Table of Contents Author Guidelines Submit a Manuscript
Journal of Control Science and Engineering
Volume 2015, Article ID 703753, 7 pages
http://dx.doi.org/10.1155/2015/703753
Research Article

A Robust Control Method for Synchronization between Different Dimensional Integer-Order and Fractional-Order Chaotic Systems

1Department of Mathematics and Computer Science, University of Tebessa, 12002 Tebessa, Algeria
2Department of Health Informatics, College of Public Health and Health Informatics, King Saud Bin Abdulaziz University for Health Science, Riyadh 11481, Saudi Arabia

Received 8 November 2015; Accepted 17 December 2015

Academic Editor: Xiao He

Copyright © 2015 Adel Ouannas and Raghib Abu-Saris. 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. I. Petráš, “A note on the fractional-order Chua's system,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 140–147, 2008. View at Publisher · View at Google Scholar · View at Scopus
  2. D. Cafagna, “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
  3. P. J. Torvik and R. L. Bagley, “On the appearance of the fractional derivative in the behavior of real materials,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 294–298, 1984. View at Publisher · View at Google Scholar · View at Scopus
  4. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  5. S. Westerlund, Dead Matter Has Memory!, Causal Consulting, Kalmar, Sweden, 2002.
  6. A. Oustaloup, La Derivation Non Entiere: Theorie, Synthese et Applications, Hermès, Paris, France, 1995.
  7. G.-Q. Si, Z.-Y. Sun, and Y.-B. Zhang, “A general method for synchronizing an integer-order chaotic system and a fractional-order chaotic system,” Chinese Physics B, vol. 20, no. 8, Article ID 080505, 2011. View at Publisher · View at Google Scholar · View at Scopus
  8. Z. Ping, Y.-M. Cheng, and K. Fei, “Synchronization between fractional-order chaotic systems and integer orders chaotic systems (fractional-order chaotic systems),” Chinese Physics B, vol. 19, no. 9, Article ID 090503, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. D.-Y. Chen, R.-F. Zhang, X.-Y. Ma, and J. Wang, “Synchronization between a novel class of fractional-order and integer-order chaotic systems via a sliding mode controller,” Chinese Physics B, vol. 21, no. 12, Article ID 120507, 2012. View at Publisher · View at Google Scholar
  10. D. Chen, R. Zhang, J. C. Sprott, H. Chen, and X. Ma, “Synchronization between integer-order chaotic systems and a class of fractional-order chaotic systems via sliding mode control,” Chaos, vol. 22, Article ID 023130, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. Y.-P. Wu and G.-D. Wang, “Synchronization between fractional-order and integer-order hyperchaotic systems via sliding mode controller,” Journal of Applied Mathematics, vol. 2013, Article ID 151025, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  12. Y. Wu and G. Wang, “Synchronization of a class of fractional-order and integer order hyperchaotic systems,” Journal of Vibration and Control, vol. 20, no. 10, pp. 1584–1588, 2013. View at Publisher · View at Google Scholar · View at Scopus
  13. L.-X. Jia, H. Dai, and M. Hui, “Nonlinear feedback synchronisation control between fractional-order and integer-order chaotic systems,” Chinese Physics B, vol. 19, no. 11, Article ID 110509, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. I. El Gammoudi and M. Feki, “Synchronization of integer order and fractional order Chua's systems using robust observer,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 3, pp. 625–638, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. D. Chen, C. Wu, H. H. Iu, and X. Ma, “Circuit simulation for synchronization of a fractional-order and integer-order chaotic system,” Nonlinear Dynamics, vol. 73, no. 3, pp. 1671–1686, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. A. Khan and P. Tripathi, “Synchronization between a fractional order chaotic system and an integer order chaotic system,” Nonlinear Dynamics and Systems Theory, vol. 14, no. 4, 2013. View at Google Scholar
  17. L.-X. Yang, W.-S. He, and X.-J. Liu, “Synchronization between a fractional-order system and an integer order system,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4708–4716, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. Y. Wu and G. Wang, “Synchronization and anti-synchronization between a class of fractional-order and integer-order chaotic systems with only one controller term,” Journal of Theoretical & Applied Information Technology, vol. 48, no. 1, pp. 145–151, 2013. View at Google Scholar · View at Scopus
  19. Z. Ping and Y.-X. Cao, “Function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems,” Chinese Physics B, vol. 19, no. 10, Article ID 100507, 2010. View at Publisher · View at Google Scholar
  20. H.-J. Liu, Z.-L. Zhu, H. Yu, and Q. Zhu, “Modified function projective synchronization of fractional order chaotic systems with different dimensions,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 763564, 7 pages, 2013. View at Publisher · View at Google Scholar
