Nonlinear Dynamics

, Volume 84, Issue 2, pp 661–667 | Cite as

Bounded synchronization of the general complex dynamical network with delay feedback controller

  • Yuhua Xu
  • Yajuan Lu
  • Weixiang Yan
  • Wuneng Zhou
  • Jian’an Fang
Original Paper


This paper is concerned with bounded synchronization for a class of the complex dynamical network with delay feedback controller. In order to realize bounded synchronization of complex networks, several new inequality theorems are proposed, and bounded synchronization criteria of complex dynamical networks are also derived based on inequality theorems. Finally, two numerical examples are provided to verify the theoretical results established in this paper.


Complex network Bounded synchronization Delay feedback 



The authors would like to thank the referees and the editor for their valuable comments and suggestions. This research is supported by the Youth Fund Project of the Humanities and Social Science Research for the Ministry of Education of China (14YJCZH173), the Science and Technology Research Key Program for the Education Department of Hubei Province of China (D20156001).


  1. 1.
    Lu, J., Ho, D.W.C.: Globally exponential synchronization and synchronizability for general dynamical networks. IEEE Trans. Syst. Man Cybern. B 40, 350–361 (2010)CrossRefGoogle Scholar
  2. 2.
    Zhou, L., Wang, Z., Hu, X., Chu, B., Zhou, W.: Adaptive almost sure asymptotically stability for neutral-type neural networks with stochastic perturbation and Markovian switching. Neurocomputing 156, 151–156 (2015)CrossRefGoogle Scholar
  3. 3.
    Zhang, Z., Park, J., Shao, H.: Adaptive synchronization of uncertain unified chaotic systems via novel feedback controls. Nonlinear Dyn. 81, 695–706 (2015)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bian, Q., Yao, H.: Adaptive synchronization of bipartite dynamical networks with distributed delays and nonlinear derivative coupling. Commun. Nonlinear Sci. Numer. Simul. 16, 4089–4098 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Shi, T.: Finite-time control of linear systems under time-varying sampling. Neurocomputing 151, 1327–1331 (2015)CrossRefGoogle Scholar
  6. 6.
    Wu, Z., Fu, X.: Complex projective synchronization in coupled chaotic complex dynamical systems. Nonlinear Dyn. 69, 771–779 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Xu, Y., Zhou, W., Fang, J., Su, W., Pan, L.: Adaptive synchronization of stochastic time-varying delay dynamical networks with complex-variable systems. Nonlinear Dyn. 81, 1717–1726 (2015)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Ma, J., Qin, H., Song, X., Chu, R.: Pattern selection in neuronal network driven by electric autapses with diversity in time delays. Int. J. Mod. Phys. B 29(1), 1450239 (2015)CrossRefGoogle Scholar
  9. 9.
    Ma, J., Hu, B., Wang, C.: Simulating the formation of spiral wave in the neuronal system. Nonlinear Dyn. 73, 73–83 (2013)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Ma, J., Liu, Q., Ying, H., Wu, Y.: Emergence of spiral wave induced by defects block. Commun. Nonlinear Sci. Numer. Simul. 18(7), 1665–1675 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Ma, J., Wu, X., Chu, R.: Selection of multi-scroll attractors in Jerk circuits and their verification using pspice. Nonlinear Dyn. 76, 1951–1962 (2014)CrossRefGoogle Scholar
  12. 12.
    Tang, Y., Gao, H., Zou, W., Kurths, J.: Distributed synchronization in networks of agent systems with nonlinearities and random switchings. IEEE Trans. Cybern. 43, 358–370 (2013)CrossRefGoogle Scholar
  13. 13.
    Shen, H., Park, J., Wu, Z.: Finite-time synchronization control for uncertain Markov jump neural networks with input constraints. Nonlinear Dyn. 77(4), 1709–1720 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Lu, J., Kurths, J., Cao, J., Mahdavi, N., Huang, C.: Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy. IEEE Trans. Neural Netw. Learn. Syst. 23(2), 285–292 (2012)CrossRefGoogle Scholar
  15. 15.
    José, V., Emerson, G., Elder, M.: Robust adaptive synchronization of a hyperchaotic finance system. Nonlinear Dyn. 80, 239–248 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Tang, Y., Wong, W.: Distributed synchronization of coupled neural networks via randomly occurring control. IEEE Trans. Neural Netw. Learn. Syst. 24, 435–447 (2013)CrossRefGoogle Scholar
  17. 17.
