Nonlinear Dynamics

, Volume 88, Issue 4, pp 2671–2691 | Cite as

Delay-range-dependent local adaptive and robust adaptive synchronization approaches for time-delay chaotic systems

  • Muhammad Siddique
  • Muhammad Rehan
  • M. K. L. Bhatti
  • Shakeel Ahmed
Original Paper


This paper studies the local adaptive and robust adaptive control methodologies for the synchronization of the chaotic drive and the response systems with finite time lags, bounded delay rates, unknown parameters, and perturbations. A local adaptive coherence control condition under bounded initial condition is developed to synthesize a feedback controller capable of executing successful synchronization of the drive and the response systems with uncertain parameters and delay belonging to a range (with either zero or nonzero lower bound). A sufficiency condition for the adaptive control approach is inculcated through an amendment in the conventional Lyapunov–Krasovskii functional, which enables to develop the adaptation laws for the unknown parameters associated with the nonlinearity in the dynamical behavior of the drive and the response systems. Further, a robust adaptive synchronization control condition for asymptotic convergent behavior of the synchronization error to a bounded region under perturbations as well as disturbances is provided. Indifferent to the existing works, delay-range-dependent adaptive synchronization methodologies are derived in this paper, and the rendered controller design conditions are valid for both the slow and the fast variations in time delays. Numerical simulations for the chaotic delayed Hopfield neural networks are presented to elaborate the efficacy of the proposed delay-range-dependent adaptive and robust adaptive synchronization control methodologies.


Adaptive synchronization Robust adaptive synchronization Time-varying delays Lyapunov–Krasovskii functional Local control approach 



This work was supported by Higher Education Commission (HEC) of Pakistan by supporting PhD studies of the first author through indigenous PhD scholarship program (phase II, batch II, 2013).


