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December  2015, 10(4): 787-807. doi: 10.3934/nhm.2015.10.787

Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics

Seung-Yeal Ha 1, , Se Eun Noh 2, and  Jinyeong Park 3,

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747
2. 

Department of Mathematics, Myongji University, Yong-In, 449-728, South Korea
3. 

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

Received  September 2014 Revised  June 2015 Published  October 2015

We study the practical synchronization of the Kuramoto dynamics of units distributed over networks. The unit dynamics on the nodes of the network are governed by the interplay between their own intrinsic dynamics and Kuramoto coupling dynamics. We present two sufficient conditions for practical synchronization under homogeneous and heterogeneous forcing. For practical synchronization estimates, we employ the configuration diameter as a Lyapunov functional, and derive a Gronwall-type differential inequality for this value.
Keywords: practical synchronization., external force, intrinsic dynamics, Kuramoto model.
Mathematics Subject Classification: Primary: 70F99; Secondary: 92B2.
Citation: Seung-Yeal Ha, Se Eun Noh, Jinyeong Park. Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks & Heterogeneous Media, 2015, 10 (4) : 787-807. doi: 10.3934/nhm.2015.10.787
References:
[1]

J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena,, Rev. Mod. Phys., 77 (2005), 137.  doi: 10.1103/RevModPhys.77.137.  Google Scholar

[2]

T. M. Antonsen, R. T. Faghih, M. Girvan, E. Ott and J. Platig, External periodic driving of large systems of globally coupled phase oscillators,, Chaos, 18 (2008).  doi: 10.1063/1.2952447.  Google Scholar

[3]

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[4]

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

[5]

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

[6]

L. M. Childs and S. H. Strogatz, Stability diagram for the forced Kuramoto model,, Chaos, 18 (2008).  doi: 10.1063/1.3049136.  Google Scholar

[7]

Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model,, Physica D, 241 (2012), 735.  doi: 10.1016/j.physd.2011.11.011.  Google Scholar

[8]

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

[9]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators,, IEEE Trans. Automatic Control, 54 (2009), 353.  doi: 10.1109/TAC.2008.2007884.  Google Scholar

[10]

X. Dong, J. Xi, Z. Shi and Y. Zhong, Consensus for High-Order Time-Delayed Swarm Systems With Uncertainties and External Disturbances,, in Proceedings of the 30th Chinese Control Conference, (2011).   Google Scholar

[11]

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

[12]

R. Femat and G. Solis-Perales, On the chaos synchronization phenomena,, Physics Letters A, 262 (1999), 50.  doi: 10.1016/S0375-9601(99)00667-2.  Google Scholar

[13]

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

[14]

S.-Y. Ha, E. Jeong and M.-J. Kang, Emergent behavior of a generalized Viscek-type flocking model,, Nonlinearity, 23 (2010), 3139.  doi: 10.1088/0951-7715/23/12/008.  Google Scholar

[15]

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

[16]

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., (2004).   Google Scholar

[17]

J. Kim, J. Yang, J. Kim and H. Shim, Practical Consensus for Heterogeneous Linear Time-Varying Multi-Agent Systems,, in Proceedings of 12th International Conference on Control, (2012).   Google Scholar

[18]

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

[19]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International symposium on mathematical problems in mathematical physics,, Lecture notes in theoretical physics, 39 (1975), 420.   Google Scholar

[20]

P. Louodop, H. Fotsin, E. Megam Ngouonkadi, S. Bowong and H. Cerdeira, Effective Synchronization of a Class of Chua's Chaotic Systems Using an Exponential Feedback Coupling,, Abstr. Appl. Anal., 2013 (2013).   Google Scholar

[21]

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

[22]

M. Ma, J. 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).   Google Scholar

[23]

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.  doi: 10.1007/s00332-006-0806-x.  Google Scholar

[24]

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.  doi: 10.1016/j.physd.2005.01.017.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