  21. Z. Wu, X. Xu, G. Chen, and X. Fu, “Generalized matrix projective synchronization of general colored networks with different-dimensional node dynamics,” Journal of the Franklin Institute, vol. 351, no. 9, pp. 4584–4591, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. G. Zhang, Z. Liu, and Z. Ma, “Generalized synchronization of different dimensional chaotic dynamical systems,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 773–779, 2007. View at Publisher · View at Google Scholar · View at Scopus
  23. Y. Yu, H.-X. Li, and J. Yu, “Generalized synchronization of different dimensional chaotic systems based on parameter identification,” Modern Physics Letters B, vol. 23, no. 22, pp. 2593–2606, 2009. View at Publisher · View at Google Scholar
  24. X. He, C. Li, J. Huang, and L. Xiao, “Generalized synchronization of arbitrary-dimensional chaotic systems,” Optik, vol. 126, pp. 454–461, 2015. View at Google Scholar
  25. A. Ouannas and Z. Odibat, “Generalized synchronization of different dimensional chaotic dynamical systems in discrete time,” Nonlinear Dynamics, vol. 81, no. 1-2, pp. 765–771, 2015. View at Publisher · View at Google Scholar
  26. M. Hu, Z. Xu, R. Zhang, and A. Hu, “Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order,” Physics Letters A, vol. 365, no. 4, pp. 315–327, 2007. View at Publisher · View at Google Scholar
  27. A. M. El-Sayed, H. M. Nour, A. Elsaid, A. E. Matouk, and A. Elsonbaty, “Circuit realization, bifurcations, chaos and hyperchaos in a new 4D system,” Applied Mathematics and Computation, vol. 239, pp. 333–345, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. S. Ogunjo, “Increased and reduced order synchronization of 2D and 3D dynamical systems,” International Journal of Nonlinear Science, vol. 16, no. 2, pp. 105–112, 2013. View at Google Scholar · View at MathSciNet
  29. M. M. Al-Sawalha and M. S. M. Noorani, “Adaptive increasing-order synchronization and anti-synchronization of chaotic systems with uncertain parameters,” Chinese Physics Letters, vol. 28, no. 11, Article ID 110507, 2011. View at Publisher · View at Google Scholar · View at Scopus
  30. K. S. Ojo, S. T. Ogunjo, A. N. Njah, and I. A. Fuwape, “Increased-order generalized synchronization of chaotic and hyperchaotic systems,” Pramana, vol. 84, no. 1, pp. 33–45, 2015. View at Publisher · View at Google Scholar · View at Scopus
  31. H. Manfeng and X. Zhenyuan, “A general scheme for Q-S synchronization of chaotic systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 4, pp. 1091–1099, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. Q. Wang and Y. Chen, “Generalized Q–S (lag, anticipated and complete) synchronization in modified Chua’s circuit and Hindmarsh–rose systems,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 48–56, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. Z. L. Wang and X.-R. Shi, “Adaptive Q-S synchronization of non-identical chaotic systems with unknowns parameters,” Nonlinear Dynamics, vol. 59, no. 4, pp. 559–567, 2010. View at Publisher · View at Google Scholar
  34. Z. Yan, “Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems—a symbolic-numeric computation approach,” Chaos, vol. 15, no. 2, Article ID 023902, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. Y. Yang and Y. Chen, “The generalized Q-S synchronization between the generalized Lorenz canonical form and the Rössler system,” Chaos, Solitons and Fractals, vol. 39, no. 5, pp. 2378–2385, 2009. View at Publisher · View at Google Scholar · View at Scopus
  36. J. Zhao and T. Ren, “Q–S synchronization between chaotic systems with double scaling functions,” Nonlinear Dynamics, vol. 62, no. 3, pp. 665–672, 2010. View at Publisher · View at Google Scholar
  37. J. Zhao and K. Zhang, “A general scheme for Q-S synchronization of chaotic systems with unknown parameters and scaling functions,” Applied Mathematics and Computation, vol. 216, no. 7, pp. 2050–2057, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  38. Z. Yan, “Q-S synchronization in 3D Hénon-like map and generalized Hénon map via a scalar controller,” Physics Letters A, vol. 342, no. 4, pp. 309–317, 2005. View at Publisher · View at Google Scholar · View at Scopus
  39. L.-X. Yang and W.-S. He, “Adaptive Q-S synchronization of fractional-order chaotic systems with nonidentical structures,” Abstract and Applied Analysis, vol. 2013, Article ID 367506, 8 pages, 2013. View at Publisher · View at Google Scholar
  40. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  41. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993. View at MathSciNet
  42. M. Caputo, “Linear models of dissipation whose Q is almost frequency independent. II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967. View at Publisher · View at Google Scholar
  43. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  44. R. Goren and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., Springer, New York, NY, USA, 1997. View at Google Scholar
  45. E. N. Lorenz, “Deterministic nonperiodic flow,” Journal of the Atmospheric Sciences, vol. 20, no. 2, pp. 130–141, 1963. View at Publisher · View at Google Scholar
  46. Z. Wang, X. Huang, Y.-X. Li, and X.-N. Song, “A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system,” Chinese Physics B, vol. 22, no. 1, Article ID 010504, 2013. View at Publisher · View at Google Scholar · View at Scopus