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International Journal of Stochastic Analysis
Volume 2010, Article ID 502803, 13 pages
http://dx.doi.org/10.1155/2010/502803
Research Article

Synchronization of Dissipative Dynamical Systems Driven by Non-Gaussian Lévy Noises

1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
3Institut für Mathematik, Johann Wolfgang Goethe-Universität, D-60054, Frankfurt am Main, Germany

Received 17 September 2009; Accepted 15 January 2010

Academic Editor: Salah-Eldin Mohammed

Copyright © 2010 Xianming Liu 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. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion Books, New York, NY, USA, 2003. View at MathSciNet
  2. T. Caraballo and P. E. Kloeden, “The persistence of synchronization under environmental noise,” Proceedings of The Royal Society of London. Series A, vol. 461, no. 2059, pp. 2257–2267, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  3. 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
  4. D. Applebaum, Lévy Processes and Stochastic Calculus, vol. 93 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2004. View at MathSciNet
  5. D. Schertzer, M. Larchevêque, J. Duan, V. V. Yanovsky, and S. Lovejoy, “Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises,” Journal of Mathematical Physics, vol. 42, no. 1, pp. 200–212, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. Kunita, “Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms,” in Real and Stochastic Analysis, Trends Math., pp. 305–373, Birkhäuser, Boston, Mass, USA, 2004. View at Google Scholar · View at MathSciNet
  7. L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer, Berlin, Germany, 1998. View at MathSciNet
  8. P. Imkeller and I. Pavlyukevich, “First exit times of SDEs driven by stable Lévy processes,” Stochastic Processes and Their Applications, vol. 116, no. 4, pp. 611–642, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. P. Imkeller, I. Pavlyukevich, and T. Wetzel, “First exit times for Lévy-driven diffusions with exponentially light jumps,” The Annals of Probability, vol. 37, no. 2, pp. 530–564, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  10. Z. Yang and J. Duan, “An intermediate regime for exit phenomena driven by non-Gaussian Lévy noises,” Stochastics and Dynamics, vol. 8, no. 3, pp. 583–591, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. 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
  12. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Stochastic Modeling, Chapman & Hall, New York, NY, USA, 1994. View at MathSciNet
  13. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, NY, USA, 1968. View at MathSciNet
  14. R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, NY, USA, 2005. View at MathSciNet
  15. B. Schmalfuss, “Attractors for nonautonomous and random dynamical systems perturbed by impulses,” Discrete and Continuous Dynamical Systems. Series A, vol. 9, no. 3, pp. 727–744, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. H. Crauel and F. Flandoli, “Attractors for random dynamical systems,” Probability Theory and Related Fields, vol. 100, no. 3, pp. 365–393, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. B. Schmalfuß, “The random attractor of the stochastic Lorenz system,” Zeitschrift für Angewandte Mathematik und Physik, vol. 48, no. 6, pp. 951–975, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. P. E. Kloeden, “Nonautonomous attractors of switching systems,” Dynamical Systems, vol. 21, no. 2, pp. 209–230, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractor, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2001. View at MathSciNet
  20. 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
  21. O. E. Barndorff-Nielsen, J. L. Jensen, and M. Sørensen, “Some stationary processes in discrete and continuous time,” Advances in Applied Probability, vol. 30, no. 4, pp. 989–1007, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet