Nonlinear Dynamics

, Volume 88, Issue 4, pp 2723–2733 | Cite as

Finite-time synchronization analysis for general complex dynamical networks with hybrid couplings and time-varying delays

Original Paper


This paper focuses on the finite-time synchronization problem for a kind of general complex networks with intrinsic time-varying delays and hybrid couplings (i.e., containing current-state couplings and time-varying delay couplings). By designing a simple discontinuous state feedback controller and using strict analytical techniques, several synchronization criteria are proposed to guarantee that the complex dynamical networks can be synchronized onto an isolated chaotic system in finite time. Besides, the upper bound of the synchronization time could be estimated, which is dependent on the initial values of the system as well as the delays. Then, some finite-time synchronization criteria about special cases of the complex networks are also obtained. Here, the coupling configuration matrices are not required to be symmetric or irreducible in all cases. Finally, numerical examples are provided to demonstrate the correctness of our theoretical results.


Complex dynamical network Finite-time synchronization Hybrid coupling Time-varying delay 



The authors are grateful to the reviewers and editors for their valuable comments and suggestions to improve the presentation of this paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 61273220, 61373087, 61472257, 61603260).


  1. 1.
    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Pagani, G.A., Aiello, M.: The power grid as a complex network: a survey. Phys. A 392, 2688–2700 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Tang, J.J., Wang, Y.H., Liu, F.: Characterzing traffic time series based on complex network theory. Phys. A 392, 4192–4201 (2013)CrossRefGoogle Scholar
  5. 5.
    Rakkiyappan, R., Dharani, S., Zhu, Q.X.: Synchronization of reaction-diffusion neural networks with time-varying delays via stochastic sampled-data controller. Nonlinear Dyn. 79(1), 485–500 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Mahmoud, G.M., Mahmoud, E.E.: Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dyn. 62(4), 875–882 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Liang, J.L., Wang, Z.D., Liu, Y.R., Liu, X.H.: Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks. IEEE Trans. Neural Netw. 19(11), 1910–1921 (2008)CrossRefGoogle Scholar
  8. 8.
    Rakkiyappan, R., Sivaranjani, K.: Sampled-data synchronization and state estimation for nonlinear singularly perturbed complex networks with time-delays. Nonlinear Dyn. 84(3), 1623–1636 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Sheng, L., Yang, H.Z.: Exponential synchronization of a class of neural networks with mixed time-varying delays and impulsive effects. Neurocomputing 71(16–18), 3666–3674 (2008)CrossRefGoogle Scholar
  10. 10.
    Rakkiyappan, R., Sakthivel, N.: Cluster synchronization for TCS fuzzy complex networks using pinning control with probabilistic time-varying delays. Complexity 21(1), 59–77 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wang, J.Y., Feng, J.W., Xu, C., Zhao, Y.: Cluster synchronization of nonlinearly-coupled complex networks with nonidentical nodes and asymmetrical coupling matrix. Nonlinear Dyn. 67(2), 1635–1646 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Miao, Q.Y., Tang, Y., Lu, S.J., Fang, J.A.: Lag synchronization of a class of chaotic systems with unknown parameters. Nonlinear Dyn. 57(1), 107–112 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Shahverdiev, E.M., Sivaprakasam, S., Shore, K.A.: Lag synchronization in time-delayed systems. Phys. Lett. A 292(6), 320–324 (2002)CrossRefMATHGoogle Scholar
  14. 14.
    Koronovskii, A.A., Moskalenko, O.I., Hramov, A.E.: Generalized synchronization in complex networks. Tech. Phys. Lett. 38(10), 924–927 (2012)CrossRefGoogle Scholar
  15. 15.
