Skip to main content
Log in

Adaptive projective synchronization of dynamical networks with distributed time delays

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, the adaptive projective synchronization of dynamical network with distributed time delays is investigated. Network with unknown topology and network with both unknown topology and system parameters of node dynamics are considered respectively. Based on Lyapunov stability theory and LaSalle’s invariance principle, the sufficient conditions for achieving projective synchronization are obtained. Numerical examples are provided to show the effectiveness of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)

    Article  Google Scholar 

  2. Barabasi, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)

    Article  MathSciNet  Google Scholar 

  3. Li, C., Sun, W., Kurths, J.: Synchronization between two coupled complete networks. Phys. Rev. E 76, 046204 (2007)

    Google Scholar 

  4. Zhou, J., Lu, J.A., Lü, J.: Pinning adaptive synchronization of a general complex dynamical network. Automatica 44, 996–1003 (2008)

    Article  Google Scholar 

  5. Tang, H., Chen, L., Lu, J.A., Tse, C.: Adaptive synchronization between two complex networks with nonidentical topological structures. Physica A 387, 5623–5630 (2008)

    Article  Google Scholar 

  6. He, G., Yang, J.: Adaptive synchronization in nonlinearly coupled dynamical networks. Chaos Solitons Fractals 38, 1254–1259 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lu, J., Cao, J.: Adaptive synchronization in tree-like dynamical networks. Nonlinear Anal., Real World Appl. 8, 1252–1260 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Zhou, J., Xiang, L., Liu, Z.: Global synchronization of generalized complex networks with mixed coupling delays. Physica A 385, 729–742 (2007)

    Article  MathSciNet  Google Scholar 

  9. Sharma, B.B., Kar, I.N.: Observer-based synchronization scheme for a class of chaotic systems using contraction theory. Nonlinear Dyn. 63, 429–445 (2011)

    Article  MathSciNet  Google Scholar 

  10. Amritkar, R.E., Hu, C.K.: Synchronized state of coupled dynamics on time-varying networks. Chaos 16, 015117 (2006)

    Article  MathSciNet  Google Scholar 

  11. Li, X., Chen, G.: Synchronization and desynchronization of complex dynamical networks. IEEE Trans. Circuits Syst. I 50, 1381–1390 (2003)

    Article  Google Scholar 

  12. Wang, X., Chen, G.: Synchronization in small-world dynamical networks. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 187–192 (2002)

    Article  Google Scholar 

  13. Mahmoud, G.M., Mahmoud, E.E.: Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dyn. 62, 875–882 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lü, L., Li, C.: Generalized synchronization of spatiotemporal chaos in a weighted complex network. Nonlinear Dyn. 63, 699–710 (2011)

    Article  Google Scholar 

  15. Mainieri, R., Rehacek, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82, 3042–3045 (1999)

    Article  Google Scholar 

  16. Jia, Z., Lu, J.A., Deng, G., Zhang, Q.: Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters. Chin. Phys. B 16, 1246–1251 (2007)

    Article  Google Scholar 

  17. Li, R.H., Xu, W., Li, S.: Adaptive generalized projective synchronization in different chaotic systems based on parameter identification. Phys. Lett. A 367, 199–206 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Feng, C.F.: Projective synchronization between two different time-delayed chaotic systems using active control approach. Nonlinear Dyn. 62, 453–459 (2010)

    Article  MATH  Google Scholar 

  19. Li, C.D., Liao, X.F., Yang, X.F., Huang, T.W.: New algebraic conditions for global exponential stability of delayed recurrent neural networks. Chaos 15, 043103 (2005)

    Article  MathSciNet  Google Scholar 

  20. Chavez, M., Hwang, D.-U., Amann, A., Hentschel, H.G.E., Boccaletti, S.: Synchronization is enhanced in weighted complex networks. Phys. Rev. Lett. 94, 218701 (2005)

    Article  Google Scholar 

  21. Zhou, J., Chen, T.: Synchronization in general complex delayed dynamical networks. IEEE Trans. Circuits Syst. I 53, 733–744 (2006)

    Article  Google Scholar 

  22. Zheng, S., Bi, Q., Cai, G.: Adaptive projective synchronization in complex networks with time-varying coupling delay. Phys. Lett. A 373, 1553–1559 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Li, C.H., Yang, S.Y.: Synchronization in linearly coupled dynamical networks with distributed time delays. Int. J. Bifurc. Chaos Appl. Sci. Eng. 18, 2039–2047 (2008)

    Article  MATH  Google Scholar 

  24. Liu, H., Lu, J., Zhang, Q.: Projectively lag synchronization and uncertain parameters identification of a new hyperchaotic system. Nonlinear Dyn. 62, 427–435 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hassan, K.K.: Nonlinear Systems. Prentice Hall, New York (2002)

    MATH  Google Scholar 

  26. Robinson, R.C.: An Introduction to Dynamical Systems: Continuous and Discrete. Pearson Education, Upper Saddle River (2004)

    MATH  Google Scholar 

  27. Jia, Y.: Robust H Control. Science Press, Beijing (2007)

    Google Scholar 

  28. Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 659–661 (2002)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhaoyan Wu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rao, P., Wu, Z. & Liu, M. Adaptive projective synchronization of dynamical networks with distributed time delays. Nonlinear Dyn 67, 1729–1736 (2012). https://doi.org/10.1007/s11071-011-0100-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0100-9

Keywords

Navigation