• Previous Article
    Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition
  • DCDS Home
  • This Issue
  • Next Article
    No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$
May  2013, 33(5): 1891-1903. doi: 10.3934/dcds.2013.33.1891

Continuous limit and the moments system for the globally coupled phase oscillators

1. 

Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan

Received  December 2011 Revised  July 2012 Published  December 2012

The Kuramoto model, which describes synchronization phenomena, is a system of ordinary differential equations on $N$-torus defined as coupled harmonic oscillators. The order parameter is often used to measure the degree of synchronization. In this paper, the moments systems are introduced for both of the Kuramoto model and its continuous model. It is shown that the moments systems for both systems take the same form. This fact allows one to prove that the order parameter of the $N$-dimensional Kuramoto model converges to that of the continuous model as $N\to \infty$.
Citation: Hayato Chiba. Continuous limit and the moments system for the globally coupled phase oscillators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1891-1903. doi: 10.3934/dcds.2013.33.1891
References:
[1]

J. A. Acebron, 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.   Google Scholar

[2]

N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis,", Hafner Publishing Co., (1965).   Google Scholar

[3]

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

[4]

H. Chiba and I. Nishikawa, Center manifold reduction for a large population of globally coupled phase oscillators,, Chaos, 21 (2011).  doi: 10.1063/1.3647317.  Google Scholar

[5]

H. Chiba and D. Pazó, Stability of an $[N/2]$-dimensional invariant torus in the Kuramoto model at small coupling,, Physica D, 238 (2009), 1068.  doi: 10.1016/j.physd.2009.03.005.  Google Scholar

[6]

J. D. Crawford and K. T. R. Davies, Synchronization of globally coupled phase oscillators: Singularities and scaling for general couplings,, Phys. D, 125 (1999), 1.  doi: 10.1016/S0167-2789(98)00235-8.  Google Scholar

[7]

H. Daido, Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators,, J. Statist. Phys., 60 (1990), 753.  doi: 10.1007/BF01025993.  Google Scholar

[8]

H. Daido, Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: Bifurcation of the order function,, Phys. D, 91 (1996), 24.  doi: 10.1016/0167-2789(95)00260-X.  Google Scholar

[9]

M. Frontini and A. Tagliani, Entropy-convergence in Stieltjes and Hamburger moment problem,, Appl. Math. Comput., 88 (1997), 39.  doi: 10.1016/S0096-3003(96)00305-0.  Google Scholar

[10]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators,, International Symposium on Mathematical Problems in Theoretical Physics, 39 (1975), 420.   Google Scholar

[11]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984).  doi: 10.1007/978-3-642-69689-3.  Google Scholar

[12]

Y. Maistrenko, O. Popovych, O. Burylko and P. A. Tass, Mechanism of desynchronization in the finite-dimensional Kuramoto model,, Phys. Rev. Lett., 93 (2004).   Google Scholar

[13]

Y. L. Maistrenko, O. V. Popovych and P. A. Tass, Chaotic attractor in the Kuramoto model,, Int. J. of Bif. and Chaos, 15 (2005), 3457.  doi: 10.1142/S0218127405014155.  Google Scholar

[14]

E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott , P. So and T. M. Antonsen, Exact results for the Kuramoto model with a bimodal frequency distribution,, Phys. Rev. E, 79 (2009).  doi: 10.1103/PhysRevE.79.026204.  Google Scholar

[15]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model,, J. Nonlinear Sci., 17 (2007), 309.  doi: 10.1007/s00332-006-0806-x.  Google Scholar

[16]

C. J. Perez and F. Ritort, A moment-based approach to the dynamical solution of the Kuramoto model,, J. Phys. A, 30 (1997), 8095.  doi: 10.1088/0305-4470/30/23/010.  Google Scholar

[17]

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

[18]

J. A. Shohat and J. D. Tamarkin, "The Problem of Moments,", American Mathematical Society, (1943).   Google Scholar

[19]

B. Simon, The classical moment problem as a self-adjoint finite difference operator,, Adv. Math., 137 (1998), 82.  doi: 10.1006/aima.1998.1728.  Google Scholar

[20]

S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,, Phys. D, 143 (2000), 1.  doi: 10.1016/S0167-2789(00)00094-4.  Google Scholar

[21]

S. H. Strogatz, R. E. Mirollo and P. C. Matthews, Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping,, Phys. Rev. Lett., 68 (1992), 2730.  doi: 10.1103/PhysRevLett.68.2730.  Google Scholar

[22]

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

show all references

References:
[1]

J. A. Acebron, 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.   Google Scholar

[2]

N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis,", Hafner Publishing Co., (1965).   Google Scholar

[3]

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

[4]

H. Chiba and I. Nishikawa, Center manifold reduction for a large population of globally coupled phase oscillators,, Chaos, 21 (2011).  doi: 10.1063/1.3647317.  Google Scholar

[5]

H. Chiba and D. Pazó, Stability of an $[N/2]$-dimensional invariant torus in the Kuramoto model at small coupling,, Physica D, 238 (2009), 1068.  doi: 10.1016/j.physd.2009.03.005.  Google Scholar

[6]

J. D. Crawford and K. T. R. Davies, Synchronization of globally coupled phase oscillators: Singularities and scaling for general couplings,, Phys. D, 125 (1999), 1.  doi: 10.1016/S0167-2789(98)00235-8.  Google Scholar

[7]