    Wu, X.Q., Zheng, W.X., Zhou, J.: Generalized outer synchronization between complex dynamical networks. Chaos 19, 013109 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Du, H.B., Li, S.H., Lin, X.Z.: Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica 47(8), 1706–1712 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liu, X.Y., Su, H.S., Michael, Z.Q.C.: A switching approach to designing finite-time synchronization controllers of coupled neural networks. IEEE Trans. Neural Netw. Learn. Syst. 27(2), 471–482 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mei, J., Jiang, M.H., Xu, W.M., Wang, B.: Finite-time synchronization control of complex dynamical networks with time delay. Commun. Nonlinear Sci. Numer. Simul. 18, 2462–2478 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Vincent, U.E., Guo, R.K.: Finite-time synchronization for a class of chaotic and hyperchaotic systems via adaptive feedback controller. Phys. Lett. A 375(24), 2322–2326 (2011)CrossRefMATHGoogle Scholar
  21. 21.
    Aghababa, M.P., Khanmohammadi, S., Alizadeh, G.: Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. Appl. Math. Model. 35(6), 3080–3091 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lu, W.L., Chen, T.P., Chen, G.R.: Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay. Phys. D 221(2), 118–134 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Balasubramaniam, P., Chandran, R., Theesar, S.J.S.: Synchronization of chaotic nonlinear continuous neural networks with time-varying delay. Cogn. Neurodyn. 5(4), 361–371 (2011)CrossRefGoogle Scholar
  24. 24.
    Li, D., Cao, J.D.: Finite-time synchronization of coupled networks with one single time-varying delay coupling. Neurocomputing 166, 265–270 (2015)CrossRefGoogle Scholar
  25. 25.
    Yang, X.S.: Can neural networks with arbitrary delays be finite-timely synchronized? Neurocomputing 143, 275–281 (2014)CrossRefGoogle Scholar
  26. 26.
    Yang, X.S., Song, Q., Liang, J.L., He, B.: Finite-time synchronization of coupled discontinuous neural networks with mixed delays and nonidentical perturbations. J. Frank. Inst. 352(10), 4382–4406 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Abdurahman, A., Jiang, H.J., Teng, Z.D.: Finite-time synchronization for memristor-based neural networks with time-varying delays. Neural Netw. 69, 20–28 (2015)CrossRefGoogle Scholar
  28. 28.
    Abdurahman, A., Jiang, H.J., Hu, C., Teng, Z.D.: Parameter identification based on finite-time synchronization for Cohen-Grossberg neural networks with time-varying delays. Nonlinear Anal. Model. Control 20(3), 348–366 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Velmurugana, G., Rakkiyappan, R., Cao, J.D.: Finite-time synchronization of fractional-order memristor-based neural networks with time delays. Neural Netw. 73, 36–46 (2016)CrossRefGoogle Scholar
  30. 30.
    Ma, Q., Wang, Z., Lu, J.W.: Finite-time synchronization for complex dynamical networks with time-varying delays. Nonlinear Dyn. 70(1), 841–848 (2012)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Li, B.C.: Finite-time synchronization for complex dynamical networks with hybrid coupling and time-varying delay. Nonlinear Dyn. 76, 1603–1610 (2014)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Gilli, M.: Strange attractors in delayed cellular neural networks. IEEE Trans. Circ. Syst. I 40(11), 849–853 (1993)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Song, Q., Liu, F., Cao, J.D., Yu, W.W.: Pinning-controllability analysis of complex networks: an M-matrix approach. IEEE Trans. Circ. Syst. I 59(11), 2692–2701 (2012)MathSciNetGoogle Scholar
  34. 34.
    Li, N., Feng, J.W., Zhao, Y.: Finite-time synchronization for nonlinearly coupled networks with time-varying delay. 28th Chinese Control and Decision Conference, Yinchuan, pp. 95–100 (2016)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Jianwen Feng
    • 1
  • Na Li
    • 1
  • Yi Zhao
    • 1
  • Chen Xu
    • 1
  • Jingyi Wang
    • 1
  1. 1.College of Mathematics and StatisticsShenzhen UniversityShenzhenPeople’s Republic of China

Personalised recommendations