Skip to main content
SpringerLink
Log in

Synchronization for non-dissipatively coupled time-varying complex dynamical networks with delayed coupling nodes

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

Abstract

This paper investigates the synchronization problem for the non-dissipatively coupled time-varying complex dynamical networks via decentralized state feedback controllers. In our network model, the outer coupling configuration matrices are not restricted by dissipatively coupled condition, time invariance, symmetry, irreducibility or certainty. Besides, different time-varying coupling delays are put into consideration. Moreover, the similarities possessed by the nodes in our network model are revealed based on the nodes’ dynamics and are applied to synthesize the controllers. Furthermore, it is the common bound, not the exact information, of the outer coupling coefficients that is used to design the synchronization controllers. It is worth pointing out that the uncertain outer coupling coefficients’ common bound is admissible in our synchronization schemes, and for this case, adaptive control mechanism is introduced to design the synchronization controllers. Several proper simulation examples are given to verify the effectiveness and feasibility of our theoretical results.

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

Similar content being viewed by others

References

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

    Article  Google Scholar 

  2. DeLellis, P., Bernardo, M., Russo, G.: On QUAD, Lipschitz, and contracting vector fields for consensus and synchronization of networks. IEEE Tran. Circuits Syst. I Regul. Pap. 58, 576–583 (2011)

    Article  Google Scholar 

  3. Guan, Z.H., Liu, Z.W., Feng, G., Wang, Y.W.: Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control. IEEE Trans. Circuits Syst. I. Regul. Pap. 57, 2182–2194 (2010)

    Article  MathSciNet  Google Scholar 

  4. DeLellis, P., Bernardo, M., Gorochowski, T.E., Russo, G.: Synchronization and control of complex networks via contraction, adaptation and evolution. IEEE Circuits Syst. Mag. Complex Netw. Appl. Circuits Syst. 3, 64–82 (2010)

    Article  Google Scholar 

  5. Li, Z., Duan, Z., Chen, G.: Cost and effects of pinning control for network synchronization. Chin. Phys. 18, 106–118 (2009)

    Article  Google Scholar 

  6. Wang, Q.Y., Duan, Z.S., Chen, G.R., Feng, Z.S.: Synchronization in a class of weighted complex networks with coupling delays. Phys. A 387, 5616–5622 (2008)

    Article  Google Scholar 

  7. Yang, S., Cao, J.: Hybrid adaptive and impulsive synchronization of uncertain complex networks with delays and general uncertain perturbations. Appl. Math. Comput. 227, 480–493 (2014)

    Article  MathSciNet  Google Scholar 

  8. Wong, W.K., Li, H., Leung, S.Y.S.: Robust synchronization of fractional-order complex dynamical networks with parametric uncertainties. Commun. Nonlinear Sci. Numer. Simul. 17, 4877–4890 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mei, J., Jiang, M., Xu, W., Wang, B.: Finite-time synchronization control of complex dynamical networks with time delay. Commun. Nonlinear Sci. Numer. Simul. 18, 2462–2478 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, H.: \({\rm H}_{\infty }\) cluster synchronization and state estimation for complex dynamical networks with mixed time delays. Appl. Math. Model. 37, 7223–7244 (2013)

    Article  MathSciNet  Google Scholar 

  11. Wang, J., Zhang, H., Wang, Z., Wang, B.: Local exponential synchronization in complex dynamical networks with time-varying delay and hybrid coupling. Appl. Math. Comput. 225, 16–32 (2013)

    Article  MathSciNet  Google Scholar 

  12. Lee, D.W., Yoo, W.J., Ji, D.H., Park, J.H.: Integral control for synchronization of complex dynamical networks with unknown non-identical nodes. Appl. Math. Comput. 224, 140–149 (2013)

    Article  MathSciNet  Google Scholar 

  13. Lee, T.H., Park, J.H., Ji, D.H., Kmon, O.M., Lee, S.M.: Guaranteed cost synchronization of a complex dynamical network via dynamic feedback control. Appl. Math. Comput. 218, 6469–6481 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Solís-Perales, G., Ruiz-Velázquez, E., Valle-Rodríguez, D.: Synchronization in complex networks with distinct chaotic nodes. Commun. Nonlinear Sci. Numer. Simul. 14, 2528–2535 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Park, M.J., Kwon, O.M., Park, J.H., Lee, S.M., Cha, E.J.: On synchronization criterion of coupled discrete-time neural networks with interval time-varying delays. Neurocomputing 99, 188–196 (2013)

    Article  Google Scholar 

  16. Li, H., Ning, Z., Yin, Y., Tang, Y.: Synchronization and state estimation for singular complex dynamical networks with time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 18, 194–208 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lee, T.H., Wu, Z.G., Park, J.H.: Synchronization of a complex dynamical network with coupling time-varying delays via sampled-data control. Appl. Math. Comput. 219, 1354–1366 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Park, M.J., Kwon, O.M., Park, J.H., Lee, S.M., Cha, E.J.: Synchronization criteria of fuzzy complex dynamical networks with interval time-varying delays. Appl. Math. Comput. 218, 11634–11647 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, Y., Fan, Y., Wang, Q., Zhang, Y.: Stabilization and synchronization of complex dynamical networks with different dynamics of nodes via decentralized controllers. IEEE Trans. Circuits Syst. I Regul. Pap. 59, 1786–1794 (2012)

