10.1016/j.jde.2011.04.004.  Google Scholar [22] A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proc. American Control Conf., 5 (2004), 4296-4301. Google Scholar [23] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar [24] Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, 39 (1975), 420-422.  Google Scholar [25] C. R. Laing, Chimera states in heterogeneous networks, Chaos, 19 (2009), 013113. doi: 10.1063/1.3068353.  Google Scholar [26] Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011), 016231. Google Scholar [27] S. Lück and A. Pikovsky, Dynamics of multi-frequency oscillator ensembles with resonant coupling, Phys. Lett. A, 375 (2011), 2714-2719. Google Scholar [28] R. E. 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 [29] 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 [30] R. E. 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 [31] E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, EPL, 83 (2008), 68003. doi: 10.1209/0295-5075/83/68003.  Google Scholar [32] K. Park, S. W. Rhee and M. Y. Choi, Glass synchronization in the network of oscillators with random phase shift, Phys. Rev. E, 57 (1998), 5030-5035. doi: 10.1103/PhysRevE.57.5030.  Google Scholar [33] 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 [34] H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Prog. Theor. Phys., 76 (1986), 576-581. doi: 10.1143/PTP.76.576.  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] T. Tanaka, T. Aoki and T. Aoyagi, Dynamics in co-evolving networks of active elements, Forma, 24 (2009), 17-22.  Google Scholar [37] 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 [38] 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 [39] Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2011), 703-707. Google Scholar

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##### References:
 [1] J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. 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. A. Rogge, Stability of phase locking and existence of entrainment in networks of globally coupled oscillators, in Proc. 6th IFAC Symposium on Nonlinear Control Systems, 3 (2004), 1031-1036. Google Scholar [3] P. Ashwin and J. W. Swift, The dynamics of $n$ weakly coupled identical oscillators, J. Nonlinear Sci., 2 (1992), 69-108. doi: 10.1007/BF02429852.  Google Scholar [4] L. L. Bonilla, J. C. Neu and R. Spigler, Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators, J. Stat. Phys., 67 (1992), 313-330. doi: 10.1007/BF01049037.  Google Scholar [5] H. Chiba, A proof of the Kuramoto's conjecture for a bifurcation structure of the infinite dimensional Kuramoto model,, preprint, ().   Google Scholar [6] Y. Choi, S.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D, 240 (2010), 32-44. doi: 10.1016/j.physd.2010.08.004.  Google Scholar [7] Y. Choi, S.-Y. Ha, S.-E. 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 [8] N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Autom. Control, 54 (2009), 353-357. doi: 10.1109/TAC.2008.2007884.  Google Scholar [9] H. Daido, Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett., 68 (1992), 1073-1076. doi: 10.1103/PhysRevLett.68.1073.  Google Scholar [10] F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Physica D, 234 (2007), 81-89. doi: 10.1016/j.physd.2007.06.025.  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 oscillator networks: A survey, submitted, (2013). 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, M. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010. doi: 10.1073/pnas.1212134110.  Google Scholar [15] 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 [16] S.-Y. Ha and M.-J. Kang, Fast and slow relaxations to bi-cluster configurations for the ensemble of Kuramoto oscillators, Quart. Appl. Math., 71 (2013), 707-728. doi: 10.1090/S0033-569X-2013-01302-0.  Google Scholar [17] S.-Y. Ha, T. 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. Ha, C. Lattanzio, B. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.  Google Scholar [19] S.-Y. Ha, and Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Commun. Math. Sci., 12 (2014), 485-508. doi: 10.4310/CMS.2014.v12.n3.a5.  Google Scholar [20] S.-Y. Ha, Z. 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 [21] S.-Y. Ha and M. Slemrod, A fast-slow dynamical systems theory for the Kuramoto phase model, J. Differential Equations, 251 (2011), 2685-2695. doi: 10.1016/j.jde.2011.04.004.  Google Scholar [22] A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proc. American Control Conf., 5 (2004), 4296-4301. Google Scholar [23] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar [24] Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, 39 (1975), 420-422.  Google Scholar [25] C. R. Laing, Chimera states in heterogeneous networks, Chaos, 19 (2009), 013113. doi: 10.1063/1.3068353.  Google Scholar [26] Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011), 016231. Google Scholar [27] S. Lück and A. Pikovsky, Dynamics of multi-frequency oscillator ensembles with resonant coupling, Phys. Lett. A, 375 (2011), 2714-2719. Google Scholar [28] R. E. 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 [29] 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 [30] R. E. 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 [31] E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, EPL, 83 (2008), 68003. doi: 10.1209/0295-5075/83/68003.  Google Scholar [32] K. Park, S. W. Rhee and M. Y. Choi, Glass synchronization in the network of oscillators with random phase shift, Phys. Rev. E, 57 (1998), 5030-5035. doi: 10.1103/PhysRevE.57.5030.  Google Scholar [33] 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 [34] H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Prog. Theor. Phys., 76 (1986), 576-581. doi: 10.1143/PTP.76.576.  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] T. Tanaka, T. Aoki and T. Aoyagi, Dynamics in co-evolving networks of active elements, Forma, 24 (2009), 17-22.  Google Scholar [37] 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 [38] 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 [39] Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2011), 703-707. Google Scholar
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