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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

 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, 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. Google Scholar [2] N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis," Hafner Publishing Co., New York, (1965), x+253 pp.  Google Scholar [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.  Google Scholar [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.  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), 1068-1081. 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), 1-46. 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), 753-800. 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), 24-66. 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), 39-51. 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, 420-422. Lecture Notes in Phys., 39. Springer, Berlin, (1975).  Google Scholar [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.  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), 084102. 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), 3457-3466. 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), 026204. 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), 309-347. 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), 8095-8103. 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, Cambridge, 2001. doi: 10.1017/CBO9780511755743.  Google Scholar [18] J. A. Shohat and J. D. Tamarkin, "The Problem of Moments," American Mathematical Society, New York, 1943.  Google Scholar [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.  Google Scholar [20] S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20. 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), 2730-2733. 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), 613-635. doi: 10.1007/BF01029202.  Google Scholar

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##### 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. Google Scholar [2] N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis," Hafner Publishing Co., New York, (1965), x+253 pp.  Google Scholar [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.  Google Scholar [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.  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), 1068-1081. 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), 1-46. 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), 753-800. 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), 24-66. 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), 39-51. 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, 420-422. Lecture Notes in Phys., 39. Springer, Berlin, (1975).  Google Scholar [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.  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), 084102. 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), 3457-3466. 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), 026204. 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), 309-347. 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), 8095-8103. 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, Cambridge, 2001. doi: 10.1017/CBO9780511755743.  Google Scholar [18] J. A. Shohat and J. D. Tamarkin, "The Problem of Moments," American Mathematical Society, New York, 1943.  Google Scholar [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.  Google Scholar [20] S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20. 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), 2730-2733. 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), 613-635. doi: 10.1007/BF01029202.  Google Scholar
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