Kybernetika 55 no. 1, 203-215, 2019

Synchronization of fractional chaotic complex networks with delays

Jian-Bing Hu, Hua Wei, Ye-Feng Feng and Xiao-Bo YangDOI: 10.14736/kyb-2019-1-0203

Abstract:

The synchronization of fractional-order complex networks with delay is investigated in this paper. By constructing a novel Lyapunov-Krasovskii function $V$ and taking integer derivative instead of fractional derivative of the function, a sufficient criterion is obtained in the form of linear matrix inequalities to realize synchronizing complex dynamical networks. Finally, a numerical example is shown to illustrate the feasibility and effectiveness of the proposed method.

Keywords:

synchronization, delays, fractional complex networks, Lyapunov-Krasovskii theorem

Classification:

34D06, 93D05

References:

  1. B. Ahmad, S. K. Ntouyas, J. Tariboon, A. Alsaedi and H. H. Alsulami: Impulsive fractional q-integro-difference equations with separated boundary conditions. Appl. Math. Comput. 281 (2016), 199-213.   DOI:10.1016/j.amc.2016.01.051
  2. F. An, X. Y. Gao, J. H. Guan, H. J. Li and Q. Liu: An evolution analysis of executive-based listed company relationships using complex networks. Physica A: Statist. Mechanics and Its Appl. 447 (2016), 276-285.   DOI:10.1016/j.physa.2015.12.050
  3. N. Aguila-Camacho, M. A. Duarte-Mermoud and J. A. Gallegos: Lyapunov functions for fractional order systems. Commu. Nonlinear Science Numer. Simul. 19 (2014), 2951-2957.   DOI:10.1016/j.cnsns.2014.01.022
  4. D. Baleanu, A. Ranjbar, S. J. Sadati, R. H. Delavari, T. Abdeljawad and V. Gejji: Lyapunov-Krasovskii stability theorem for fractional systems with delay. Romanian J. Phys. 56 (2011), 636-643.   CrossRef
  5. Y. Chen and J. Lü: Delay-induced discrete-time consensus. Automatica 85 (2017), 356-361.   DOI:10.1016/j.automatica.2017.07.059
  6. L. P. Chen, W. Pan, R. C. Wu, J. A. T. Machado and A. M. Lopes: Design and implementation of grid multi-scroll fractional-order chaotic attractors. Chaos 26 (2016), 8, 084303.   DOI:10.1063/1.4958717
  7. H. Dai, G. Q. Si, L. X. Jia and Y. B. Zhang: Adaptive generalized function matrix projective lag synchronization between fractional-order and integer-order complex networks with delayed coupling and different dimensions. Physica Scripta 88 (2013), 5, 055006.   DOI:10.1088/0031-8949/88/05/055006
  8. S. A. David, J. A. T. Machado, D. D. Quintino and J. M. Balthazar: Partial chaos suppression in a fractional order macroeconomic model. Math. Computers Simul. 122 (2016), 55-68.   DOI:10.1016/j.matcom.2015.11.004
  9. J. B. Hu, G. P. Lu and L. D. Zhao: Synchronization of fractional chaotic complex networks with distributed delays. Nonlinear dynamics 83 (2016), 1101-1108.   DOI:10.1007/s11071-015-2390-9
  10. J. B. Hu, H. Wei and L. D. Zhao: Synchronization of fractional-order chaotic systems with multiple delays by a new approach. Kybernetika 51 (2015), 1068-1083.   DOI:10.14736/kyb-2015-6-1068
  11. B. C. Li: Pinning adaptive hybrid synchronization of two general complex dynamical networks with mixed coupling. Appl. Math. Modell. 40 (2016), 2983-2998.   DOI:10.1016/j.apm.2015.09.092
  12. Y. Li, X. Wu, J. A. Lu and J. Lü: Synchronizability of duplex networks. IEEE Trans. Circuits Systems II Express Briefs 63 (2016), 206-210.   DOI:10.1109/tcsii.2015.2468924
  13. S. Liang, R. C. Wu and L. P. Chen: Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay. Physica a-Statistical Mechanics and Its Applications 444(2016), 49-62.   DOI:10.1016/j.physa.2015.10.011
  14. K. Liu, L. Wu, J. Lü and H. Zhu: Finite-time adaptive consensus of a class of multi-agent systems. Science China Technol. Sci. 59 (2016), 22-32.   DOI:10.1007/s11431-015-5989-7
  15. K. Liu, H. Zhu and J. Lü: Cooperative Stabilization of a class of LTI plants with distributed observers. IEEE Trans. Circuits Systems I Regular Papers 64(2017), 1891-1902.   DOI:10.1109/tcsi.2017.2675922
