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Mathematical Problems in Engineering
Volume 2013, Article ID 685798, 10 pages
http://dx.doi.org/10.1155/2013/685798
Research Article

Synchronization of Coupled Stochastic Systems Driven by -Stable Lévy Noises

College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China

Received 9 October 2012; Revised 25 December 2012; Accepted 28 December 2012

Academic Editor: Weihai Zhang

Copyright © 2013 Anhui Gu. 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. V. S. Afraimovich, S.-N. Chow, and J. K. Hale, “Synchronization in lattices of coupled oscillators,” Physica D, vol. 103, no. 1–4, pp. 442–451, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. V. S. Afraimovich and W.-W. Lin, “Synchronization in lattices of coupled oscillators with Neumann/periodic boundary conditions,” Dynamics and Stability of Systems, vol. 13, no. 3, pp. 237–264, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. V. S. Afraĭmovich, N. N. Verichev, and M. I. Rabinovich, “Stochastic synchronization of oscillations in dissipative systems,” Izvestiya Vysshikh Uchebnykh Zavedeniĭ, vol. 29, no. 9, pp. 1050–1060, 1986. View at Google Scholar · View at MathSciNet
  4. S. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion Press, 2003. View at MathSciNet
  5. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization, A Universal Concept in Nonlinear Sciences, Cambridge University Press, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  6. L. Glass, “Synchronization and rhythmic processes in physiology,” Nature, vol. 410, pp. 277–284, 2001. View at Google Scholar
  7. V. S. Afraimovich and H. M. Rodrigues, “Uniform dissipativeness and synchronization of nonau-tonomous equation,” in Proceedings of the International Conference on Differential Equations, pp. 3–17, World Scientific, Lisbon, Portugal, 1995.
  8. P. E. Kloeden, “Synchronization of nonautonomous dynamical systems,” Electronic Journal of Differential Equations, vol. 39, pp. 1–10, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. N. Carvalho, H. M. Rodrigues, and T. Dłotko, “Upper semicontinuity of attractors and synchronization,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 13–41, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. H. M. Rodrigues, “Abstract methods for synchronization and applications,” Applicable Analysis, vol. 62, no. 3-4, pp. 263–296, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. T. Caraballo and P. E. Kloeden, “The persistence of synchronization under environmental noise,” Proceedings of The Royal Society of London A, vol. 461, no. 2059, pp. 2257–2267, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. T. Caraballo, P. E. Kloeden, and A. Neuenkirch, “Synchronization of systems with multiplicative noise,” Stochastics and Dynamics, vol. 8, no. 1, pp. 139–154, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. T. Caraballo, I. D. Chueshov, and P. E. Kloeden, “Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain,” SIAM Journal on Mathematical Analysis, vol. 38, no. 5, pp. 1489–1507, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  14. I. Chueshov and B. Schmalfuß, “Master-slave synchronization and invariant manifolds for coupled stochastic systems,” Journal of Mathematical Physics, vol. 51, no. 10, Article ID 102702, 23 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  15. Z. W. Shen, S. F. Zhou, and X. Y. Han, “Synchronization of coupled stochastic systems with multiplicative noise,” Stochastics and Dynamics, vol. 10, no. 3, pp. 407–428, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. X. M. Liu, J. Q. Duan, J. C. Liu, and P. E. Kloeden, “Synchronization of dissipative dynamical systems driven by non-Gaussian Lévy noises,” International Journal of Stochastic Analysis, vol. 2010, Article ID 502803, 13 pages, 2010. View at Google Scholar
  17. X. M. Liu, J. Q. Duan, J. C. Liu, and P. E. Kloeden, “Synchronization of systems of Marcus canonical equations driven by α-stable noises,” Nonlinear Analysis, vol. 11, no. 5, pp. 3437–3445, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. H. Gu, “Synchronization of coupled stochastic systems driven by non-Gaussian Lévy noises,” Stochastic and Dynamics, submitted.
  19. D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  20. S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Léevy Processes, Cambridge University Press, Cambridge, UK, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  21. K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1999. View at MathSciNet
  22. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall, New York, NY, USA, 1994. View at MathSciNet
  23. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, NY, USA, 1968. View at MathSciNet
  24. L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer, 1998. View at MathSciNet
  25. S. I. Marcus, “Modeling and approximation of stochastic differential equations driven by semimartingales,” Stochastics, vol. 4, no. 3, pp. 223–245, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  26. M. Errami, F. Russo, and P. Vallois, “Itô's formula for C1,λ-functions of a càdlàg process and related calculus,” Probability Theory and Related Fields, vol. 122, no. 2, pp. 191–221, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  27. J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, UK, 2001. View at Publisher · View at Google Scholar · View at MathSciNet