Table of Contents
Journal of Complex Systems
Volume 2013 (2013), Article ID 591513, 8 pages
http://dx.doi.org/10.1155/2013/591513
Research Article

Synchronization of Nonidentical Coupled Phase Oscillators in the Presence of Time Delay and Noise

Department of Chemistry, Visva-Bharati, Santiniketan 731235, India

Received 10 May 2013; Accepted 19 July 2013

Academic Editor: Julian Candia

Copyright © 2013 Somrita Ray et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. H. Strogatz, “From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators,” Physica D. Nonlinear Phenomena, vol. 143, no. 1–4, pp. 1–20, 2000. View at Publisher · View at Google Scholar
  2. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization—A Universal Concept in Nonlinear Sciences, vol. 12, Cambridge University Press, Cambridge, UK, 2001. View at Publisher · View at Google Scholar
  3. R. Albert and A. L. Barabási, “Statistical mechanics of complex networks,” Reviews of Modern Physics, vol. 74, no. 1, pp. 47–97, 2002. View at Publisher · View at Google Scholar
  4. H. Haken, Brain Dynamics: Synchronization and Activity Patterns in Pulse-Coupled Neural Nets with Delays and Noise, Springer, Berlin, Germany, 2002.
  5. S. C. Manrubia, A. S. Mikhailov, and D. H. Zanette, Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems, World Scientific, Singapore, 2004.
  6. J. J. Acebron, L. L. Bonilla, C. J. Perez-Vicente, F. Ritort, R. Spigler, and The Kuramoto model:, “A simple paradigm for synchronization phenomena,” Reviews of Modern Physics, vol. 77, no. 1, pp. 137–185, 2005. View at Publisher · View at Google Scholar
  7. A. T. Winfree, “Biological rhythms and the behavior of populations of coupled oscillators,” Journal of Theoretical, vol. 16, no. 1, pp. 15–42, 1967. View at Publisher · View at Google Scholar
  8. A. T. Winfree, The Geometry of Biological Time, Springer, New York, NY, USA, 1980.
  9. Y. Kuramoto, “Cooperative dynamics of oscillator community—a study based on lattice of rings,” Progress of Theoretical Physics, vol. 79, pp. 223–240, 1984. View at Publisher · View at Google Scholar
  10. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, New York, NY, USA, 1984.
  11. Y. Kuramoto and I. Nishikawa, “Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities,” Journal of Statistical Physics, vol. 49, no. 3-4, pp. 569–605, 1987. View at Publisher · View at Google Scholar
  12. K. Kaneko, “Globally coupled chaos violates the law of large numbers but not the central-limit theorem,” Physical Review Letters, vol. 65, no. 12, pp. 1391–1394, 1990. View at Publisher · View at Google Scholar
  13. L. M. Pecora and T. L. Caroll, “Master stability functions for synchronized coupled systems,” Physical Review Letters, vol. 80, no. 10, pp. 2109–2112, 1998. View at Publisher · View at Google Scholar
  14. H. Hong, M. Y. Choi, and B. J. Kim, “Phase ordering on small-world networks with nearest-neighbor edges,” Physical Review E, vol. 65, no. 4, Article ID 047104, 4 pages, 2002. View at Publisher · View at Google Scholar
  15. P. Tass, J. Kurths, M. G. Rosenblum, G. Gausti, and H. Hefter, “Delay-induced transitions in visually guided movements,” Physical Review E, vol. 54, no. 3, pp. R2224–R2227, 1996. View at Publisher · View at Google Scholar
  16. M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science, vol. 197, no. 4300, pp. 287–289, 1977. View at Publisher · View at Google Scholar