    Tang, Y., Wang, Z., Gao, H., Swift, S., Kurths, J.: A constrained evolutionary computation method for detecting controlling regions of cortical networks. IEEE/ACMTrans. Comput. Biol. Bioinform. 9, 1569–1581 (2012)CrossRefGoogle Scholar
  18. 18.
    Wang, X., Fang, J., Mao, H., Dai, A.: Finite-time global synchronization for a class of Markovian jump complex networks with partially unknown transition rates under feedback control. Nonlinear Dyn. 79, 47–61 (2015)CrossRefzbMATHGoogle Scholar
  19. 19.
    Liu, X., Yu, X., Xi, H.: Finite-time synchronization of neutral complex networks with Markovian switching based on pinning controller. Neurocomputing 153, 148–158 (2015)CrossRefGoogle Scholar
  20. 20.
    Soldatos, A.G., Corless, M.: Stabilizing uncertain systems with bounded control. Dyn. Control 1, 227–238 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    El-Farra, N.H., Mhaskar, P.D.: Uniting bounded control and MPC for stabilization of constrained linear systems. Automatica 40, 101–110 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Bounit, H., Hammouri, H.: Bounded feedback stabilization and global separation principle of distributed parameter systems. IEEE Trans. Automat. Control 42, 414–419 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Florian, D., Francesco, B.: Synchronization and transient stability in power networks and non-uniform kuramoto oscillators. In Proceedings of the 2010 American Control Conference (2010)Google Scholar
  24. 24.
    Zhai, S., Yang, X.: Bounded synchronisation of singularly perturbed complex network with an application to power systems. IET Control Theory Appl. 8, 61–66 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Wang, L., Qian, W., Wang, Q.: Bounded synchronisation of a time-varying dynamical network with nonidentical nodes. Int. J. Syst. Sci. 46, 1234–1245 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Yu, W., Lü, J., Wang, Z., Cao, J., Zhou, Q.: Robust \(H_\infty \) control and uniformly bounded control for genetic regulatory network with stochastic disturbance. IET Control Theory Appl. 4, 1687–1706 (2010)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Shen, B., Wang, Z., Liu, X.: Bounded \(H_\infty \) Synchronization and state estimation for discrete time-varying stochastic complex networks over a finite horizon. IEEE Trans. Neural Netw. 22, 145–157 (2011)CrossRefGoogle Scholar
  28. 28.
    Zhao, J., David, J., Liu, T.: Global bounded synchronization of general dynamical networks with nonidentical nodes. IEEE Trans. Autom. Control 57, 2656–2662 (2012)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Wang, L., Chen, S., Wang, Q.: Eigenvalue based approach to bounded synchronization of asymmetrically coupled networks. Commun. Nonlinear Sci. Numer. Simulat. 22, 769–779 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Holyst, J., Urbanowicz, K.: Chaos control in economical model by time-delayed feedback method. Phys. A 287, 587–598 (2000)CrossRefGoogle Scholar
  31. 31.
    Pyragas, K.: Control of chaos via an unstable delayed feedback controller. Phys. Rev. Lett. 86, 2265–2268 (2001)CrossRefGoogle Scholar
  32. 32.
    Chen, W.: Dynamics and control of a financial system with time-delayed feedbacks. Chaos Solitons Fractals 37, 1198–1207 (2008)CrossRefzbMATHGoogle Scholar
  33. 33.
    Huang, C., Cao, J.: On pth moment exponential stability of stochastic Cohen–Grossberg neural networks with time varying delays. Neurocomputing 73, 986–990 (2010)CrossRefGoogle Scholar
  34. 34.
    Leonov, G.A., Kuznetsov, N.V.: On differences and similarities in the analysis of Lorenz, Chen and Lu systems. Appl. Math. Comput. 256, 334–343 (2015)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Hidden attractor in smooth Chua systems. Phys. D 241, 1482–1486 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–148 (1963)CrossRefGoogle Scholar
  37. 37.
    Chen, Y., Chang, C.: Impulsive synchronization of Lipschitz chaotic systems. Chaos Solitons Fractals 40, 1221–1228 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Zhou, J., Lu, J.: Topology identification of weighted complex dynamical network. Phys. A 386, 481–491 (2007)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Yuhua Xu
    • 1
  • Yajuan Lu
    • 1
  • Weixiang Yan
    • 2
  • Wuneng Zhou
    • 3
  • Jian’an Fang
    • 3
  1. 1.School of FinanceNanjing Audit UniversityNanjingChina
  2. 2.School of Economics and TradeNanjing Audit UniversityNanjingChina
  3. 3.College of Information Science and TechnologyDonghua UniversityShanghaiChina

Personalised recommendations