American Institute of Mathematical Sciences

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

Asymptotic synchronous behavior of Kuramoto type models with frustrations

Seung-Yeal Ha 1, , Yongduck Kim 2, and  Zhuchun Li 3,

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.
Keywords: frustration, Kuramoto model, synchronization., bicluster configurations.
Mathematics Subject Classification: 92D25, 34C15, 76N1.
Citation: Seung-Yeal Ha, Yongduck Kim, Zhuchun Li. Asymptotic synchronous behavior of Kuramoto type models with frustrations. Networks and 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.

[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.

[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.

[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.

[5]

H. Chiba, A proof of the Kuramoto's conjecture for a bifurcation structure of the infinite dimensional Kuramoto model,, preprint, (). 

[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.

[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.

[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.

[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.

[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.

[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.

[12]

F. Dörfler and F. Bullo, Synchronization in complex oscillator networks: A survey, submitted, (2013).

[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.

[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.

[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.

[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.

[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.

[18]

S.-Y. Ha, C. Lattanzio, B. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.

[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.

[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.

[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.

[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.

[23]

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

[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.

[25]

C. R. Laing, Chimera states in heterogeneous networks, Chaos, 19 (2009), 013113. doi: 10.1063/1.3068353.

[26]

Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011), 016231.

[27]

S. Lück and A. Pikovsky, Dynamics of multi-frequency oscillator ensembles with resonant coupling, Phys. Lett. A, 375 (2011), 2714-2719.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[36]

T. Tanaka, T. Aoki and T. Aoyagi, Dynamics in co-evolving networks of active elements, Forma, 24 (2009), 17-22.

[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.

[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.

[39]

Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2011), 703-707.

show all references

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.

[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.

[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.

[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.

[5]

H. Chiba, A proof of the Kuramoto's conjecture for a bifurcation structure of the infinite dimensional Kuramoto model,, preprint, (). 

[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.

[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.

[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.

[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.

[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.

[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.

[12]

F. Dörfler and F. Bullo, Synchronization in complex oscillator networks: A survey, submitted, (2013).

[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.

[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.

[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.

[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.

[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.

[18]

S.-Y. Ha, C. Lattanzio, B. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103.

[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.

[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.

[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.

[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.

[23]

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

[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.

[25]

C. R. Laing, Chimera states in heterogeneous networks, Chaos, 19 (2009), 013113. doi: 10.1063/1.3068353.

[26]

Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011), 016231.

[27]

S. Lück and A. Pikovsky, Dynamics of multi-frequency oscillator ensembles with resonant coupling, Phys. Lett. A, 375 (2011), 2714-2719.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[36]

T. Tanaka, T. Aoki and T. Aoyagi, Dynamics in co-evolving networks of active elements, Forma, 24 (2009), 17-22.

[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.

[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.

[39]

Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2011), 703-707.

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