    Article  MathSciNet  Google Scholar 

  20. Fan, Y.Q., Wang, Y.H., Zhang, Y., Wang, Q.R.: The synchronization of complex dynamical networks with similar nodes and coupling time-delay. Appl. Math. Comput. 219, 6719–6728 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, L.L., Wang, Y.H., Wang, Q.Y., Wang, Q.R., Zhang, Y.: Synchronisation of complex dynamical networks with different dynamics of nodes via decentralised dynamical compensation controllers. Int. J. Control 86, 1766–1776 (2013)

    Article  Google Scholar 

  22. Li, Z., Jiao, L., Lee, J.J.: Robust adaptive global synchronization of complex dynamical networks by adjusting time-varying coupling strength. Phas. A Stat. Mech. Appl. 387, 1369–1380 (2008)

    Article  Google Scholar 

  23. Wang, S., Yao, H., Zheng, S., Xie, Y.: A novel criterion for cluster synchronization of complex dynamical networks with coupling time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 17, 2997–3004 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, K., Zhou, J., Yu, W., Small, M., Fu, X.: Adaptive cluster synchronization in networks with time-varying and distributed coupling delays. Appl. Math. Model. 38, 1300–1314 (2014)

    Article  MathSciNet  Google Scholar 

  25. Anzo, A., Barafas-Ramírez, J.G.: Synchronization in complex networks under structural evolution. J. Franklin Inst. 351, 358–372 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhao, J., Hill, D.J., Liu, T.: Global bounded synchronization of general dynamical networks with nonidentical nodes. IEEE Trans. Autom. Control 57, 2656–2662 (2012)

    Article  MathSciNet  Google Scholar 

  27. Lü, J., Chen, G.: A time-varying dynamical network model and its controlled synchronization criteria. IEEE Trans. Autom. Control 50, 841–846 (2005)

    Article  Google Scholar 

  28. Li, P., Yi, Z.: Synchronization analysis of delayed complex networks with time-varying couplings. Phys. A 387, 3729–3737 (2008)

    Article  Google Scholar 

  29. Zheng, S.: Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling. Nonlinear Dyn. 67, 2621–2630 (2012)

  30. Wu, X., Lu, H.: Outer synchronization of uncertain general complex delayed networks with adaptive coupling. Neurocomputing 82, 157–166 (2012)

    Article  Google Scholar 

  31. Araujo, C.S., Castro, J.C.: Application of power system stabilizers in a plant with identical units. IEEE Proc. C 138, 11–18 (1991)

    Google Scholar 

  32. Bakule, L., Lunze, J.: Decentralized design of feedback control for large-scale systems. Kybernetika 24, 3–96 (1988)

    MathSciNet  Google Scholar 

  33. Bakule, L.: Decentralized control: an overview. Ann. Rev. Control 32, 87–98 (2008)

    Article  Google Scholar 

  34. Wang, Y.H., Zhang, S.Y.: Robust control for nonlinear similar composite systems with uncertain parameters. IEEE Proc. Control Theory Appl. 147, 80–86 (2000)

    Article  Google Scholar 

  35. Zhang, S.Y.: The structures of symmetry and similarity in complex control systems. Control Theory Appl. 11, 231–237 (1994)

    Google Scholar 

  36. Wang, Y., Cao, J.: Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems. Nonlinear Anal. Real World Appl. 14, 842–851 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Khalil, H.K.: Nonlinear Systems. Prentice-Hall, Englewood Cliffs (2002)

    MATH  Google Scholar 

  38. Ozkaynak, F., Yavuz, S.: Security problems for a pseudorandom sequence generator based on the Chen chaotic system. Comput. Phys. Commun. 184, 2178–2181 (2013)

    Article  Google Scholar 

  39. Sudheer, K.S., Sabir, M.: Adaptive modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system with uncertain parameters. Phys. Lett. A 373, 3743–3748 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Li, D., Lu, J., Wu, X., Chen, G.: Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system. J. Math. Anal. Appl. 323, 844–853 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Science Foundation of China (61273219, 61305098), the National Science Foundation of Guangdong Province of China (S2013010015768, S2012040007700), Project Program of KLGHEI of China (2013CXZDA015), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20134420110003) and the Important Program of the Youth Foundation of Guangdong University of Technology (15ZK0036).

Author information

Authors and Affiliations

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510006, People’s Republic of China

    Lili Zhang

  2. School of Automation, Guangdong University of Technology, Guangzhou, 510006, People’s Republic of China

    Lili Zhang, Yinhe Wang & Yuanyuan Huang

Authors
  1. Lili Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yinhe Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuanyuan Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lili Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, L., Wang, Y. & Huang, Y. Synchronization for non-dissipatively coupled time-varying complex dynamical networks with delayed coupling nodes. Nonlinear Dyn 82, 1581–1593 (2015). https://doi.org/10.1007/s11071-015-2262-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2262-3

Keywords

Search

Navigation