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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

Hayato Chiba 1,

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 $N\to \infty$.
Keywords: synchronization, Kuramoto model, coupled oscillators., moments system, continuous limit.
Mathematics Subject Classification: 37L05, 35Q7.
Citation: Hayato Chiba. Continuous limit and the moments system for the globally coupled phase oscillators. Discrete and Continuous Dynamical Systems, 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), 137-185.

[2]

N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis," Hafner Publishing Co., New York, (1965), x+253 pp.

[3]

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

[4]

H. Chiba and I. Nishikawa, Center manifold reduction for a large population of globally coupled phase oscillators, Chaos, 21 (2011), 043103. doi: 10.1063/1.3647317.

[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), 1068-1081. doi: 10.1016/j.physd.2009.03.005.

[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), 1-46. doi: 10.1016/S0167-2789(98)00235-8.

[7]

H. Daido, Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators, J. Statist. Phys., 60 (1990), 753-800. doi: 10.1007/BF01025993.

[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), 24-66. doi: 10.1016/0167-2789(95)00260-X.

[9]

M. Frontini and A. Tagliani, Entropy-convergence in Stieltjes and Hamburger moment problem, Appl. Math. Comput., 88 (1997), 39-51. doi: 10.1016/S0096-3003(96)00305-0.

[10]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, 420-422. Lecture Notes in Phys., 39. Springer, Berlin, (1975).

[11]

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

[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), 084102.

[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), 3457-3466. doi: 10.1142/S0218127405014155.

[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), 026204. doi: 10.1103/PhysRevE.79.026204.

[15]

R. 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.

[16]

C. J. Perez and F. Ritort, A moment-based approach to the dynamical solution of the Kuramoto model, J. Phys. A, 30 (1997), 8095-8103. doi: 10.1088/0305-4470/30/23/010.

[17]

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

[18]

J. A. Shohat and J. D. Tamarkin, "The Problem of Moments," American Mathematical Society, New York, 1943.

[19]

B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math., 137 (1998), 82-203. doi: 10.1006/aima.1998.1728.

[20]

S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4.

[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), 2730-2733. doi: 10.1103/PhysRevLett.68.2730.

[22]

S. H. Strogatz and R. E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635. doi: 10.1007/BF01029202.

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), 137-185.

[2]

N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis," Hafner Publishing Co., New York, (1965), x+253 pp.

[3]

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

[4]

H. Chiba and I. Nishikawa, Center manifold reduction for a large population of globally coupled phase oscillators, Chaos, 21 (2011), 043103. doi: 10.1063/1.3647317.

[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), 1068-1081. doi: 10.1016/j.physd.2009.03.005.

[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), 1-46. doi: 10.1016/S0167-2789(98)00235-8.

[7]

H. Daido, Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators, J. Statist. Phys., 60 (1990), 753-800. doi: 10.1007/BF01025993.

[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), 24-66. doi: 10.1016/0167-2789(95)00260-X.

[9]

M. Frontini and A. Tagliani, Entropy-convergence in Stieltjes and Hamburger moment problem, Appl. Math. Comput., 88 (1997), 39-51. doi: 10.1016/S0096-3003(96)00305-0.

[10]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, 420-422. Lecture Notes in Phys., 39. Springer, Berlin, (1975).

[11]

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

[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), 084102.

[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), 3457-3466. doi: 10.1142/S0218127405014155.

[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), 026204. doi: 10.1103/PhysRevE.79.026204.

[15]

R. 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.

[16]

C. J. Perez and F. Ritort, A moment-based approach to the dynamical solution of the Kuramoto model, J. Phys. A, 30 (1997), 8095-8103. doi: 10.1088/0305-4470/30/23/010.

[17]

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

[18]

J. A. Shohat and J. D. Tamarkin, "The Problem of Moments," American Mathematical Society, New York, 1943.

[19]

B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math., 137 (1998), 82-203. doi: 10.1006/aima.1998.1728.

[20]

S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4.

[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), 2730-2733. doi: 10.1103/PhysRevLett.68.2730.

[22]

S. H. Strogatz and R. E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635. doi: 10.1007/BF01029202.

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