  17. T. D. Frank, A. Daffertshofer, P. J. Beek, and H. Haken, “Impacts of noise on a field theoretical model of the human brain,” Physica D, vol. 127, no. 3-4, pp. 233–249, 1999. View at Google Scholar · View at Scopus
  18. J. J. Wright and D. T. J. Liley, “Simulation of electrocortical waves,” Biological Cybernetics, vol. 72, no. 4, pp. 347–356, 1995. View at Publisher · View at Google Scholar · View at Scopus
  19. E. Niebur, H. G. Schuster, and D. M. Kammen, “Collective frequencies and metastability in networks of limit-cycle oscillators with time delay,” Physical Review Letters, vol. 67, no. 20, pp. 2753–2756, 1991. View at Publisher · View at Google Scholar · View at Scopus
  20. Y. Nakamura, F. Tominaga, and T. Munakata, “Clustering behavior of time-delayed nearest-neighbor coupled oscillators,” Physical Review E, vol. 49, no. 6, pp. 4849–4856, 1994. View at Publisher · View at Google Scholar · View at Scopus
  21. M. Y. Choi, H. J. Kim, D. Kim, and H. Hong, “Synchronization in a system of globally coupled oscillators with time delay,” Physical Review E, vol. 61, no. 1, pp. 371–381, 2000. View at Publisher · View at Google Scholar
  22. M. K. S. Yeung and S. H. Strogatz, “Time delay in the Kuramoto model of coupled oscillators,” Physical Review Letters, vol. 82, no. 3, pp. 648–651, 1999. View at Publisher · View at Google Scholar
  23. M. K. Sen, B. C. Bag, K. G. Petrosyan, and C. K. Hu, “Effect of time delay on the onset of synchronization of the stochastic Kuramoto model,” Journal of Statistical Mechanics, vol. 2010, Article ID P08018, 2010. View at Publisher · View at Google Scholar
  24. E. Montbrió, D. Pazó, and J. Schmidt, “Time delay in the Kuramoto model with bimodal frequency distribution,” Physical Review E, vol. 74, no. 5, Article ID 056201, 5 pages, 2006. View at Publisher · View at Google Scholar
  25. L. Borland, “Ito-Langevin equations within generalized thermostatistics,” Physics Letters A, vol. 245, no. 1-2, pp. 67–72, 1998. View at Publisher · View at Google Scholar
  26. H. S. Wio and R. Toral, “Effect of non-Gaussian noise sources in a noise-induced transition,” Physica D, vol. 193, no. 1–4, pp. 161–168, 2004. View at Publisher · View at Google Scholar · View at Scopus
  27. S. Bouzat and H. S. Wio, “Current and efficiency enhancement in Brownian motors driven by non Gaussian noises,” European Physical Journal B, vol. 41, no. 1, pp. 97–105, 2004. View at Publisher · View at Google Scholar · View at Scopus
  28. S. Bouzat and H. S. Wio, “New aspects on current enhancement in Brownian motors driven by non-Gaussian noises,” Physica A, vol. 351, no. 1, pp. 69–78, 2005. View at Publisher · View at Google Scholar · View at Scopus
  29. P. Hanggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: fifty years after Kramers,” Reviews of Modern Physics, vol. 62, no. 2, pp. 251–341, 1990. View at Publisher · View at Google Scholar
  30. P. Hänggi and P. Jung, “Colored noise in dynamical systems,” in Advances in Chemical Physics, vol. 89, pp. 239–326, 1995. View at Publisher · View at Google Scholar
  31. B. C. Bag, “Colored non-Gaussian noise driven systems: mean first passage time,” European Physical Journal B, vol. 34, no. 1, pp. 115–118, 2003. View at Publisher · View at Google Scholar
  32. P. Majee, G. Goswami, and B. C. Bag, “Colored non-Gaussian noise induced resonant activation,” Chemical Physics Letters, vol. 416, no. 4–6, pp. 256–260, 2005. View at Publisher · View at Google Scholar · View at Scopus
