Skip to main content
SpringerLink
Log in

Analyzing projective synchronization on different scaling factors in a drive-response coupling dynamical network with time-varying delays

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

Abstract

In this paper, projective synchronization of drive-response coupled dynamical network with delayed system nodes and coupling time-varying delay is investigated via impulsive control, where the scaling factors are different from each other. Different controllers are designed to achieve the projective synchronization: only impulsive control is used when the scaling factors need extra limitation, while an extra controller, that is, a simple linear feedback controller, is added when the scaling factors don’t need extra limitation. Based on the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization of such coupled network are established, and an estimate of the upper bound of impulsive intervals ensuring global exponential synchronization of drive-response coupled dynamical network is also given. Numerical examples on the time-delay Lorenz chaotic systems are presented finally to illustrate the effectiveness and advantage of the proposed synchronization criteria.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Huberman, B.A., Adamic, L.A.: Growth dynamics of the world-wide-web. Nature 401, 131–132 (1999)

    Google Scholar 

  2. Strogatz, S.H.: Exploring complex networks. Nature 410, 268–276 (2001)

    Article  Google Scholar 

  3. Wang, F., Sun, Y.: Self-organizing peer-to-peer social networks. Comput. Intell. 24, 213–233 (2008)

    Article  Google Scholar 

  4. 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 

  5. Wu, C.W.: Synchronization in Complex Networks of Nonlinear Dynamical Systems. World Scientific, Singapore (2007)

    Book  MATH  Google Scholar 

  6. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.: Phys. Rep. 424, 175–308 (2006)

    Article  MathSciNet  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yu, W., Cao, J., Lü, J.: Global synchronization of linearly hybrid coupled networks with time-varying delay. SIAM J. Appl. Dyn. Syst. 7, 108–133 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sorrentino, F., Ott, E.: Adaptive synchronization of dynamics on evolving complex networks. Phys. Rev. Lett. 100, 114101 (2008)

    Article  Google Scholar 

  10. Gorochowski, T., deBernardo, M., Grierson, C.: Evolving enhanced topologies for the synchronization of dynamical complex networks. Phys. Rev. E 81, 056212 (2010)

    Article  Google Scholar 

  11. Zhu, Q., Cao, J.: Adaptive synchronization under almost every initial data for stochastic neural networks with time-varying delays and distributed delays. Commun. Nonlinear Sci. Numer. Simul. 16, 2139–2159 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. DeLellis, P., diBernardo, M., Garofalo, F.: Synchronization of complex networks through local adaptive coupling. Chaos 18, 037110 (2008)

    Article  MathSciNet  Google Scholar 

  13. DeLellis, P., diBernardo, M., Garofalo, F.: Novel decentralized adaptive strategies for the synchronization of complex networks. Automatica 45, 1312–1318 (2009)

    Article  MathSciNet  Google Scholar 

  14. Lu, J., Cao, J.: Adaptive synchronization of uncertain dynamical networks with delayed coupling. Nonlinear Dyn. 53, 107–115 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhu, Q., Cao, J.: Adaptive synchronization of chaotic Cohen–Crossberg neural networks with mixed time delays. Nonlinear Dyn. 61, 517–534 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 16, 229–238 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhou, J., Xiang, L., Liu, Z.: Synchronization in complex delayed dynamical networks with impulsive effects. Physica A 384, 684–692 (2007)

    Article  MathSciNet  Google Scholar 

  18. Li, P., Cao, J., Wang, Z.: Robust impulsive synchronization of coupled delayed neural networks with uncertainties. Physica A 373, 261–272 (2007)

    Article  Google Scholar 

  19. Dai, Y., Cai, Y., Xu, X.: Synchronisation analysis and impulsive control of complex networks with coupling delays. IET Control Theory Appl. 3, 1167–1174 (2009)

