Nonlinear Dynamics

, Volume 67, Issue 4, pp 2621–2630 | Cite as

Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling

  • Song Zheng
Original Paper


This paper investigates the adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling, in which the weights of links between two connected nodes are time varying. By the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization are obtained, and a hybrid controller, that is, an adaptive feedback controller with impulsive control effects is designed. The numerical examples are presented to illustrate the effectiveness and advantage of the proposed synchronization criteria.


Drive-response delayed networks Adaptive-impulsive control Projective synchronization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 440–442 (1998) CrossRefGoogle Scholar
  2. 2.
    Barbaasi, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Li, X., Chen, G.: Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint. IEEE Trans Circuits Syst. I 50, 1381–1390 (2003) CrossRefGoogle Scholar
  4. 4.
    Wu, C.W.: Synchronization in Complex Networks of Nonlinear Dynamical Systems. World Scientific, Singapore (2007) zbMATHGoogle Scholar
  5. 5.
    Wang, X., Chen, G.: Pinning control of scale-free dynamical networks. Physica A 310, 521–531 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Li, X., Wang, X., Chen, G.: Pinning a complex dynamical network to its equilibrium. IEEE Trans. Circuits Syst. I 51, 2074–2087 (2004) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, T., Liu, X., Lu, W.: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I 54, 1317–1326 (2007) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yu, W., Chen, G.: On pinning synchronization of complex dynamical networks. Automatica 45, 429–435 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    De Lellis, P., Bernardo, M., di Garofalo, F.: Synchronization of complex networks through local adaptive coupling. Chaos 18, 037110 (2008) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cao, J., Lu, J.: Adaptive synchronization of neural networks with or without time-varying delays. Chaos 16, 013133 (2006) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lu, J., Cao, J.: Adaptive synchronization of uncertain dynamical networks with delayed coupling. Nonlinear Dyn. 53, 107–115 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Zheng, S., Dong, G., Bi, Q.: Adaptive projective synchronization in complex networks with time-varying coupling delay. Phys. Lett. A 373, 1553–1559 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Zhou, J., Xiang, L., Liu, Z.: Synchronization in complex delayed dynamical networks with impulsive effects. Physica A 384, 684–692 (2007) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zhang, G., Liu, Z., Ma, Z.: Synchronization of complex dynamical networks via impulsive control. Chaos 17, 043126 (2007) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Li, P., Cao, J., Wang, Z.: Robust impulsive synchronization of coupled delayed neural networks with uncertainties. Physica A 373, 261–272 (2007) CrossRefGoogle Scholar
  16. 16.
    Song, Q., Cao, J., Liu, F.: Synchronization of complex dynamical networks with nonidentical nodes. Phys. Lett. A 374, 544–551 (2010) CrossRefGoogle Scholar
  17. 17.
    Jiang, H., Bi, Q.: Impulsive synchronization of networked nonlinear dynamical systems. Phys. Lett. A 374, 2723–2729 (2010) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tang, Y., Leung, S.Y.S., Wong, W.K., Fang, J.: Impulsive pinning synchronization of stochastic discrete-time networks. Neurocomputing 73, 2132–2139 (2010) CrossRefGoogle Scholar
  19. 19.
    Jiang, H., Bi, Q., Zheng, S.: Impulsive consensus in directed networks of identical nonlinear oscillators with switching topologies. Commun. Nonlinear Sci. Numer. Simul. (2011). doi: 10.1016/j.cnsns.2011.04.030 Google Scholar
  20. 20.
    Mainieri, R., Rehacek, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82, 3042 (1999) CrossRefGoogle Scholar
  21. 21.
    Jia, Q.: Projective synchronization of a new hyperchaotic Lorenz system. Phys. Lett. A 370, 40–45 (2007) zbMATHCrossRefGoogle Scholar
  22. 22.
    Wang, Z.: Projective synchronization of hyperchaotic Lü system and Liu system. Nonlinear Dyn. 59, 455–462 (2010) zbMATHCrossRefGoogle Scholar
  23. 23.
    Feng, C.: Projective synchronization between two different time-delayed chaotic systems using active control approach. Nonlinear Dyn. 62, 453–459 (2010) zbMATHCrossRefGoogle Scholar
  24. 24.
    Hu, M., Yang, Y., Xu, Z., Zhang, R., Guo, L.: Projective synchronization in drive-response dynamical networks. Physica A 381, 457–466 (2007) CrossRefGoogle Scholar
  25. 25.
    Guo, L., Xu, Z., Hu, M.: Projective synchronization in drive-response networks via impulsive control. Chin. Phys. Lett. 25, 2816 (2008) CrossRefGoogle Scholar
  26. 26.
    Zhao, Y. Yang: Y.: The impulsive control synchronization of the drive-response dynamical network system. Phys. Lett. A 372, 7165–7171 (2008) zbMATHCrossRefGoogle Scholar
  27. 27.
    Sun, M., Zeng, C., Tian, L.X.: Projective synchronization in drive-response dynamical networks of partially linear systems with time-varying coupling delay. Phys. Lett. A 372, 6904–6908 (2008) zbMATHCrossRefGoogle Scholar
  28. 28.
    Wu, C.W.: Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 52, 282–286 (2005) Google Scholar
  29. 29.
    Lü, J., Chen, G.: A time-varying complex dynamical network model and its controlled synchronization criteria. IEEE Trans. Autom. Control 50, 84–846 (2005) Google Scholar
  30. 30.
    Stilwell, D., Bollt, E., Roberson, D.: Sufficient conditions for fast switching synchronization in time-varying network topologies. SIAM J. Appl. Dyn. Syst. 5, 140–156 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Leyva, I., Sendina-Nadal, I., Almendral, J.A., Sanjuan, M.A.F.: Sparse repulsive coupling enhances synchronization in complex networks. Phys. Rev. E 74, 056112 (2006) CrossRefGoogle Scholar
  32. 32.
    Chen, M.: Synchronization in time-varying networks: a matrix measure approach. Phys. Rev. E 76, 016104 (2007) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Li, P., Yi, Z.: Synchronization analysis of delayed complex networks with time-varying couplings. Physica A 387, 3729–3737 (2008) CrossRefGoogle Scholar
  34. 34.
    Wan, X., Sun, J.: Adaptive-impulsive synchronization of chaotic systems. Math. Comput. Simul. 81, 1609–1617 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Guan, Z., Hill, D.J., Yao, J.: A hybrid impulsive and switching control strategy for synchronization of nonlinear systems and application to Chua’s chaotic circuit. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16, 229–238 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Li, K., Lai, C.H.: Adaptive-impulsive synchronization of uncertain complex dynamical networks. Phys. Lett. A 372, 1601–1606 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Yang, M., Wang, Y., Wang, H.O., Tanaka, K., Guan, Z.: Delay independent synchronization of complex network via hybrid control. In: 2008 American Control Conference, Seatle, USA, 11–13 June, pp. 2266–2271 (2008) CrossRefGoogle Scholar
  38. 38.
    Cao, J., Ho, D.W.C., Yang, Y.: Projective synchronization of a class of delayed chaotic systems via impulsive control. Phys. Lett. A 373, 3128–3133 (2009) MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yan, J., Shen, J.: Impulsive stabilization of functional differential equations by Lyapunov-Razumikhin functions. Nonlinear Anal. 37, 245–255 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Newman, M.J.E., Watts, D.J.: Renormalization group analysis of the small-world network model. Phys. Lett. A 263, 341–346 (1999) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhejiang University of Finance & EconomicsHangzhouP.R. China

Personalised recommendations