H. Daido, Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators,, J. Statist. Phys., 60 (1990), 753.  doi: 10.1007/BF01025993.  Google Scholar

[8]

H. Daido, Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: Bifurcation of the order function,, Phys. D, 91 (1996), 24.  doi: 10.1016/0167-2789(95)00260-X.  Google Scholar

[9]

M. Frontini and A. Tagliani, Entropy-convergence in Stieltjes and Hamburger moment problem,, Appl. Math. Comput., 88 (1997), 39.  doi: 10.1016/S0096-3003(96)00305-0.  Google Scholar

[10]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators,, International Symposium on Mathematical Problems in Theoretical Physics, 39 (1975), 420.   Google Scholar

[11]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Springer Series in Synergetics, 19 (1984).  doi: 10.1007/978-3-642-69689-3.  Google Scholar

[12]

Y. Maistrenko, O. Popovych, O. Burylko and P. A. Tass, Mechanism of desynchronization in the finite-dimensional Kuramoto model,, Phys. Rev. Lett., 93 (2004).   Google Scholar

[13]

Y. L. Maistrenko, O. V. Popovych and P. A. Tass, Chaotic attractor in the Kuramoto model,, Int. J. of Bif. and Chaos, 15 (2005), 3457.  doi: 10.1142/S0218127405014155.  Google Scholar

[14]

E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott , P. So and T. M. Antonsen, Exact results for the Kuramoto model with a bimodal frequency distribution,, Phys. Rev. E, 79 (2009).  doi: 10.1103/PhysRevE.79.026204.  Google Scholar

[15]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model,, J. Nonlinear Sci., 17 (2007), 309.  doi: 10.1007/s00332-006-0806-x.  Google Scholar

[16]

C. J. Perez and F. Ritort, A moment-based approach to the dynamical solution of the Kuramoto model,, J. Phys. A, 30 (1997), 8095.  doi: 10.1088/0305-4470/30/23/010.  Google Scholar

[17]

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

[18]

J. A. Shohat and J. D. Tamarkin, "The Problem of Moments,", American Mathematical Society, (1943).   Google Scholar

[19]

B. Simon, The classical moment problem as a self-adjoint finite difference operator,, Adv. Math., 137 (1998), 82.  doi: 10.1006/aima.1998.1728.  Google Scholar

[20]

S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,, Phys. D, 143 (2000), 1.  doi: 10.1016/S0167-2789(00)00094-4.  Google Scholar

[21]

S. H. Strogatz, R. E. Mirollo and P. C. Matthews, Coupled nonlinear oscillators below the synchronization threshold: Relaxation by generalized Landau damping,, Phys. Rev. Lett., 68 (1992), 2730.  doi: 10.1103/PhysRevLett.68.2730.  Google Scholar

[22]

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

[1]

Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li. Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks & Heterogeneous Media, 2017, 12 (1) : 1-24. doi: 10.3934/nhm.2017001

[2]

Chun-Hsiung Hsia, Chang-Yeol Jung, Bongsuk Kwon. On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3319-3334. doi: 10.3934/dcdsb.2018322

[3]

R. Yamapi, R.S. MacKay. Stability of synchronization in a shift-invariant ring of mutually coupled oscillators. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 973-996. doi: 10.3934/dcdsb.2008.10.973

[4]

Xiaoxue Zhao, Zhuchun Li, Xiaoping Xue. Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring. Networks & Heterogeneous Media, 2018, 13 (2) : 323-337. doi: 10.3934/nhm.2018014

[5]

Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2893-2905. doi: 10.3934/dcds.2014.34.2893

[6]

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

[7]

Igor Chueshov, Peter E. Kloeden, Meihua Yang. Synchronization in coupled stochastic sine-Gordon wave model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2969-2990. doi: 10.3934/dcdsb.2016082

[8]

Klas Modin, Olivier Verdier. Integrability of nonholonomically coupled oscillators. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1121-1130. doi: 10.3934/dcds.2014.34.1121

[9]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019208

[10]

Long Hu, Tatsien Li, Bopeng Rao. Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type. Communications on Pure & Applied Analysis, 2014, 13 (2) : 881-901. doi: 10.3934/cpaa.2014.13.881

[11]

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

[12]

Richard H. Rand, Asok K. Sen. A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators. Communications on Pure & Applied Analysis, 2003, 2 (4) : 567-577. doi: 10.3934/cpaa.2003.2.567

[13]

Marzia Bisi, Giampiero Spiga. A Boltzmann-type model for market economy and its continuous trading limit. Kinetic & Related Models, 2010, 3 (2) : 223-239. doi: 10.3934/krm.2010.3.223

[14]

Martin Burger, Peter Alexander Markowich, Jan-Frederik Pietschmann. Continuous limit of a crowd motion and herding model: Analysis and numerical simulations. Kinetic & Related Models, 2011, 4 (4) : 1025-1047. doi: 10.3934/krm.2011.4.1025

[15]

William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971

[16]

Simone Fiori. Synchronization of first-order autonomous oscillators on Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1725-1741. doi: 10.3934/dcdsb.2018233

[17]

Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118

[18]

V. Afraimovich, J.-R. Chazottes, A. Cordonet. Synchronization in directionally coupled systems: Some rigorous results. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 421-442. doi: 10.3934/dcdsb.2001.1.421

[19]

Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial & Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87

[20]

Michal Fečkan. Blue sky catastrophes in weakly coupled chains of reversible oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 193-200. doi: 10.3934/dcdsb.2003.3.193

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]