  33. G. Goswami, P. Majee, P. K. Ghosh, and B. C. Bag, “Colored multiplicative and additive non-Gaussian noise-driven dynamical system: mean first passage time,” Physica A, vol. 374, no. 2, pp. 549–558, 2007. View at Publisher · View at Google Scholar
  34. B. C. Bag and C. K. Hu, “Escape through an unstable limit cycle driven by multiplicative colored non-Gaussian and additive white Gaussian noises,” Physical Review E, vol. 75, no. 4, Article ID 042101, 4 pages, 2007. View at Publisher · View at Google Scholar
  35. M. K. Sen, A. Baura, and B. C. Bag, “Noise induced escape through an unstable limit cycle in the presence of a fluctuating barrier,” Journal of Statistical Mechanics, vol. 2009, no. 11, Article ID P11004, 2009. View at Publisher · View at Google Scholar
  36. B. C. Bag, K. G. Petrosyan, and C. K. Hu, “Influence of noise on the synchronization of the stochastic Kuramoto model,” Physical Review E, vol. 76, no. 5, Article ID 056210, 6 pages, 2007. View at Publisher · View at Google Scholar
  37. P. K. Ghosh, M. K. Sen, and B. C. Bag, “Kinetics of self-induced aggregation of Brownian particles: non-Markovian and non-Gaussian features,” Physical Review E, vol. 78, no. 5, Article ID 051103, 7 pages, 2008. View at Publisher · View at Google Scholar
  38. B. C. Bag and C. K. Hu, “Current inversion induced by colored non-Gaussian noise,” Journal of Statistical Mechanics, vol. 2009, no. 2, Article ID P02003, 2009. View at Publisher · View at Google Scholar
  39. M. K. Sen and B. C. Bag, “Generalization of barrier crossing rate for coloured non Gaussian noise driven open systems,” The European Physical Journal B, vol. 68, no. 2, pp. 253–259, 2009. View at Publisher · View at Google Scholar
  40. A. K. Baura, M. K. Sen, G. Goswami, and B. C. Bag, “Colored non-Gaussian noise driven open systems: generalization of Kramers’ theory with a unified approach,” Journal of Chemical Physics, vol. 134, no. 4, Article ID 044126, 10 pages, 2011. View at Publisher · View at Google Scholar
  41. M. A. Fuentes, H. S. Wio, and R. Toral, “Effective Markovian approximation for non-Gaussian noises: a path integral approach,” Physica A, vol. 303, no. 1-2, pp. 91–104, 2002. View at Publisher · View at Google Scholar
  42. X. Luo and S. Zhu, “Stochastic resonance driven by two different kinds of colored noise in a bistable system,” Physical Review E, vol. 67, no. 2, Article ID 021104, 13 pages, 2003. View at Publisher · View at Google Scholar
  43. P. Majee and B. C. Bag, “The effect of interference of coloured additive and multiplicative white noises on escape rate,” Journal of Physics A, vol. 37, no. 10, pp. 3353–3361, 2004. View at Publisher · View at Google Scholar · View at Scopus
  44. K. Lindenberg and B. J. West, The Nonequilibrium Statistitical Mechanics of Open and Closed Systems, VCH, New York, NY, USA, 1990.
  45. H. Hong, H. Chate, H. Park, and L. H. Tang, “Entrainment transition in populations of random frequency oscillators,” Physical Review Letters, vol. 99, no. 18, Article ID 184101, 4 pages, 2007. View at Publisher · View at Google Scholar
  46. R. Tönjes, “Synchronization transition in the Kuramoto model with colored noise,” Physical Review E, vol. 81, no. 5, Article ID 055201, 4 pages, 2010. View at Publisher · View at Google Scholar
  47. R. Toral, “Computational field theory and pattern formation,” in Computational Physics, P. Garrido and J. Marro, Eds., vol. 448 of Lecture Notes in Physics, pp. 1–65, Springer. View at Publisher · View at Google Scholar
  48. R. Mannella, “Numerical integration of stochastic differential equations,” http://arxiv.org/abs/condmat/9709326.