Skip to main content
SpringerLink
Log in

Adaptive synchronization of uncertain dynamical networks with delayed coupling

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

Abstract

We propose a simple scheme for the synchronization of an uncertain complex dynamical network with delayed coupling. Based on the Lyapunov stability theory of functional differential equations, certain controllers can be designed for ensuring the states of uncertain dynamical network with coupling delays to globally asymptotically synchronize by combining the adaptive method and linear feedback with the updated feedback strength. Different update gains η i will lead to different rates toward synchrony, the choice of which depends on the concrete systems and network models. This strategy can be applied to any complex dynamical network (regular, small-world, scale-free or random). Numerical examples with respectively nearest-neighbor coupling and scale-free structure are given to demonstrate the effectiveness of our presented scheme.

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

Similar content being viewed by others

References

  1. Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)

    Article  Google Scholar 

  2. Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  4. Lu, J.Q., He, J., Cao, J.D., Gao, Z.Q.: Topology influences performance in the associative memory neural networks. Phys. Lett. A 354(5–6), 335–343 (2006)

    Article  Google Scholar 

  5. Liu, Y., Takiguchi, Y., Davis, P., Aida, T., Saito, S., Liu, J.M.: Injection locking and synchronization of chaos in semiconductor lasers. Appl. Phys. Lett. 80, 4306–4308 (2002)

    Article  Google Scholar 

  6. Kim, K.T., Kim, M.S., Chong, Y., Niemeyer, J.: Simulations of collective synchronization in Josephson junction arrays. Appl. Phys. Lett. 88, 062501 (2006)

    Article  Google Scholar 

  7. Lu, J.Q., Cao, J.D.: Adaptive synchronization in tree-like dynamical networks. Nonlinear Anal. Real World Appl. 8(4), 1252–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lu, J.Q., Ho, D.W.C.: Local and global synchronization in general complex dynamical networks with delay coupling. Chaos Solitons Fractals (2006). doi:10.1016/j.chaos.2006.10.030

    Google Scholar 

  9. Baek, S.J., Ott, E.: Onset of synchronization in systems of globally coupled chaotic maps. Phys. Rev. E 69(6), 66210 (2004)

    Article  MathSciNet  Google Scholar 

  10. Donetti, L., Hurtado, P.I., Munoz, M.A.: Entangled networks, synchronization, and optimal network topology. Phys. Rev. Lett. 95, 188701 (2005)

    Article  Google Scholar 

  11. Belykh, I., de Lange, E., Hasler, M.: Synchronization of bursting neurons: what matters in the network topology. Phys. Rev. Lett. 94(18), 188101 (2005)

    Article  Google Scholar 

  12. Zhou, J., Lu, J., Lü, J.: Adaptive synchronization of an uncertain complex dynamical network. IEEE Trans. Autom. Control 51(4), 652–656 (2006)

    Article  Google Scholar 

  13. Wang, W., Cao, J.: Synchronization in an array of linearly coupled networks with time-varying delay. Physica A: Stat. Mech. Appl. 366, 197–211 (2006)

    Article  Google Scholar 

  14. Wang, X.F.: Complex networks: topology, dynamics and synchronization. Int. J. Bifurc. Chaos 12(5), 885–916 (2002)

    Article  MATH  Google Scholar 

  15. Lu, J.Q., Ho, D.W.C., Liu, M.: Globally exponential synchronization in an array of asymmetric coupled neural networks. Phys. Lett. A 369, 444–451 (2007)

    Article  MathSciNet  Google Scholar 

  16. Pikovsky, A., Rosenblum, M., Kurths, J., Hilborn, R.C.: Synchronization: a universal concept in nonlinear science. Am. J. Phys. 70, 655 (2002)

    Article  Google Scholar 

  17. Li, C., Chen, L., Aihara, K.: Synchronization of coupled nonidentical genetic oscillators. Phys. Biol. 3, 37–44 (2006)