    Article  MathSciNet  Google Scholar 

  20. Zhou, J., Wu, Q.J., Xiang, L., Cai, S.M., Liu, Z.R.: Impulsive synchronization seeking in general complex delayed dynamical networks. Nonlinear Anal. Hybrid Syst. 5, 513–524 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, X., Chen, G.: Pinning control of scale-free dynamical networks. Physica A 310, 521–531 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, X., Wang, X., Chen, G.: Pinning a complex dynamical network to its equilibrium. IEEE Trans. Circuits Syst. I 51, 2074–2087 (2004)

    Article  MathSciNet  Google Scholar 

  23. Chen, T., Liu, X., Lu, W.: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I 54, 1317–1326 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  25. Yu, W., Chen, G., Lü, J.: On pinning synchronization of complex dynamical networks. Automatica 45, 429–435 (2009)

    Article  MATH  Google Scholar 

  26. Xia, W., Cao, J.: Pinning synchronization of delayed dynamical networks via periodically intermittent control. Chaos 19, 013120 (2009)

    Article  MathSciNet  Google Scholar 

  27. Cai, S., Liu, Z., Xu, F., Shen, J.: Periodically intermittent controlling complex dynamical networks with time-varying delays to a desired orbit. Phys. Lett. A 373, 3846–3854 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  29. Hu, M., Xu, Z.: Adaptive feedback controller for projective synchronization. Nonlinear Anal., Real World Appl. 9, 1253–1260 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Vasegh, N., Khellat, F.: Projective synchronization of chaotic time-delayed systems via sliding mode controller. Chaos Solitons Fractals 42, 1054–1061 (2009)

    Article  MATH  Google Scholar 

  31. Wang, X., Wang, M.: Projective synchronization of nonlinear-coupled spatiotemporal chaotic systems. Nonlinear Dyn. 62, 567–571 (2010)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, Q., Zhao, J.: Projective and lag synchronization between general complex networks via impulsive control. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-0164-6

    Google Scholar 

  34. Hu, M., Yang, Y., Xu, Z., Zhang, R., Guo, L.: Projective synchronization in drive-response dynamical networks. Physica A 381, 457–466 (2007)

    Article  Google Scholar 

  35. Hu, M., Xu, Z., Yang, Y.: Projective cluster synchronization in drive-response dynamical networks. Physica A 387, 3759–3768 (2008)

    Article  Google Scholar 

  36. Zhao, Y., Yang, Y.: The impulsive control synchronization of the drive-response complex system. Phys. Lett. A 372, 7165–7171 (2008)

    Article  MATH  Google Scholar 

  37. 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)

    Article  MATH  Google Scholar 

  38. Chen, J., Jiao, L., Wu, J., Wang, X.: Projective synchronization with different scale factors in a driven-response complex network and its application in image encryption. Nonlinear Anal., Real World Appl. 11, 3045–3058 (2010)

    Article  MATH  Google Scholar 

  39. Xu, X., Gao, Y., Zhao, Y., Yang, Y.: The impulsive control of the projective synchronization in the drive-response dynamical networks with coupling delay. Adv. Neural Netw. 6063, 520–527 (2010)

    Google Scholar 

  40. Zheng, S.: Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-0175-3

    Google Scholar 

  41. 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)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang, Z., Xu, D.: Stability analysis and design of impulsive control systems with time delay. IEEE Trans. Autom. Control 52, 1448–1454 (2007)

    Article  Google Scholar 

Download references

Acknowledgements

This work was jointly supported by the Startup Fund for Ph.D. of Zhejiang University of Finance & Economics, the National Science Foundation of China (Grant 11102076) and the Society Science Foundation from Ministry of Education of China (Grant 2012JA790057).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhejiang University of Finance & Economics, Hangzhou, Zhejiang, 310018, P.R. China

    Song Zheng

  2. Institute of Applied Mathematics, Zhejiang University of Finance & Economics, Hangzhou, Zhejiang, 310018, P.R. China

    Song Zheng

Authors
  1. Song Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Song Zheng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zheng, S. Analyzing projective synchronization on different scaling factors in a drive-response coupling dynamical network with time-varying delays. Nonlinear Dyn 70, 709–719 (2012). https://doi.org/10.1007/s11071-012-0489-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0489-9

Keywords

Search

Navigation