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Continuous limit and the moments system for the globally coupled phase oscillators
1. | Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan |
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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