Phase clustering and transition to phase synchronization in a large number of coupled nonlinear oscillators

Zonghua Liu; Ying-Cheng Lai; Frank C. Hoppensteadt

doi:10.1103/PhysRevE.63.055201

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Phase clustering and transition to phase synchronization in a large number of coupled nonlinear oscillators

Zonghua Liu¹, Ying-Cheng Lai^1,2,3, and Frank C. Hoppensteadt^1,2

¹Department of Mathematics, Arizona State University, Tempe, Arizona 85287
²Department of Electrical Engineering, Center for Systems Science and Engineering Research, Arizona State University, Tempe, Arizona 85287
³Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287

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Phys. Rev. E 63, 055201(R) – Published 9 April, 2001

DOI: https://doi.org/10.1103/PhysRevE.63.055201

Abstract

The transition to phase synchronization in systems consisting of a large number $(N)$ of coupled nonlinear oscillators via the route of phase clustering (phase synchronization among subsets of oscillators) is investigated. We elucidate the mechanism for the merger of phase clusters and find an algebraic scaling between the critical coupling parameter required for phase synchronization and N. Our result implies that, in realistic situations, phase clustering may be more prevalent than full phase synchronization.

References (19)

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In the frequency-modulation theory of neural networks, subpopulations that are synchronized can interchange information using phase and frequency modulation. Therefore, the extent of phase clustering could bear on the recruitment of network elements to populations that can interact. These can be reprogrammed by changing attributes of the elements, for example through conditioning by chemical pools in which they reside. See, for example, F. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks (Springer-Verlag, New York, 1997).
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[c1] M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett. 76, 1804 (1996).

[c2] A. S. Pikovsky, M. G. Rosenblum, and J. Kurths, Europhys. Lett. 34, 165 (1996).

[c3] A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, Physica D 104, 219 (1997).

[c4] C. Schäfer, M. G. Rosenblum, J. Kurths, and H.-H. Abel, Nature (London) 392, 239 (1998).

[c5] K. J. Lee, Y. Kwak, and T. K. Lim, Phys. Rev. Lett. 81, 321 (1998).

[c6] E. Rosa, E. Ott, and M. H. Hess, Phys. Rev. Lett. 80, 1642 (1998).

[c7] Z. Zheng, G. Hu, and B. Hu, Phys. Rev. Lett. 81, 5318 (1998).

[c8] P. Tass, M. G. Rosenblum, J. Weule, J. Kurths, A. Pikovsky, J. Volkmann, A. Schnitzler, and H.-J. Freund, Phys. Rev. Lett. 81, 3291 (1998).

[c9] B. Blasius, A. Huppert, and L. Stone, Nature (London) 399, 354 (1999).

[c10] V. Andrade, R. L. Davidchack, and Y.-C. Lai, Phys. Rev. E 61, 3230 (2000).

[c11] H. Fujigaki, M. Nishi, and T. Shimada, Phys. Rev. E 53, 3192 (1996); M. Palus, Phys. Lett. A 235, 341 (1997); Z. Liu and S. Chen, Phys. Rev. E 56, 7297 (1997); V. Makarenko and R. Llinas, Proc. Natl. Acad. Sci. U.S.A. 26, 15 747 (1998); D. E. Postnov, T. E. Vadivasova, O. V. Sosnovtseva, A. G. Balanov, V. S. Anishchenko, and E. Mosekilde, Chaos 9, 227 (1999); D. E. Postnov, A. G. Balanov, N. B. Janson, and E. Mosekilde, Phys. Rev. Lett. 83, 1942 (1999); J. W. Shuai and D. M. Durand, Phys. Lett. A 264, 289 (1999); A. Neiman, L. Schimansky-Geier, A. Cornell-Bell, and F. Moss, Phys. Rev. Lett. 83, 4896 (1999); B. Hu and C. Zhou, Phys. Rev. E 61, R1001 (2000).

[c12] T. Yalcinkaya and Y.-C. Lai, Phys. Rev. Lett. 79, 3885 (1997).

[c13] U. Parlitz, L. Junge, W. Lauterborn, and L. Kocarev, Phys. Rev. E 54, 2115 (1996).

[c14] P. Horowitz and W. Hill, The Art of Electronics (Cambridge University Press, Cambridge, England, 1989).

[c15] In the frequency-modulation theory of neural networks, subpopulations that are synchronized can interchange information using phase and frequency modulation. Therefore, the extent of phase clustering could bear on the recruitment of network elements to populations that can interact. These can be reprogrammed by changing attributes of the elements, for example through conditioning by chemical pools in which they reside. See, for example, F. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks (Springer-Verlag, New York, 1997).

[c16] O. E. Rössler, Phys. Lett. A 71, 155 (1979).

[c17] C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 50, 935 (1983); Ergod. Theory Dyn. Syst. 5, 341 (1985); Y.-C. Lai, C. Grebogi, J. A. Yorke, and S. C. Venkataramani, Phys. Rev. Lett. 77, 55 (1996).

[c18] In general, for chaotic flows with a well defined rotational structure, we expect this separation in the time scales of phase and amplitude variables to be approximately true. For other cases, this approximation may or may not be valid.

[c19] Y. Kuramoto, Prog. Theor. Phys. Suppl. 79, 223 (1974).

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Phase clustering and transition to phase synchronization in a large number of coupled nonlinear oscillators

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