Abstract.
We analyze the phenomenon of frequency clustering in a system of coupled phase oscillators. The oscillators, which in the absence of coupling have uniformly distributed natural frequencies, are coupled through a small-world network, built according to the Watts-Strogatz model. We study the time evolution and determine variations in the transient times depending on the disorder of the network and on the coupling strength. We investigate the effects of fluctuations in the average frequencies, and discuss the definition of the threshold for synchronization. We characterize the structure of clusters and the distribution of cluster sizes in the synchronization transition, and define suitable order parameters to describe the aggregation of the oscillators as the network disorder and the coupling strength change. The non-monotonic behavior observed in some order parameters is related to fluctuations in the mean frequencies.
Similar content being viewed by others
References
A.T. Winfree, The Geometry of Biological Time (Springer, New York, 1980)
Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984)
S.H. Strogatz, Physica D 143, 1 (2000)
D. Domínguez, H.A. Cerdeira, Phys. Rev. Lett. 71, 3359 (1993)
N. Khrustova, G. Veser, A. Mikhailov, R. Imbihl, Phys. Rev. Lett. 75, 3564 (1995)
H. Sakaguchi, S. Shinomoto, Y. Kuramoto, Prog. Theor. Phys. 77, 1005 (1987)
H. Daido, Phys. Rev. Lett. 68, 1073 (1992)
J.C. Stiller, G. Radons, Phys. Rev. E 58, 1789 (1998)
A.S. Pikovsky, M.G. Rosenblum, J. Kurths, Synchronization: a universal concept in nonlinear sciences (Cambridge University Press, 2001)
S.C. Manrubia, A.S. Mikhailov, D.H. Zanette, Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems (World Scientific, Singapore, 2004)
S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, Phys. Rep. 366, 1 (2002)
D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)
M.E.J. Newman, D.J. Watts, Phys. Rev. E 60, 7332 (1999)
M. Kuperman, G. Abramson, Phys. Rev. Lett. 86, 2909 (2001)
D.H. Zanette, Phys. Rev. E 65, 041908 (2002)
L.G. Morelli, G. Abramson, M.N. Kuperman, Eur. Phys. J. B 38, 495 (2004)
L.G. Morelli, H.A. Cerdeira, Phys. Rev. E 69, 051107 (2004)
H. Hong, M.Y. Choi, B.J. Kim, Phys. Rev. E 65, 026139 (2002)
H. Hong, M.Y. Choi, B.J. Kim, Phys. Rev. E 65, 047104 (2002)
M. Barahona, L.M. Pecora, Phys. Rev. Lett. 89, 054101 (2002)
Z. Zheng, G. Hu, B. Hu, Phys. Rev. Lett. 81, 5318 (1998)
El-Nashar et al., Int. J. Bifurcat. Chaos 12, 2945 (2002)
El-Nashar et al., Chaos 13, 1216 (2003)
D.H. Zanette, A.S. Mikhailov, Phys. Rev. E. 57, 276 (1998)
S.C. Manrubia, A.S. Mikhailov, Phys. Rev. E. 60, 1579 (1999)
D.H. Zanette, Europhys. Lett. 60, 945 (2002)
Z. Hou, H. Xin, Phys. Rev. E 68, 055103 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Morelli, L., Cerdeira, H. & Zanette, D. Frequency clustering of coupled phase oscillators on small-world networks. Eur. Phys. J. B 43, 243–250 (2005). https://doi.org/10.1140/epjb/e2005-00046-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2005-00046-2