Nonlinear Dynamics

, Volume 75, Issue 1–2, pp 209–216 | Cite as

Adaptive impulsive synchronization of uncertain drive-response complex-variable chaotic systems

  • Danfeng Liu
  • Zhaoyan Wu
  • Qingling Ye
Original Paper


In this paper, impulsive synchronization of drive-response complex-variable chaotic systems is investigated. The drive-response systems with known parameters is considered via impulsive control and adaptive scheme as well as systems with unknown parameters. Noticeably, adaptive strategy is adopted to relax the restriction on the impulsive interval, and the system parameters need not to be known beforehand. According to the Lyapunov stability theory, some synchronization criteria are derived and verified by several numerical simulations.


Synchronization Complex-variable Chaotic system Impulsive control 



The author would like to thank the referees and editors for their valuable comments and suggestions on improving this article. This research is supported jointly by the Tianyuan Special Funds of the NSFC under Grant No. 11226242 and the Natural Science Foundation of Jiangxi Province of China under Grants Nos. 20122BAB211006 and 20132BAB201016.


  1. 1.
    Mainieri, R., Rehacek, J.: Projective synchronization in the three-dimensional chaotic system. Phys. Rev. Lett. 82, 3042–3045 (1999) CrossRefGoogle Scholar
  2. 2.
    Xu, D.L., Chee, C.Y.: Controlling the ultimate state of projective synchronization in chaotic systems of arbitrary dimension. Phys. Rev. E 66, 046218 (2002) CrossRefGoogle Scholar
  3. 3.
    Ma, Z.J., Liu, Z.R., Zhang, G.: A new method to realize cluster synchronization in connected chaotic networks. Chaos 16, 023103 (2006) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Sun, W.G., Yang, Y.Y., Li, C.P., Liu, Z.R.: Synchronization inside complex dynamical networks with double time-delays and nonlinear inner-coupling functions. Int. J. Mod. Phys. B 25, 1531–1541 (2011) CrossRefGoogle Scholar
  5. 5.
    Xu, Y.H., Zhou, W.N., Fang, J.A., Sun, W.: Adaptive synchronization of uncertain chaotic systems with adaptive scaling function. J. Franklin Inst. 348, 2406–2416 (2011) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Rao, P.C., Wu, Z.Y., Liu, M.: Adaptive projective synchronization of dynamical networks with distributed time delays. Nonlinear Dyn. 67, 1729–1736 (2012) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Guan, J.B.: Function projective synchronization of a class of chaotic systems with uncertain parameters. Math. Probl. Eng. 2012, 431752 (2012) Google Scholar
  8. 8.
    Yu, W.W., Chen, G.R., Cao, J.D.: Adaptive synchronization of certain and uncertain coupled complex networks. Asian J. Control 13, 418–429 (2011) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Sun, W.G., Zhang, J.Y., Li, C.P.: Synchronization analysis of two coupled complex networks with time delays. Discrete Dyn. Nat. Soc. 2011, 209321 (2011) MathSciNetGoogle Scholar
  10. 10.
    Wu, Z.Y., Fu, X.C.: Cluster mixed synchronization via pinning control and adaptive coupling strength in community networks with nonidentical nodes. Commun. Nonlinear Sci. Numer. Simul. 17, 1628–1636 (2012) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Yu, W.W., DeLellis, P., Chen, G.R., Bernardo, M., Kurths, J.: Distributed adaptive control of synchronization in complex networks. IEEE Trans. Autom. Control 57, 2153–2158 (2012) CrossRefGoogle Scholar
  12. 12.
    Chen, Y.S., Hwang, R.R., Chang, C.C.: Adaptive impulsive synchronization of uncertain chaotic systems. Phys. Lett. A 374, 2254–2258 (2010) CrossRefMATHGoogle Scholar
  13. 13.
    Deng, L., Wu, Z.: Impulsive cluster synchronization in community network with nonidentical nodes. Commun. Theor. Phys. 58, 525–530 (2012) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Zhang, Q., Zhao, J.: Projective and lag synchronization between general complex networks via impulsive control. Nonlinear Dyn. 67, 2519–2525 (2012) CrossRefMATHGoogle Scholar
  15. 15.
    Sun, W., Chen, Z., Lü, J., Chen, S.: Outer synchronization of complex networks with delay via impulse. Nonlinear Dyn. 69, 1751–1764 (2012) CrossRefMATHGoogle Scholar
  16. 16.
    Ma, C., Wang, X.Y.: Impulsive control and synchronization of a new unified hyperchaotic system with varying control gains and impulsive intervals. Nonlinear Dyn. 70, 551–558 (2012) CrossRefMATHGoogle Scholar
  17. 17.
    Fowler, A.C., Gibbon, J.D., McGuinness, M.J.: The complex Lorenz equations. Physica D 4, 139–163 (1982) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Gibbon, J.D., McGuinnes, M.J.: The real and complex Lorenz equations in rotating fluids and lasers. Physica D 5, 108–122 (1982) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Ning, C.Z., Haken, H.: Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations. Phys. Rev. A 41, 3826–3837 (1990) CrossRefGoogle Scholar
  20. 20.
    Mahmoud, G.M., Bountis, T., Mahmoud, E.E.: Active control and global synchronization for complex Chen and Lü systems. Int. J. Bifurc. Chaos 17, 4295–4308 (2007) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Mahmoud, G.M., Aly, S.A., Farghaly, A.A.: On chaos synchronization of a complex two coupled dynamos system. Chaos Solitons Fractals 33, 178–187 (2007) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hu, M.F., Yang, Y.Q., Xu, Z.Y., Guo, L.X.: Hybrid projective synchronization in a chaotic complex nonlinear system. Math. Comput. Simul. 79, 449–457 (2008) CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Nian, F.Z., Wang, X.Y., Niu, Y.J., Lin, D.: Module-phase synchronization in complex dynamic system. Appl. Math. Comput. 217, 2481–2489 (2010) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Wu, Z.Y., Chen, G.R., Fu, X.C.: Synchronization of a network coupled with complex-variable chaotic systems. Chaos 22, 023127 (2012) CrossRefGoogle Scholar
  25. 25.
    Wu, Z.Y., Duan, J.Q., Fu, X.C.: Complex projective synchronization in coupled chaotic complex dynamical systems. Nonlinear Dyn. 69, 771–779 (2012) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Wu, Z.Y.: Complex hybrid synchronization in drive-response complex-variable chaotic systems. Int. J. Nonlinear Sci. Numer. Simul. 13, 469–478 (2012) Google Scholar
  27. 27.
    Yu, W.W., Chen, G.R., Cao, J.D., Lü, J.H., Parlitz, U.: Parameter identification of dynamical systems from time series. Phys. Rev. E 75, 067201 (2007) CrossRefGoogle Scholar
  28. 28.
    Guan, J.B.: Synchronization control of two different chaotic systems with known and unknown parameters. Chin. Phys. Lett. 27, 020502 (2010) CrossRefGoogle Scholar
  29. 29.
    Wu, Z.Y., Fu, X.C.: Adaptive function projective synchronization of discrete chaotic systems with unknown parameters. Chin. Phys. Lett. 27, 050502 (2010) CrossRefGoogle Scholar
  30. 30.
    Peng, H.P., Li, L.X., Yang, Y.X., Sun, F.: Conditions of parameter identification from time series. Phys. Rev. E 83, 036202 (2011) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceJiangxi Normal UniversityNanchangChina

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