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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
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show all references

References:
[1]

Rev. Mod. Phys., 77 (2005), 137-185. Google Scholar

[2]

Hafner Publishing Co., New York, (1965), x+253 pp.  Google Scholar

[3]

Phys. D, 143 (2000), 21-55. doi: 10.1016/S0167-2789(00)00095-6.  Google Scholar

[4]

Chaos, 21 (2011), 043103. doi: 10.1063/1.3647317.  Google Scholar

[5]

Physica D, 238 (2009), 1068-1081. doi: 10.1016/j.physd.2009.03.005.  Google Scholar

[6]

Phys. D, 125 (1999), 1-46. doi: 10.1016/S0167-2789(98)00235-8.  Google Scholar

[7]

J. Statist. Phys., 60 (1990), 753-800. doi: 10.1007/BF01025993.  Google Scholar

[8]

Phys. D, 91 (1996), 24-66. doi: 10.1016/0167-2789(95)00260-X.  Google Scholar

[9]

Appl. Math. Comput., 88 (1997), 39-51. doi: 10.1016/S0096-3003(96)00305-0.  Google Scholar

[10]

International Symposium on Mathematical Problems in Theoretical Physics, 420-422. Lecture Notes in Phys., 39. Springer, Berlin, (1975).  Google Scholar

[11]

Springer Series in Synergetics, 19. Springer-Verlag, Berlin, 1984 doi: 10.1007/978-3-642-69689-3.  Google Scholar

[12]

Phys. Rev. Lett., 93 (2004), 084102. Google Scholar

[13]

Int. J. of Bif. and Chaos, 15 (2005), 3457-3466. doi: 10.1142/S0218127405014155.  Google Scholar

[14]

Phys. Rev. E, 79 (2009), 026204. doi: 10.1103/PhysRevE.79.026204.  Google Scholar

[15]

J. Nonlinear Sci., 17 (2007), 309-347. doi: 10.1007/s00332-006-0806-x.  Google Scholar

[16]

J. Phys. A, 30 (1997), 8095-8103. doi: 10.1088/0305-4470/30/23/010.  Google Scholar

[17]

Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.  Google Scholar

[18]

American Mathematical Society, New York, 1943.  Google Scholar

[19]

Adv. Math., 137 (1998), 82-203. doi: 10.1006/aima.1998.1728.  Google Scholar

[20]

Phys. D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00094-4.  Google Scholar

[21]

Phys. Rev. Lett., 68 (1992), 2730-2733. doi: 10.1103/PhysRevLett.68.2730.  Google Scholar

[22]

J. Statist. Phys., 63 (1991), 613-635. doi: 10.1007/BF01029202.  Google Scholar

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