Table of Contents Author Guidelines Submit a Manuscript
Complexity
Volume 2017, Article ID 4020548, 9 pages
https://doi.org/10.1155/2017/4020548
Research Article

Fixed-Time Complex Modified Function Projective Lag Synchronization of Chaotic (Hyperchaotic) Complex Systems

School of Electrical Engineering, University of Ulsan, 93 Daehak-ro, Nam-gu, Ulsan 680-749, Republic of Korea

Correspondence should be addressed to Hee-Jun Kang; rk.ca.naslu@gnakjh

Received 21 February 2017; Revised 11 April 2017; Accepted 7 May 2017; Published 13 July 2017

Academic Editor: Sigurdur F. Hafstein

Copyright © 2017 Xuan-Toa Tran and Hee-Jun Kang. 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. J. D. Gibbon and M. J. McGuinness, “The real and complex Lorenz equations in rotating fluids and lasers,” Physica D. Nonlinear Phenomena, vol. 5, no. 1, pp. 108–122, 1982. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. F. Zhang and S. Liu, “Self-time-delay synchronization of time-delay coupled complex chaotic system and its applications to communication,” International Journal of Modern Physics C. Computational Physics and Physical Computation, vol. 25, no. 3, Article ID 1350102, 1350102, 13 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. X. Wu, C. Zhu, and H. Kan, “An improved secure communication scheme based passive synchronization of hyperchaotic complex nonlinear system,” Applied Mathematics and Computation, vol. 252, pp. 201–214, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. X. B. Zhou, M. R. Jiang, and X. M. Cai, “Synchronization of a novel hyperchaotic complex-variable system based on finite-time stability theory,” Entropy, vol. 15, no. 10, pp. 4334–4344, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  5. J. Sun and Y. Shen, “Adaptive anti-synchronization of chaotic complex systems and chaotic real systems with unknown parameters,” Journal of Vibration and Control, vol. 22, no. 13, pp. 2992–3003, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. J. Sun, Y. Shen, Q. Yin, and C. Xu, “Compound synchronization of four memristor chaotic oscillator systems and secure communication,” Chaos. An Interdisciplinary Journal of Nonlinear Science, vol. 23, no. 1, Article ID 013140, 013140, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. P. Liu, H. J. Song, and X. Li, “Observe-based projective synchronization of chaotic complex modified van der pol-duffing oscillator with application to secure communication,” Journal of Computational and Nonlinear Dynamics, vol. 10, no. 5, Article ID 051015, 2015. View at Publisher · View at Google Scholar
  8. J. Sun, Y. Shen, and X. Zhang, “Modified projective and modified function projective synchronization of a class of real nonlinear systems and a class of complex nonlinear systems,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 78, no. 3, pp. 1755–1764, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. J. Sun, Y. Shen, X. Wang, and J. Chen, “Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 76, no. 1, pp. 383–397, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. X. Wang and H. Zhang, “Backstepping-based lag synchronization of a complex permanent magnet synchronous motor system,” Chinese Physics B, vol. 22, no. 4, Article ID 048902, 2013. View at Publisher · View at Google Scholar · View at Scopus
  11. J. Sun, Y. Wang, L. Yao, Y. Shen, and G. Cui, “General hybrid projective complete dislocated synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 39, no. 20, pp. 6150–6164, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. G. M. Mahmoud and E. E. Mahmoud, “Modified projective lag synchronization of two nonidentical hyperchaotic complex nonlinear systems,” International Journal of Bifurcation and Chaos, vol. 21, no. 8, pp. 2369–2379, 2011. View at Publisher · View at Google Scholar · View at Scopus
  13. X. Wang and N. Wei, “Modified function projective lag synchronization of hyperchaotic complex systems with parameter perturbations and external perturbations,” Journal of Vibration and Control, vol. 21, no. 16, pp. 3266–3280, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. F. Nian, X. Wang, Y. Niu, and D. Lin, “Module-phase synchronization in complex dynamic system,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2481–2489, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. E. E. Mahmoud, “Complex complete synchronization of two nonidentical hyperchaotic complex nonlinear systems,” Mathematical Methods in the Applied Sciences, vol. 37, no. 3, pp. 321–328, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. J. Sun, Y. Wang, Y. Wang, and Y. Shen, “Finite-time synchronization between two complex-variable chaotic systems with unknown parameters via nonsingular terminal sliding mode control,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 85, no. 2, pp. 1105–1117, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. Z. Wu, J. Duan, and X. Fu, “Complex projective synchronization in coupled chaotic complex dynamical systems,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 69, no. 3, pp. 771–779, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. J. Sun, G. Cui, Y. Wang, and Y. Shen, “Combination complex synchronization of three chaotic complex systems,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 79, no. 2, pp. 953–965, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. G. M. Mahmoud and E. E. Mahmoud, “Complex modified projective synchronization of two chaotic complex nonlinear systems,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 73, no. 4, pp. 2231–2240, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. F. Zhang and S. Liu, “Adaptive complex function projective synchronization of uncertain complex chaotic systems,” Journal of Computational and Nonlinear Dynamics, vol. 11, no. 1, Article ID 011013, 2016. View at Publisher · View at Google Scholar · View at Scopus
  21. J. W. Lee, S. M. Lee, and S. C. Won, “Complex function projective synchronization of general networked chaotic systems by using complex adaptive fuzzy logic,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 81, no. 4, pp. 2095–2106, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. S. P. Bhat and D. S. Bernstein, “Finite-time stability of continuous autonomous systems,” SIAM Journal on Control and Optimization, vol. 38, no. 3, pp. 751–766, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 57, no. 8, pp. 2106–2110, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. M. Basin, C. B. Panathula, and Y. Shtessel, “Adaptive uniform finite-/fixed-time convergent second-order sliding-mode control,” International Journal of Control, vol. 89, no. 9, pp. 1777–1787, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. M. Basin, Y. Shtessel, and F. Aldukali, “Continuous finite- and fixed-time high-order regulators,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 353, no. 18, pp. 5001–5012, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. A. Polyakov, D. Efimov, and W. Perruquetti, “Robust stabilization of MIMO systems in finite/fixed time,” International Journal of Robust and Nonlinear Control, vol. 26, no. 1, pp. 69–90, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  27. A. Levant, “Higher-order sliding modes, differentiation and output-feedback control,” International Journal of Control, vol. 76, no. 9-10, pp. 924–941, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. M. T. Angulo, J. A. Moreno, and L. Fridman, “Robust exact uniformly convergent arbitrary order differentiator,” Automatica. A Journal of IFAC, the International Federation of Automatic Control, vol. 49, no. 8, pp. 2489–2495, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. A. Polyakov and L. Fridman, “Stability notions and Lyapunov functions for sliding mode control systems,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 351, no. 4, pp. 1831–1865, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. E. Bechkenback and R. Bell Man, Inequalities, Springer, Berlin, Germany, 1971.
  31. J. J. E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 1991.
  32. H. K. Khalil, Nonlinear Systems, Prentice–Hall, Upsaddle River, NJ, USA, thrd edition, 2002.