E. Steur, L. Kodde and H. Nijmeijer, Synchronization of Diffusively Coupled Electronic Hindmarsh-Rose Oscillators,, in Dynamics and control of hybrid mechanical systems (eds. G. Leonov, (2010), 195.  doi: 10.1142/9789814282321_0013.  Google Scholar

[30]

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

[31]

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

[32]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators,, J. Theor. Biol., 16 (1987), 15.  doi: 10.1016/0022-5193(67)90051-3.  Google Scholar

[33]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen,, Math. Ann., 71 (1912), 441.  doi: 10.1007/BF01456804.  Google Scholar

show all references

References:
[1]

J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena,, Rev. Mod. Phys., 77 (2005), 137.  doi: 10.1103/RevModPhys.77.137.  Google Scholar

[2]

T. M. Antonsen, R. T. Faghih, M. Girvan, E. Ott and J. Platig, External periodic driving of large systems of globally coupled phase oscillators,, Chaos, 18 (2008).  doi: 10.1063/1.2952447.  Google Scholar

[3]

R. Bhatia, Matrix Analysis,, Graduate Text in Mathematics, (1997).  doi: 10.1007/978-1-4612-0653-8.  Google Scholar

[4]

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

[5]

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

[6]

L. M. Childs and S. H. Strogatz, Stability diagram for the forced Kuramoto model,, Chaos, 18 (2008).  doi: 10.1063/1.3049136.  Google Scholar

[7]

Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model,, Physica D, 241 (2012), 735.  doi: 10.1016/j.physd.2011.11.011.  Google Scholar

[8]

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

[9]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators,, IEEE Trans. Automatic Control, 54 (2009), 353.  doi: 10.1109/TAC.2008.2007884.  Google Scholar

[10]

X. Dong, J. Xi, Z. Shi and Y. Zhong, Consensus for High-Order Time-Delayed Swarm Systems With Uncertainties and External Disturbances,, in Proceedings of the 30th Chinese Control Conference, (2011).   Google Scholar

[11]

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

[12]

R. Femat and G. Solis-Perales, On the chaos synchronization phenomena,, Physics Letters A, 262 (1999), 50.  doi: 10.1016/S0375-9601(99)00667-2.  Google Scholar

[13]

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

[14]

S.-Y. Ha, E. Jeong and M.-J. Kang, Emergent behavior of a generalized Viscek-type flocking model,, Nonlinearity, 23 (2010), 3139.  doi: 10.1088/0951-7715/23/12/008.  Google Scholar

[15]

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

[16]

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., (2004).   Google Scholar

[17]

J. Kim, J. Yang, J. Kim and H. Shim, Practical Consensus for Heterogeneous Linear Time-Varying Multi-Agent Systems,, in Proceedings of 12th International Conference on Control, (2012).   Google Scholar

[18]

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

[19]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International symposium on mathematical problems in mathematical physics,, Lecture notes in theoretical physics, 39 (1975), 420.   Google Scholar

[20]

P. Louodop, H. Fotsin, E. Megam Ngouonkadi, S. Bowong and H. Cerdeira, Effective Synchronization of a Class of Chua's Chaotic Systems Using an Exponential Feedback Coupling,, Abstr. Appl. Anal., 2013 (2013).   Google Scholar

[21]

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

[22]

M. Ma, J. 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).   Google Scholar

[23]

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.  doi: 10.1007/s00332-006-0806-x.  Google Scholar

[24]

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.  doi: 10.1016/j.physd.2005.01.017.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

E. Steur, L. Kodde and H. Nijmeijer, Synchronization of Diffusively Coupled Electronic Hindmarsh-Rose Oscillators,, in Dynamics and control of hybrid mechanical systems (eds. G. Leonov, (2010), 195.  doi: 10.1142/9789814282321_0013.  Google Scholar

[30]

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

[31]

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

[32]

A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators,, J. Theor. Biol., 16 (1987), 15.  doi: 10.1016/0022-5193(67)90051-3.  Google Scholar

[33]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen,, Math. Ann., 71 (1912), 441.  doi: 10.1007/BF01456804.  Google Scholar

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