    Article  Google Scholar 

  18. Li, C., Chen, L., Aihara, K.: Stochastic synchronization of genetic oscillator networks. BMC Syst. Biol. 1, 6 (2007)

    Article  Google Scholar 

  19. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)

    Article  MathSciNet  Google Scholar 

  20. Li, C., Chen, G.: Synchronization in general complex dynamical networks with coupling delays. Physica A: Stat. Mech. Appl. 343, 263–278 (2004)

    Article  Google Scholar 

  21. Cao, J.D., Lu, J.Q.: Adaptive synchronization of neural networks with or without time-varying delay. Chaos 16(1), 013133 (2006)

    Article  MathSciNet  Google Scholar 

  22. Jiang, Y.: Globally coupled maps with time delay interactions. Phys. Lett. A 267(5–6), 342–349 (2000)

    Article  Google Scholar 

  23. Masoller, C., Martı, A.C., Zanette, D.H.: Synchronization in an array of globally coupled maps with delayed interactions. Physica A: Stat. Mech. Appl. 325(1–2), 186–191 (2003)

    Article  MATH  Google Scholar 

  24. Choi, M.Y., Kim, H.J., Kim, D., Hong, H.: Synchronization in a system of globally coupled oscillators with time delay. Phys. Rev. E 61(1), 371–381 (2000)

    Article  Google Scholar 

  25. Heil, T., Fischer, I., Elsässer, W., Mulet, J., Mirasso, C.R.: Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers. Phys. Rev. Lett. 86(5), 795–798 (2001)

    Article  Google Scholar 

  26. Earl, M.G., Strogatz, S.H.: Synchronization in oscillator networks with delayed coupling: a stability criterion. Phys. Rev. E 67(3), 36204 (2003)

    Article  Google Scholar 

  27. Atay, F.M., Jost, J., Wende, A.: Delays, connection topology, and synchronization of coupled chaotic maps. Phys. Rev. Lett. 92(14), 144101 (2004)

    Article  Google Scholar 

  28. Wünsche, H.J., Bauer, S., Kreissl, J., Ushakov, O., Korneyev, N., Henneberger, F., Wille, E., Erzgräber, H., Peil, M., Elsäßer, W., et al.: Synchronization of delay-coupled oscillators: a study of semiconductor lasers. Phys. Rev. Lett. 94(16), 163901 (2005)

    Article  Google Scholar 

  29. Cao, J., Li, P., Wang, W.: Global synchronization in arrays of delayed neural networks with constant and delayed coupling. Phys. Lett. A 353(4), 318–325 (2006)

    Article  Google Scholar 

  30. Chen, M., Zhou, D.: Synchronization in uncertain complex networks. Chaos 16(1), 013101 (2006)

    Article  MathSciNet  Google Scholar 

  31. Li, Z., Chen, G.: Robust adaptive synchronization of uncertain dynamical networks. Phys. Lett. A 324(2-3), 166–178 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hale, J.K.: Diffusive coupling, dissipation, and synchronization. J. Dyn. Differ. Equ. 9(1), 1–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wu, C.W.: Synchronization in Coupled Chaotic Circuits and Systems. World Scientific, Singapore (2002)

    MATH  Google Scholar 

  34. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  35. Leonov, G., Bunin, A., Koksch, N.: Attractor localization of the Lorenz system. Z. Angew. Math. Mech. 67(2), 649–656 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Jianquan Lu & Jinde Cao

  2. Department of Mathematics, City University of Hong Kong, Hong Kong, China

    Jianquan Lu

Authors
  1. Jianquan Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinde Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinde Cao.

Additional information

This work was jointly supported by the National Natural Science Foundation of China under Grant 60574043, the 973 Program of China under Grant 2003CB317004 and the Natural Science Foundation of Jiangsu Province of China under Grant BK2006093.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lu, J., Cao, J. Adaptive synchronization of uncertain dynamical networks with delayed coupling. Nonlinear Dyn 53, 107–115 (2008). https://doi.org/10.1007/s11071-007-9299-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-007-9299-x

Keywords

Search

Navigation