Differential Equations and Dynamical Systems

, Volume 27, Issue 4, pp 413–422 | Cite as

A New Q–S Synchronization Results for Discrete Chaotic Systems

  • Adel Ouannas
  • Zaid OdibatEmail author
  • Nabil Shawagfeh
Original Research


This paper investigates the problem of Q–S synchronization for different dimensional chaotic dynamical systems in discrete-time. Based on two control laws and stability theory of dynamical systems in discrete-time, new synchronization schemes are derived. Numerical examples demonstrate the effectiveness and feasibility of the proposed control techniques.


Q–S synchronization Discrete-time Chaotic system Controller Stability 


  1. 1.
    Pecora, L., Carrol, T.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Blasius, B., Stone, L.: Chaos and phase synchronization in ecological systems. Int. J. Bifur. Chaos 10, 2361–2380 (2000)zbMATHCrossRefGoogle Scholar
  3. 3.
    Lakshmanan, M., Murali, K.: Chaos in Nonlinear Oscillators: Controlling and Synchronization. World Scientific, Singapore (1996)zbMATHCrossRefGoogle Scholar
  4. 4.
    Han, S.K., Kerrer, C., Kuramoto, Y.: Dephasing and bursting in coupled neural oscillators. Phys. Rev. Lett. 75, 3190–3193 (1995)CrossRefGoogle Scholar
  5. 5.
    Mengue, A., Essimbi, B.: Secure communication using chaotic synchronization in mutually coupled semiconductor lasers. Nonlin. Dyn. 70, 1241–1253 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, G., Yu, X.: Chaos Control: Theory and Applications. Springer, Berlin (2003)zbMATHCrossRefGoogle Scholar
  7. 7.
    Yamada, T., Fujisaka, H.: Stability theory of synchronized motion in coupled-oscillator systems. Progr. Theoret. Phys. 70, 1240–1248 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ott, E., Grebogi, C., Yorke, J.: Controling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Aziz-Alaoui, M.A.: Synchronization of chaos. In: Encyclopedia of Mathematical Physics, pp. 213–226 (2006)CrossRefGoogle Scholar
  10. 10.
    Lu, J., Wu, X., Han, X., Lü, J.: Adaptive feedback synchronization of a unified chaotic system. Phys. Lett. A 329(4), 327–333 (2004)zbMATHCrossRefGoogle Scholar
  11. 11.
    Wu, X., Lu, J.: Parameter identification and backstepping control of uncertain Lü system. Chaos Solitons Fractals 18(4), 721–729 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Yang, C., Lin, C.: Robust adaptive sliding mode control for synchronization of space-clamped Fitz–Hugh–Nagumo neurons. Nonlin. Dyn. 69, 2089–2096 (2012)CrossRefGoogle Scholar
  13. 13.
    Zhang, X., Zhu, H.: Anti-synchronization of two different hyperchaotic systems via active and adaptive control. Int. J. Nonlin. Sci. 6(3), 216–223 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Park, J.H.: Adaptive synchronization of hyperchaotic Chen system with uncertain parameters. Chaos Solitons Fract. 26(3), 959–964 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Xin, L., Yong, C.: Generalized projective synchronization between R össler system and new unified chaotic system. Commun. Theor. Phys. 48, 132–136 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Li, X., Leung, A., Han, X., Liu, X., Chu, Y.: Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control. Nonlin. Dyn. 63, 263–275 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Yao, C., Zhao, Q., Yu, J.: Complete synchronization induced by disorder in coupled chaotic lattices. Phys. Lett. A 377(5), 370–377 (2013)CrossRefGoogle Scholar
  18. 18.
    Zhang, G., Liu, Z., Ma, Z.: Generalized synchronization of different dimensional chaotic dynamical systems. Chaos Solitons Fract. 32(2), 773–779 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    He, X., Li, C., Huang, J., Xiao, L.: Generalized synchronization of arbitrary-dimensional chaotic systems. Opti-Int. J. Light Electron Optics. 126(4), 454–459 (2015)CrossRefGoogle Scholar
  20. 20.
    Qiang, J.: Projective synchronization of a new hyperchaotic Lorenz system. Phys. Lett. A 370(1), 40–45 (2007)zbMATHCrossRefGoogle Scholar
  21. 21.
    Han, M., Zhang, M., Zhang, Y.: Projective synchronization between two delayed networks of different sizes with nonidentical nodes and unknown parameters. Neurocomputing 171(1), 605–614 (2016)CrossRefGoogle Scholar
  22. 22.
    Li, X.: Generalized projective synchronization using nonlinear control method. Int. J. Nonlin. Sci. 8, 79–85 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Cai, G., Hu, P., Li, Y.: Modified function lag projective synchronization of a financial hyperchaotic system. Nonlin. Dyn. 69(3), 1457–1464 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Du, H.: Function projective synchronization in complex dynamical networks with or without external disturbances via error feedback control. Neurocomputing 173(3), 1443–1449 (2016)CrossRefGoogle Scholar
  25. 25.
    Yan, Z.: Chaos Q-S synchronization between Rössler system and the new unified chaotic system. Phys. Lett. A 334(5), 406–412 (2005)zbMATHCrossRefGoogle Scholar
  26. 26.
    Manfeng, H., Zhenyuan, X.: Nonlin. Anal. Theor. Meth. Appl. A general scheme for Q-S synchronization of chaotic systems. 69(4), 1091–1099 (2008)Google Scholar
  27. 27.
    Wang, Q., Chen, Y.: Generalized Q-S (lag, anticipated and complete) synchronization in modified Chua’s circuit and Hindmarsh–Rose systems. Appl. Math. Comput. 181(1), 48–56 (2006)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Wang, Z.L., Shi, X.R.: Adaptive Q-S synchronization of nonidentical chaotic systems with unknowns parameters. Nonlin. Dyn. 59, 559–567 (2010)zbMATHCrossRefGoogle Scholar
  29. 29.
    Yan, Z.: Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic system-A symbolic-numeric computation approach. Chaos 15(2), 023902 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Yang, Y., Chen, Y.: The generalized Q-S synchronization between the generalized Lorenz canonical form and the Rössler system. Chaos Solitons Fract. 39, 2378–2385 (2009)zbMATHCrossRefGoogle Scholar
  31. 31.
    Zhao, J., Ren, T.: Q-S synchronization between chaotic systems with double scaling functions. Nonlin. Dyn. 62(3), 665–672 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Zhao, J., Zhang, K.: A general scheme for Q-S synchronization of chaotic systems with unknown parameters and scaling functions. Appl. Math. Comput. 216(7), 2050–2057 (2010)MathSciNetzbMATHGoogle Scholar
  33. 33.
    An, H.L., Chen, Y.: The function cascade synchronization scheme for discrete-time hyperchaotic systems. Commun. Nonlin. Sci. Numer. Simulat. 14, 1494–1501 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Yan, Z.: Q-S synchronization in 3D Hénon-like map and generalized Hénon map via a scalar controller. Phys. Lett. A 342, 309–317 (2005)zbMATHCrossRefGoogle Scholar
  35. 35.
    Yan, Z.: Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: A symbolic-numeric computation approach. Chaos 16(1), 013119 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Studies in Nonlinearity. Westview Press, Boulder (2001)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2016

Authors and Affiliations

  1. 1.LAMIS Laboratory, Department of Mathematics and Computer ScienceUniversity of TebessaTebessaAlgeria
  2. 2.Department of Mathematics, Faculty of ScienceAl-Balqa’ Applied UniversitySaltJordan
  3. 3.Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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