Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 762807, 18 pages
http://dx.doi.org/10.1155/2012/762807
Research Article

Inverse Projective Synchronization between Two Different Hyperchaotic Systems with Fractional Order

1State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 400030, China
2School of Automation, Chongqing University, Chongqing 400030, China
3School of Mathematics, Anhui University, Hefei 230039, China

Received 14 July 2011; Accepted 5 October 2011

Academic Editor: Yuri Sotskov

Copyright © 2012 Yi Chai 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. R. C. Koeller, “Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics,” Acta Mechanica, vol. 58, no. 3-4, pp. 251–264, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 299–307, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. O. Heaviside, Electromagnetic theory, Chelsea, New York, NY, USA, 1971.
  4. 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
  5. F. B. M. Duarte and J. A. T. Machado, “Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 315–342, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. 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
  7. C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A, vol. 341, no. 1–4, pp. 55–61, 2004. View at Publisher · View at Google Scholar · View at Scopus
  8. C. Li and G. Peng, “Chaos in Chen's system with a fractional order,” Chaos, Solitons & Fractals, vol. 22, no. 2, pp. 443–450, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. L.-J. Sheu, H.-K. Chen, J.-H. Chen et al., “Chaos in the Newton-Leipnik system with fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 98–103, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. 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
  11. X. Gao and J. B. Yu, “Chaos in the fractional order periodically forced complex Duffing's oscillators,” Chaos, Solitons & Fractals, vol. 24, no. 4, pp. 1097–1104, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. X. Wu and Y. Lu, “Generalized projective synchronization of the fractional-order Chen hyperchaotic system,” Nonlinear Dynamics, vol. 57, no. 1-2, pp. 25–35, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. T. Wang and X. Wang, “Generalized synchronization of fractional order hyperchaotic lorenz system,” Modern Physics Letters B, vol. 23, no. 17, pp. 2167–2178, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. Y. G. Yu and H.-X. Li, “The synchronization of fractional-order Rössler hyperchaotic systems,” Physica A, vol. 387, no. 5-6, pp. 1393–1403, 2008. View at Publisher · View at Google Scholar
  15. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. G. R. Chen and X. Dong, From Chaos to Order—Perspectives, Methodologies and Applications, World Scientific, Singapore, 1998.
  17. S. K. Dana, P. K. Roy, and J. Kurths, Complex Dynamics in Physiological Systems, from Heart to Brain, Springer, New York, NY, USA, 2009.
  18. C. Li and X. F. Liao, “Complete and lag synchronization of hyperchaotic systems using small impulses,” Chaos, Solitons & Fractals, vol. 22, no. 4, pp. 857–867, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. G. H. Li, “Projective lag synchronization in chaotic systems,” Chaos, Solitons & Fractals, vol. 41, no. 5, pp. 2630–2634, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. X. D. Zhang, P. D. Zhao, and A. H. Li, “Construction of a new fractional chaotic system and generalized synchronization,” Communications in Theoretical Physics, vol. 53, no. 6, pp. 1105–1110, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. H. Jiang and Q. Bi, “Impulsive synchronization of networked nonlinear dynamical systems,” Physics Letters A, vol. 374, no. 27, pp. 2723–2729, 2010. View at Publisher · View at Google Scholar
  22. X. F. Lang, Q. S. Lu, and K. Jurge, “Phase synchronization in noise-driven bursting neurons,” Physical Review E, vol. 82, no. 2, pp. 1–6, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. I. Wedekind and U. Parlitz, “Experimental observation of synchronization and anti-synchronization of chaotic low-frequency-fluctuations in external cavity semiconductor lasers,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 11, no. 4, pp. 1141–1147, 2001. View at Publisher · View at Google Scholar · View at Scopus
  24. M. K. Chil, S. H. Rim, W. H. Kye, J. W. Ryu, and Y. J. Park, “Anti-synchronization of chaotic oscillators,” Physics Letters A, vol. 320, no. 1, pp. 39–46, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. J. Hu, S. Chen, and L. Chen, “Adaptive control for anti-synchronization of Chua's chaotic system,” Physics Letters A, vol. 339, no. 6, pp. 455–460, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. G. H. Li and S. P. Zhou, “Anti-synchronization in different chaotic systems,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 516–520, 2007. View at Publisher · View at Google Scholar · View at Scopus
  27. L. Pan, W. Zhou, J. Fang, and D. Li, “A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems,” Nonlinear Dynamics, vol. 62, no. 1-2, pp. 417–425, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. J. M. González-Miranda, “Synchronization of symmetric chaotic systems,” Physical Review E, vol. 53, no. 6, pp. 5656–5669, 1996. View at Google Scholar · View at Scopus
  29. R. Mainieri and J. Rehacek, “Projective synchronization in three-dimensional chaotic systems,” Physical Review Letters, vol. 82, no. 15, pp. 3042–3045, 1999. View at Publisher · View at Google Scholar · View at Scopus
  30. G.-H. Li, “Modified projective synchronization of chaotic system,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1786–1790, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. Q. Jia, “Projective synchronization of a new hyperchaotic Lorenz system,” Physics Letters A, vol. 370, no. 1, pp. 40–45, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  32. M. F. Hu and Z. Y. Xu, “Adaptive feedback controller for projective synchronization,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 1253–1260, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. D. Ghosh, “Projective synchronization in multiple modulated time-delayed systems with adaptive scaling factor,” Nonlinear Dynamics, vol. 62, no. 4, pp. 751–759, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. S. Shao, “Controlling general projective synchronization of fractional order Rossler systems,” Chaos, Solitons & Fractals, vol. 39, no. 4, pp. 1572–1577, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  35. G. Peng and Y. Jiang, “Generalized projective synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal,” Physics Letters A, vol. 372, no. 22, pp. 3963–3970, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. X. Wu and H. Wang, “A new chaotic system with fractional order and its projective synchronization,” Nonlinear Dynamics, vol. 61, no. 3, pp. 407–417, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. S. Wang, Y. Yu, and M. Diao, “Hybrid projective synchronization of chaotic fractional order systems with different dimensions,” Physica A, vol. 389, no. 21, pp. 4981–4988, 2010. View at Publisher · View at Google Scholar · View at Scopus
  38. L. Pan, W. Zhou, J. Fang, and D. Li, “Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3754–3760, 2010. View at Publisher · View at Google Scholar
  39. H. M. Deng, T. H. Li, Q. Wang, and H. Li, “A fractional-order hyperchaotic system and its synchronization,” Chaos, Solitons & Fractals, vol. 41, no. 2, pp. 962–969, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  40. N. Sun, H. G. Zhang, Z. L. Wang, and M. Dong, “Anti-synchronization of fractional-order hyperchaotic systems via sliding mode controller,” in Proceedings of the 29th Chinese Control Conference (CCC '10), pp. 755–759, July 2010. View at Scopus
  41. H. W. Songa, X. R. Lin, H. Yang, R. Zhen, Y. Chie, and X. Feng, “Full state hybrid projective synchronization in fractional Chen-Lee hyperchaotic system,” in Proceedings of the International Conference on Communications, Circuits and Systems (ICCCAS '10), pp. 777–780, July 2010. View at Publisher · View at Google Scholar · View at Scopus