Synchronization for a Class of Fractional-order Linear Complex Networks via Impulsive Control

  • Na Liu
  • Jie Fang
  • Wei Deng
  • Zhen-Jun Wu
  • Guo-Qiang Ding
Regular Papers Control Theory and Applications


Up to now, the research topic about fractional-order complex networks is mainly focused on the synchronization. In this paper, synchronization for a class of fractional-order linear complex networks is realized via impulsive control. The general expression of solution for a fractional-order impulsive error system is deduced by utilizing iteration algorithm. Some inequality conditions are established to guarantee that the largest Lyapunov exponents of each node are negative, which means that the corresponding error system is asymptotic stable and synchronization is realized. It is the first time to achieve the synchronization of fractional-order systems based on the largest Lyapunov exponent. Finally, examples are present to illustrate the validity and effectiveness of proposed conclusions. Numerical simulations also indicate that the fractional-order parameter has a great influence on the largest Lyapunov exponent, although it is not reflected in the theoretical analysis.


Complex networks fractional-order impulsive control synchronization 


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  1. [1]
    K. B. Shi, X. Z. Liu, H. Zhu, S. M. Zhong, Y. Zeng, and C. Yin, “Novel delay-dependent master-slave synchronization criteria of chaotic Lur’e systems with time-varying-delay feedback control,” Applied Mathematics and Computation, vol. 282, pp. 137–154, 2016.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Q. Wang and D.-L. Qi, “Synchronization for fractional order chaotic systems with uncertain parameters,” Control Theory and Applications, vol. 14, no. 1, pp. 211–216, February 2016.Google Scholar
  3. [3]
    K. B. Shi, Y. Y. Tang, X. Z. Liu, and S. M. Zhong, “Nonfragile sampled-data robust synchronization of uncertain delayed chaotic Lurie systems with randomly occurring controller gain fluctuation,” ISA Transactions, vol. 66, pp. 185–199, January 2017.CrossRefGoogle Scholar
  4. [4]
    S. Song, X.-N. Song, N. Pathak, and I. T. Balsera, “Multiswitching adaptive synchronization of two fractional-order chaotic systems with different structure and different order,” Control Theory and Applications, vol. 15, no. 4, pp. 1524–1535, August 2017.Google Scholar
  5. [5]
    K. Z. Guan, “Global power-rate synchronization of chaotic neural networks with proportional delay via impulsive control,” Neurocomputing, vol. 283, pp. 256–265, March 2018.Google Scholar
  6. [6]
    Y. Tang, F. Qian, H. J. Gao, and J. Kurths, “Synchronization in complex networks and its application-A survey of recent advances and challenges,” Annual Reviews in Control, vol. 38, no. 2, pp. 184–198, September 2014.CrossRefGoogle Scholar
  7. [7]
    X.Wu, D. Lai, and H. Lu, “Generalized synchronization of the fractional-order Chaos in weighted complex dynamical networks with nonidentical nodes,” Nonlinear Dynamics, vol. 69, no. 1–2, pp. 667–683, December 2012.MathSciNetzbMATHGoogle Scholar
  8. [8]
    J. Yu, C. Hu, H. J. Jiang, and X. L. Fan, “Projective synchronization for fractional neural networks,” Neural Network, vol. 49, pp. 87–95, January 2014.CrossRefzbMATHGoogle Scholar
  9. [9]
    J. Wang and C. Zeng, “Synchronization of fractional-order linear complex networks,” ISA Transactions, vol. 55, no. 1, pp. 129–134, March 2015.CrossRefGoogle Scholar
  10. [10]
    S. Liang, R. Wu, and L. Chen, “Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay,” Physica A: Statistical Mechanics and its Applications, vol. 44, pp. 49–62, February 2016.MathSciNetzbMATHGoogle Scholar
  11. [11]
    J. W. Wang and Y. B. Zhang, “Network synchronization in a population of star-coupled fractional nonlinear oscillators,” Physics Letters A, vol. 374, no. 13–14, pp. 1464–1468, March 2010.CrossRefzbMATHGoogle Scholar
  12. [12]
    Y. Tang and J. A. Fang, “Reply to the comment on “Synchronization of N-coupled fractional-order chaotic systems with ring connection”,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 4244–4245, December 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    S. S. Delshad, M. M. Asheghan, and M. H. Behesh, “Synchronization of N-coupled incommensurate fractionalorder chaotic systems with ring connection,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3815–3824, September 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    T. D. Ma, J. Zhang, Y. C. Zhou, and H. Y. Wang, “Adaptive hybrid projective synchronization of two coupled fractional-order complex networks with different sizes,” Neurocomputing, vol. 164, no. C, pp. 182–189, September 2015.Google Scholar
  15. [15]
    R. Rakkiyappan, J. G. Cao, and G. Velmurugan, “Existence and uniform stability analysis of fractional-order complexvalued neural networks with time delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 1, pp. 84–97, March 2014.CrossRefGoogle Scholar
  16. [16]
    T. D. Ma and J. Zhang, “Hybrid synchronization of coupled fractional-order complex networks,” Neurocomputing, vol. 157, pp. 166–172, June 2015.CrossRefGoogle Scholar
  17. [17]
    Q. X. Fang and J. G. Peng, “Synchronization of fractionalorder linear complex networks with directed coupling topology,” Physica A: Statistical Mechanics and its Applications, vol. 490, pp. 542–553, September 2018.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Y. Yang, Y. He, Y. Wang, and M. Wu, “Stability analysis of fractional-order neural networks: An LMI approach,” Neurocomputing, vol. 285, pp. 582–93, April 2018.Google Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Na Liu
    • 1
  • Jie Fang
    • 1
  • Wei Deng
    • 1
  • Zhen-Jun Wu
    • 1
  • Guo-Qiang Ding
    • 1
  1. 1.School of Electric and Information EngineeringZhengzhou University of Light IndustryZhengzhou, Henan ProvinceChina

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