American Institute of Mathematical Sciences

Advanced
  • 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

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 & Continuous Dynamical Systems, 2013, 33 (5) : 1891-1903. doi: 10.3934/dcds.2013.33.1891
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

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

[1]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208
[2]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079
[3]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933
[4]

Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248
[5]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
[6]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003
[7]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011
[8]

Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037
[9]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623
[10]

Reza Mazrooei-Sebdani, Zahra Yousefi. The coupled 1:2 resonance in a symmetric case and parametric amplification model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3737-3765. doi: 10.3934/dcdsb.2020255
[11]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
[12]

Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021056
[13]

Lu Li. On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021032
[14]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[15]

Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021098
[16]

Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016
[17]

Wen Si. Response solutions for degenerate reversible harmonic oscillators. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3951-3972. doi: 10.3934/dcds.2021023
[18]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[19]

Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3045-3062. doi: 10.3934/dcds.2020397
[20]

Wenjuan Zhao, Shunfu Jin, Wuyi Yue. A stochastic model and social optimization of a blockchain system based on a general limited batch service queue. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1845-1861. doi: 10.3934/jimo.2020049

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences