Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2010, Article ID 859685, 10 pages
http://dx.doi.org/10.1155/2010/859685
Research Article

Synchronization of Chaotic Fractional-Order WINDMI Systems via Linear State Error Feedback Control

1Nonlinear Dynamics and Chaos Group, School of Management, Tianjin University, Tianjin 300072, China
2Center for Applied Mathematics, School of Economics and Management, Shandong University of Science and Technology, Qingdao 266510, China
3Department of Mathematics, Dezhou University, Dezhou 253023, China

Received 25 April 2010; Accepted 20 August 2010

Academic Editor: Yuri Vladimirovich Mikhlin

Copyright © 2010 Baogui Xin 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. W. Horton, R. S. Weigel, and J. C. Sprott, “Chaos and the limits of predictability for the solar-wind-driven magnetosphere-ionosphere system,” Physics of Plasmas, vol. 8, no. 6, pp. 2946–2952, 2001. View at Publisher · View at Google Scholar · View at Scopus
  2. W. Horton and I. Doxas, “A low-dimensional dynamical model for the solar wind driven geotail-ionosphere system,” Journal of Geophysical Research A, vol. 103, no. A3, pp. 4561–4572, 1998. View at Publisher · View at Google Scholar · View at Scopus
  3. J. P. Smith, J.-L. Thiffeault, and W. Horton, “Dynamical range of the WINDMI model: an exploration of possible magnetospheric plasma states,” Journal of Geophysical Research A, vol. 105, no. A6, pp. 12983–12996, 2000. View at Publisher · View at Google Scholar · View at Scopus
  4. J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, New York, NY, USA, 2003.
  5. K. B. Oldham and J. Spanier, The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, NY, USA, 1974.
  6. K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. W. Deng, “Numerical algorithm for the time fractional Fokker-Planck equation,” Journal of Computational Physics, vol. 227, no. 2, pp. 1510–1522, 2007. View at Publisher · View at Google Scholar
  8. S. Wang and M. Xu, “Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 1087–1096, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. X. Jiang, M. Xu, and H. Qi, “The fractional diffusion model with an absorption term and modified Fick's law for non-local transport processes,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 262–269, 2010. View at Publisher · View at Google Scholar
  10. Y.-Q. Liu and J.-H. Ma, “Exact solutions of a generalized multi-fractional nonlinear diffusion equation in radical symmetry,” Communications in Theoretical Physics, vol. 52, no. 5, pp. 857–861, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. K. D. Papoulia, V. P. Panoskaltsis, N. V. Kurup, and I. Korovajchuk, “Rheological representation of fractional order viscoelastic material models,” Rheologica Acta, vol. 49, no. 4, pp. 381–400, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. H. Ye and Y. Ding, “Nonlinear dynamics and chaos in a fractional-order HIV model,” Mathematical Problems in Engineering, vol. 2009, Article ID 378614, 12 pages, 2009. View at Google Scholar · View at Zentralblatt MATH
  13. Y. Ding and H. Ye, “A fractional-order differential equation model of HIV infection of CD4+ T-cells,” Mathematical and Computer Modelling, vol. 50, no. 3-4, pp. 386–392, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. R. E. Gutiérrez, J. M. Rosário, and J. Tenreiro MacHado, “Fractional order calculus: basic concepts and engineering applications,” Mathematical Problems in Engineering, vol. 2010, Article ID 375858, 19 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. I. Podlubny, “Fractional-order systems and PIλDμ-controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–214, 1999. View at Publisher · View at Google Scholar
  16. J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa et al., “Some applications of fractional calculus in engineering,” Mathematical Problems in Engineering, vol. 2010, Article ID 639801, 34 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. W. H. Deng and C. P. Li, “Chaos synchronization of the fractional Lü system,” Physica A, vol. 353, no. 1–4, pp. 61–72, 2005. View at Publisher · View at Google Scholar · View at Scopus
  18. C. P. Li, W. H. Deng, and D. Xu, “Chaos synchronization of the Chua system with a fractional order,” Physica A, vol. 360, no. 2, pp. 171–185, 2006. View at Publisher · View at Google Scholar
  19. X.-Y. Wang and J.-M. Song, “Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3351–3357, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. X. Y. Wang and Y. He, “Projective synchronization of fractional order chaotic system based on linear separation,” Physics Letters A, vol. 372, no. 4, pp. 435–441, 2008. View at Publisher · View at Google Scholar · View at Scopus
  21. R.-X. Zhang, S.-P. Yang, and Y.-L. Liu, “Synchronization of fractional-order unified chaotic system via linear control,” Acta Physica Sinica, vol. 59, no. 3, pp. 1549–1553, 2010. View at Google Scholar · View at Scopus
  22. H. Zhu, S. Zhou, and Z. He, “Chaos synchronization of the fractional-order Chen's system,” Chaos, Solitons and Fractals, vol. 41, no. 5, pp. 2733–2740, 2009. View at Publisher · View at Google Scholar · View at Scopus
  23. Z. M. Odibat, N. Corson, M. A. Aziz-Alaoui, and C. Bertelle, “Synchronization of chaotic fractional-order systems via linear control,” International Journal of Bifurcation and Chaos, vol. 20, no. 1, pp. 81–97, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. X. Wu, H. Lu, and S. Shen, “Synchronization of a new fractional-order hyperchaotic system,” Physics Letters A, vol. 373, no. 27-28, pp. 2329–2337, 2009. View at Publisher · View at Google Scholar
  25. M. Shahiri, R. Ghaderi, A. Ranjbar N., S. H. Hosseinnia, and S. Momani, “Chaotic fractional-order Coullet system: synchronization and control approach,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 665–674, 2010. View at Publisher · View at Google Scholar
  26. I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  27. D. Cafagna, “Past and present—fractional calculus: a mathematical tool from the past for present engineers,” IEEE Industrial Electronics Magazine, vol. 1, no. 2, pp. 35–40, 2007. View at Publisher · View at Google Scholar · View at Scopus
  28. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proceedings of the IMACS-IEEE Multiconference on Computational Engineering in Systems Applications (CESA '96), pp. 963–968, Lille, France, 1996.
  29. Y. Yu, H.-X. Li, S. Wang, and J. Yu, “Dynamic analysis of a fractional-order Lorenz chaotic system,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1181–1189, 2009. View at Publisher · View at Google Scholar
  30. K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms, vol. 36, no. 1, pp. 31–52, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH