Lag synchronization analysis of general complex networks with multiple time-varying delays via pinning control strategy

Original Article


This paper focuses on the lag synchronization issue for a kind of general complex networks with multiple time-varying delays via the pinning control strategy. By applying the Lyaponov functional theory and mathematical analysis techniques, sufficient verifiable criteria that depend on both intrinsic time-varying delay and coupled time-varying delay are obtained to achieve lag synchronization of the networks. Moreover, the coupling configuration matrices are not required to be symmetric or irreducible, and the minimum number of pinned nodes is determined by node dynamics, coupling matrices, and the designed parameter matrices. Finally, a numerical example is given to illustrate the feasibility of the theoretical results.


Adaptive lag synchronization Complex networks Pinning control Multiple time-varying delays 



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 No. 61533006) and China Scholarship Council.

Compliance with ethical standards

Competing interests

The authors declare that they have no competing interests


  1. 1.
    Steinmetz P, Roy A, Fitzgerald P (2000) Attention modulates synchronized neuronal firing in primate somatosensory cortex. Nature 404:457–490CrossRefGoogle Scholar
  2. 2.
    Vlirollo R, Strogatz S (1990) Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 50:1645–1662MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Park JH, Lee TH (2015) Synchronization of complex dynamical networks with discontinuous coupling signals. Nonlinear Dyn 79:1353–1362MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Sheikhan M, Shahnazi R, Garoucy S (2013) Synchronization of general chaotic systems using neural controllers with application to secure communication. Neural Comput Appl 22:361– 373CrossRefGoogle Scholar
  5. 5.
    Oliveira A, Jones A (1998) Synchronization of chaotic maps by feedback control and application to secure communications using chaotic neural networks. Int J Bifur Chaos 8:2225–2237CrossRefMATHGoogle Scholar
  6. 6.
    Kuhnert L, Agladze K, Krinsky V (1989) Image processing using light-sensitive chemical waves. Nature 337:244–247CrossRefGoogle Scholar
  7. 7.
    Cheng J, Zhu H, Zhong S, Zheng F, Zeng Y (2015) Finite-time filtering for switched linear systems with a mode-dependent average dwell time. Nonlinear Anal Hybrid Syst 15:145–156MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cheng J, Park JH, Liu Y, Liu T (2016) Finite-time \(H_{\infty }\) fuzzy control of nonlinear Markovian jump delayed systems with partly uncertain transition descriptions. Fuzzy Sets Syst. doi: 10.1016/j.fss.2016.06.007
  9. 9.
    Zhu Q, Cao J (2012) pth moment exponential synchronization for stochastic delayed Cohen-Grossberg neural networks with Markovian switching. Nonlinear Dyn 67:829–845CrossRefMATHGoogle Scholar
  10. 10.
    Sheikhan M, Shahnazi R, Garoucy S (2013) Hyperchaos synchronization using PSO-optimized RBF-based controllers to improve security of communication systems. Neural Comput Appl 22:835–846CrossRefGoogle Scholar
  11. 11.
    Jiang G, Tang W, Chen G (2006) A state-observer-based approach for synchronization in complex dynamical networks. IEEE Trans Circuits Syst I Reg Papers 53:2739–2745MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lu J, Ho D, Cao J, Kurths J (2013) Single impulsive controller for globally exponential synchronization of dynamical networks. Nonlinear Anal Real World Appl 14:581–593MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Jeong S, Ji D, Park JH, Won S (2013) Adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer. Appl Math Comput 219:5984–5995MathSciNetMATHGoogle Scholar
  14. 14.
    Tang Z, Park JH, Lee TH (2016) Topology and parameters recognition of uncertain complex networks via nonidentical adaptive synchronization. Nonlinear Dyn 85:2171–2181MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wang X, She K, Zhong S, Cheng J (2016) Synchronization of complex networks with non-delayed and delayed couplings via adaptive feedback and impulsive pinning control. Nonlinear Dyn 86:165–176MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Yu W, Chen G, Lu J (2009) On pinning synchronization of complex dynamical networks. Automatica 45:429–435MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Huang C, Ho D, Lu J, Kurths J (2015) Pinning synchronization in T-S fuzzy complex networks with partial and discrete-time couplings. IEEE Trans Fuzzy Syst 23:1274–1285CrossRefGoogle Scholar
  18. 18.
    Lee TH, Ma Q, Xu S, Park JH (2015) Pinning control for cluster synchronisation of complex dynamical networks with semi-Markovian jump topology. Int J Control 88:1223–1235MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Yu W, Chen G, Lu J, Kurths J (2013) Synchronization via pinning control on general complex networks. SIAM J Control Optim 51:1395–1416MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Jin X, Yang G (2013) Adaptive pinning synchronization of a class of nonlinear coupled complex networks. Commun Nonlin Sci Numer Simul 18:316–326CrossRefMATHGoogle Scholar
  21. 21.
    Wang X, She K, Zhong S, Yang H (2016) New result on synchronization of complex dynamical networks with time-varying coupling delay and sampled-data control. Neurocomputing 214:508–515CrossRefGoogle Scholar
  22. 22.
    Sun W, Chen Z, Lu J, Chen S (2012) Outer synchronization of complex networks with delay via impulsive. Nonlinear dyn 58:525–530Google Scholar
  23. 23.
    Li C, Sun W, Kurths J (2007) Synchronization between two coupled complex networks. Phys Rev E 76:046204CrossRefGoogle Scholar
  24. 24.
    Batista C, Batista A et al (2007) Chaotic phase synchronization on scale-free networks of bursting neurons. Phys Rev E 76:016218MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xiao Y, Xu W, Li X (2007) Adaptive complete synchronization of chaotic dynamical networks with unknown and mismatched parameters. Chaos 17:033118MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Mahmoud G, Mahmoud E (2012) Lag synchronization of hyperchaotic complex nonlinear systems. Nonlinear Dyn 67:1613–1622MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Jin Y, Zhong S (2015) Function projective synchronization in complex networks with switching topology and stochastic effects. Appl Math Comput 259:730–740MathSciNetGoogle Scholar
  28. 28.
    Rulkov N, Sushchik M, Tsimring L (1995) Generalized synchronization of chaos in directionally coupled chaotic systems. Phys Rev E 51:980–994CrossRefGoogle Scholar
  29. 29.
    Yang Y, Cao J (2007) Exponential lag synchronization of a class of chaotic delayed neural networks with impulsive effects. Phys A 386:492–502CrossRefGoogle Scholar
  30. 30.
    Li C, Liao X, Wong K (2004) Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication. Phys D 194:187–202MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Zhao M, Zhang H, Wang Z, Liang H (2014) Observer-based lag synchronization between two different complex networks. Commun Nonlin Sci Numer Simul 19:2048–2059MathSciNetCrossRefGoogle Scholar
  32. 32.
    Miao Q, Tang Y, Lu S, Fang J (2009) Lag synchronization of a class of chaotic systems with unknown parameters. Nonlinear Dyn 57:107–112MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Ji D, Jeong S, Park JH, Lee S, Wonb S (2012) Adaptive lag synchronization for uncertain complex dynamical network with delayed coupling. Appl Math Comput 218:4872–4880MathSciNetMATHGoogle Scholar
  34. 34.
    Wang J, Ma X, Wen X, Sun Q (2016) Pinning lag synchronization of drive-response complex networks via intermittent control with two different switched periods. Phys A 461:278– 287MathSciNetCrossRefGoogle Scholar
  35. 35.
    Guo W (2011) Lag synchronization of complex networks via pinning control. Nonlinear Anal Real World Appl 12:2579–2585MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Sun W, Wang S, Wang G, Wu Y (2015) Lag synchronization via pinning control between two coupled networks. Nonlinear Dyn 79:2659–2666MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Li H (2011) New criteria for synchronization stability of continuous complex dynamical networks with non-delayed and delayed coupling. Commun Nonlin Sci Numer Simul 16:1027– 1043MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Zhou J, Wu Q, Xiang L, Cai S, Liu Z (2011) Impulsive synchronization seeking in general complex delayed dynamical networks. Nonlinear Anal Hybrid Syst 5:513–524MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Gu K, Kharitonov V, Chen J (2003) Stability of time-delay systems. Birkhauser, BostonCrossRefMATHGoogle Scholar
  40. 40.
    Song Q, Cao J (2010) On pinning synchronization of directed and undirected complex dynamical networks. IEEE Trans Circuits Syst I Reg Papers 57:672–680MathSciNetCrossRefGoogle Scholar
  41. 41.
    Boyd S, Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  42. 42.
    Cai S, Cao J, He Q, Liu Z Exponential synchronization of complex delayed dynamical networks via pinning periodically intermittent control. Phys Lett A 375:1965–1971Google Scholar

Copyright information

© The Natural Computing Applications Forum 2017

Authors and Affiliations

  • Xin Wang
    • 1
  • Kun She
    • 1
  • Shouming Zhong
    • 2
  • Huilan Yang
    • 2
  1. 1.School of Information and Software EngineeringUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China

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