A fast algorithm to calculate the critical coupling strength for synchronization in a chain of Kuramoto oscillators

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• 作者：Lin Zhang (1)
  Ye Wu (1) (2)
  Xia Shi (1)
  Zuguo He (1)
  Jinghua Xiao (1) (2)
  
• 关键词：Kuramoto ; Synchronization ; Critical coupling strength ; Algorithm ; Distribution
• 刊名：Nonlinear Dynamics
• 出版年：2014
• 出版时间：July 2014
• 年：2014
• 卷：77
• 期：1-2
• 页码：99-105
• 全文大小：
• 参考文献：1. Buck, J.: Synchronous rhythmic flashing of fireflies, II. Q. Rev. Biol. 63, 265 (1988) CrossRef
  2. Glass, L., Mackey, M.C.: From Clocks to Chaos: The Rhythms of Life. Princeton University Press, Princeton, NJ (1988)
  3. Rodriguez, E., George, N., Lachaux, J.P., Renault, J.M.B., Varela, F.J.: Perceptions shadow: long-distance synchronization of human brain activity. Nature 397, 430 (1999) CrossRef
  4. Kiss, I.Z., Zhai, Y., Hudson, J.L.: Emerging coherence in a population of chemical oscillators. Science 296, 1676 (2002) CrossRef
  5. Arenas, A., D鈥檌az-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469, 93 (2008) CrossRef
  6. Wu, C.W.: Synchronization in Coupled Chaotic Circuits and Systems. World Scientific, Singapore (2002)
  7. Suresh, R., Senthilkumar, D.V., Lakshmanan, M., Kurths, J.: Global phase synchronization in an array of time-delay systems. Phys. Rev. E 82, 016215 (2010) CrossRef
  8. Winfree, A.T.: Geometry of Biological Time. Springer, New York (1990)
  9. Freund, H.J.: Motor unit and muscle activity in voluntary motor control. Psychoanal. Rev. 63, 387 (1983)
  10. Levy, R.: High-frequency synchronization of neuronal activity in the subthalamic nucleus of Parkinsonian patients with limb tremor. J. Neurosci. 20, 7766 (2000)
  11. Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Dover, New York (2003)
  12. Hong, H., Park, H., Choi, M.Y.: Collective phase synchronization in locally-coupled limit-cycle oscillators. Phys. Rev. E 70, 045204 (2004) CrossRef
  13. Strogatz, S.H.: Sync: The Emerging Science of Spontaneous Order. Hyperion, New York (2003)
  14. Manrubbia, S., Mikhailov, A., Zanette, D.: Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems. World Scientific, Singapore (2004)
  15. EI-Nashar, H.F., Muruganandam, P., Ferreira, F.F., Cerdeira, H.A.: Transition to complete synchronization in phase-coupled oscillators with nearest neighbor coupling. Chaos 19, 013103 (2009) CrossRef
  16. EI-Nashar, H.F., Cerdeira, H.A.: Determination of the critical coupling for oscillators in a ring. Chaos 19, 033127 (2009)
  17. Yamapi, R., Enjieu Kadji, H.G., Filatrella, G.: Stability of the synchronization manifold in nearest neighbor nonidentical van der Pol-like oscillators. Nonlinear Dyn. 61, 275 (2010)
  18. Muruganandam, P., Ferreira, F.F., EI-Nashar, H.F., Cerdeira, H.A.: Analytical calculation of the transition to complete phase synchronization in coupled oscillators. Pramana-J. Phys. 70, 1143 (2008) CrossRef
  19. Tilles, P.F.C., Ferreira, F.F., Cerdeira, H.A.: Multistable behavior above synchronization in a locally coupled Kuramoto model. Phys. Rev. E 83, 066206 (2011) CrossRef
  20. Ma, Y., Yoshikawa, K.: Self-sustained collective oscillation generated in an array of nonoscillatory cells. Phys. Rev. E 79, 046217 (2009) CrossRef
  21. Strogatz, S.H., Mirollo, R.E.: Phase-locking and critical phenomena in lattices of coupled nonlinear oscillators with random intrinsic frequencies. Physica D 31, 143 (1988) CrossRef
  22. Liu, Z., Lai, Y., Hoppensteadt, F.C.: Phase clustering and transition to phase synchronization in a large number of coupled nonlinear oscillators. Phys. Rev. E 63, 055201 (2001) CrossRef
  23. Wu, Y., Xiao, J.H., Hu, G., Zhan, M.: Synchronizing large number of nonidentical oscillators with small coupling. Euro. Phys. Lett. 97, 40005 (2012) CrossRef
  24. Feller, W.: The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Stat. 22, 427 (1951) CrossRef
  
• 作者单位：Lin Zhang (1)
  Ye Wu (1) (2)
  Xia Shi (1)
  Zuguo He (1)
  Jinghua Xiao (1) (2)
  
  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China
  2. State Key Lab of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing, 100876, China
  
• ISSN：1573-269X

文摘

Synchronization in a one-dimensional chain of Kuramoto oscillators with periodic boundary conditions is studied. An algorithm to rapidly calculate the critical coupling strength \(K_c\) for complete frequency synchronization is presented according to the mathematical constraint conditions and the periodic boundary conditions. By this new algorithm, we have checked the relation between \(\langle K_c\rangle \) and \(N\) , which is \(\langle K_c\rangle \sim \sqrt{N}\) , not only for small \(N\) , but also for large \(N\) . We also investigate the heavy-tailed distribution of \(K_c\) for random intrinsic frequencies, which is obtained by showing that the synchronization problem is equivalent to a discretization of Brownian motion. This theoretical result was checked by generating a large sample of \(K_c\) for large \(N\) from our algorithm to get the empirical density of \(K_c\) . Finally, we derive the permutation for the maximum coupling strength and its exact expression, which grows linearly with \(N\) and would provide the theoretical support for engineering applications.