  16. M. Rivero, S. V. Rogosin, J. A. T. Machado and J. J. Trujillo: Stability of fractional order systems. Math. Problems Engrg. 2013 (2013), 1-14.   CrossRef
  17. D. T. Spasic, N. I. Kovincic and D. V. Dankuc: A new material identification pattern for the fractional kelvin-zener model describing biomaterials and human tissues. Comm. Nonlinear Sci. Numer. Simul. 37 (2016), 193-199.   DOI:10.1016/j.cnsns.2016.01.004
  18. H. W. Tang, L. Chen, J. A. Lu and C. K. Tse: Adaptive synchronization between two complex networks with nonidentical topological structures. Physica a-Statistical Mechanics and Its Applications 387 (2008) 5623-5630.   DOI:10.1016/j.physa.2008.05.047
  19. Y. Tang, H. J. Gao and J. Kurths: Distributed robust synchronization of dynamical networks with stochastic coupling. IEEE Trans. Circuits Systems I-Regular Papers, 61 (2014), 1508-1519.   DOI:10.1109/tcsi.2013.2285699
  20. A. Uncini and F. Piazza: Blind signal processing by complex domain adaptive spline neural networks. IEEE Trans. Neural Networks 14 (2003), 399-412.   DOI:10.1109/tnn.2003.809411
  21. Y. Wang and T. Z. Li: Synchronization of fractional order complex dynamical networks. Physica a-Statistical Mechanics and Its Applications 428 (2015), 1-12.   CrossRef
  22. J. W. Wang, Q. H. Ma, A. M. Chen and Z. P. Liang: Pinning synchronization of fractional-order complex networks with lipschitz-type nonlinear dynamics. ISA Trans. 57 (2015), 111-116.   DOI:10.1016/j.isatra.2015.02.002
  23. F. Wang, Y. Q. Yang, A. H. Hu and X. Y. Xu: Exponential synchronization of fractional-order complex networks via pinning impulsive control. Nonlinear Dynamics 82 (2015), 1979-1987.   DOI:10.1007/s11071-015-2292-x
  24. Z. Wang, X. Huang, Y. X. Li and X. N. Song: A new image encryption algorithm based on the fractional-order hyperchaotic lorenz system. Chinese Physics B 22 (2013), 1, 010504.   CrossRef
  25. F. Wang, Y. Q. Yang, M. F. Hu and X. Y. Xu: Projective cluster synchronization of fractional-order coupled-delay complex network via adaptive pinning control. Physica a-Statistical Mechanics and Its Applications 434 (2015), 134-143.   DOI:10.1016/j.physa.2015.03.089
  26. G. C. Wu and D. Baleanu: Discrete chaos in fractional delayed logistic maps. Nonlinear Dynamics 80 (2015), 1697-1703.   DOI:10.1007/s11071-014-1250-3
  27. G. C. Wu, D. Baleanu, Z. G. Deng and S. D. Zeng: Lattice fractional diffusion equation in terms of a riesz-caputo difference. Physica a-Statistical Mechanics and Its Applications 438 (2015), 335-339.   DOI:10.1016/j.physa.2015.06.024
  28. J. W. Yi, Y. W. Wang, J. W. Xiao and Y. H. Huang: Synchronisation of complex dynamical networks with additive stochastic time-varying delays. Int. J. Systems Sci. 47 (2016), 1221-1229.   DOI:10.1080/00207721.2014.919426
  29. W. B. Zhang, Y. Tang, Q. Y. Miao and J. A. Fang: Synchronization of stochastic dynamical networks under impulsive control with time delays. IEEE Trans. Neural Networks Learning Systems 25 (2014), 1758-1768.   DOI:10.1109/tnnls.2013.2294727
  30. L. D. Zhao, J. B. Hu, J. A. Fang, W. X. Cui, Y. L. Xu and X. Wang: Adaptive synchronization and parameter identification of chaotic system with unknown parameters and mixed delays based on a special matrix structure. ISA Trans. 52 (2013),738-743.   DOI:10.1016/j.isatra.2013.07.001
  31. W. N. Zhou, A. D. Dai, J. Yang, H. S. Liu and X. L. Liu: Exponential synchronization of markovian jumping complex dynamical networks with randomly occurring parameter uncertainties. Nonlinear Dynamics 78(2014), 15-27.   DOI:10.1007/s11071-014-1418-x
  32. Y. Zhou, C. Ionescu and J. A. T. Machado: Fractional dynamics and its applications. Nonlinear Dynamics 80 (2015), 1661-1664.   DOI:10.1007/s11071-015-2069-2
  33. J. Zhou, J. Chen, J. A. Lu and J. Lü: On applicability of auxiliary system approach to detect generalized synchronization in complex networks. IEEE Trans. Automat. Control 62 (2017), 3468-3473.   DOI:10.1109/tac.2016.2615679