  1. 1.
    Yang, J., Chen, Y., Zhu, F.: Associated observer-based synchronization for uncertain chaotic systems subject to channel noise and chaos-based secure communication. Neurocomputing 167, 587–595 (2015)CrossRefGoogle Scholar
  2. 2.
    Yu, W.T., Tang, J., Ma, J., Luo, J.M., Yang, X.Q.: Damped oscillations in a multiple delayed feedback NF-B signaling module. Eur. Biophys. J. 44(8), 677–684 (2015)CrossRefGoogle Scholar
  3. 3.
    Siddique, M., Rehan, M.: A concept of coupled chaotic synchronous observers for nonlinear and adaptive observers-based chaos synchronization. Nonlinear Dyn. 84, 2251–2272 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Shahverdiev, E.M., Shore, K.A.: Synchronization of chaos in unidirectionally and bidirectionally coupled multiple time delay laser diodes with electro-optical feedback. Opt. Commun. 282(2), 310–316 (2009)CrossRefGoogle Scholar
  5. 5.
    Wickramasinghe, M., Kiss, I.Z.: Spatially organized dynamical states in chemical oscillator networks: synchronization dynamical differentiation and chimera patterns. PloS ONE 8(11), e80586 (2013)CrossRefGoogle Scholar
  6. 6.
    Yang, X., Cao, J., Yang, Z.: Synchronization of coupled reaction–diffusion neural networks with time-varying delays via pinning-impulsive controller. SIAM J. Control Optim. 51(5), 3486–3510 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Rau, A.W., Nill, S., Eidens, R.S., Oelfke, U.: Synchronized tumor tracking with electromagnetic transponders and kV X-ray imaging: evaluation based on a thorax phantom. Phys. Med. Biol. 53(14), 3789–3805 (2008)CrossRefGoogle Scholar
  8. 8.
    Wenxiu, Z., Jun, L., Lili, P., Jian, C.: Application of RS485 for communication and synchronization in distributed electromagnetic exploration system. In: Proceedings of Electric Information and Control Engineering (ICEICE), Wuhan, pp. 4815–4818 (2011)Google Scholar
  9. 9.
    Wei, G.W., Jia, Y.Q.: Synchronization-based image edge detection. Europhys. Lett. 59(6), 814–819 (2002)CrossRefGoogle Scholar
  10. 10.
    Elfring, G.J., Pak, O.S., Lauga, E.: Two-dimensional flagellar synchronization in viscoelastic fluids. J. Fluid Mech. 646, 505–515 (2010)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Sato, D., Xie, L.H., Sovari, A.A., Tran, D.X., Morita, N., Xie, F., Karagueuzian, H., Garfinkel, A., Weiss, J.N., Qu, Z.: Synchronization of chaotic early after depolarizations in the genesis of cardiac arrhythmias. Proc. Natl. Acad. Sci. USA 106(9), 2983–2988 (2009)CrossRefGoogle Scholar
  12. 12.
    Yao, C.G., Ma, J., Li, C., He, Z.W.: The effect of process delay on dynamical behaviors in a self-feedback nonlinear oscillator. Commun. Nonlinear Sci. Numer. Simul. 39, 99–107 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Tian, J.K., Liu, Y.M.: Improved delay-dependent stability analysis for neural networks with interval time-varying delays. Math. Probl. Eng. Article ID 705367 (2015)Google Scholar
  14. 14.
    Rong, R.Z., Tian, J.K.: Improved stability analysis for neural networks with interval time-varying delays. Appl. Mech. Mater. 687, 2078–2082 (2014)Google Scholar
  15. 15.
    Guan, Z.H., Liu, Z.W., Feng, G., Wang, Y.W.: Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control. IEEE Trans. Circuits Syst. I Regul. Pap. 57(8), 2182–2195 (2010)Google Scholar
  16. 16.
    Zhu, Q., Cao, J.: Adaptive synchronization of chaotic Cohen–Crossberg neural networks with mixed time delays. Nonlinear Dyn. 61(3), 517–534 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lin, T.C., Lee, T.Y.: Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control. IEEE Trans. Fuzzy Syst. 19(4), 623–635 (2011)CrossRefGoogle Scholar
  18. 18.
    Hu, J., Wang, Z., Gao, H., Stergioulas, L.K.: Robust sliding mode control for discrete stochastic systems with mixed time delays, randomly occurring uncertainties, and randomly occurring nonlinearities. IEEE Trans. Ind. Electron. 59(7), 3008–3015 (2012)CrossRefGoogle Scholar
  19. 19.
    Li, Z., Cao, X., Ding, N.: Adaptive fuzzy control for synchronization of nonlinear teleoperators with stochastic time-varying communication delays. IEEE Trans. Fuzzy Syst. 19(4), 745–757 (2011)CrossRefGoogle Scholar
  20. 20.
    Rafique, M.A., Rehan, M., Siddique, M.: Adaptive mechanism for synchronization of chaotic oscillators with interval time-delays. Nonlinear Dyn. 81(1), 495–509 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    He, P., Jing, C.G., Fan, T., Chen, C.Z.: Robust decentralized adaptive synchronization of general complex networks with coupling delayed and uncertainties. Complexity 19(3), 10–26 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wang, T., Zhou, W., Zhao, S., Yu, W.Q.: Robust master–slave synchronization for general uncertain delayed dynamical model based on adaptive control scheme. ISA Trans. 53(2), 335–340 (2014)CrossRefGoogle Scholar
  23. 23.
    Yang, X.S., Cao, J., Long, Y., Rui, W.: Adaptive lag synchronization for competitive neural networks with mixed delays and uncertain hybrid perturbations. IEEE Trans. Neural Netw. Learn. Syst. 21(10), 1656–1667 (2010)CrossRefGoogle Scholar
  24. 24.
    Lu, J., Cao, J., Ho, D.W.: Adaptive stabilization and synchronization for chaotic Lur’e systems with time-varying delay. IEEE Trans. Circuits Syst. I Regul. Pap. 55(5), 1347–1356 (2008)Google Scholar
  25. 25.
    Farid, Y., Bigdeli, N.: Robust adaptive intelligent sliding model control for a class of uncertain chaotic systems with unknown time-delay. Nonlinear Dyn. 67(3), 2225–2240 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Yue, D., Li, H.: Synchronization stability of continuous/discrete complex dynamical networks with interval time-varying delays. Neurocomputing 73(4–6), 809–819 (2010)CrossRefGoogle Scholar
  27. 27.
    Zhang, H., Gong, D., Chen, B., Liu, Z.: Synchronization for coupled neural networks with interval delay: a novel augmented Lyapunov–Krasovskii functional method. IEEE Trans. Neural Netw. Learn. Syst. 24(1), 58–70 (2013)CrossRefGoogle Scholar
  28. 28.
    Karimi, H.R., Maass, P.: Delay-range-dependent exponential \(H_\infty \) synchronization of a class of delayed neural networks. Chaos Solitons Fractals 41(3), 1125–1135 (2009)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Shao, H.: Improved delay-dependent stability criteria for systems with a delay varying in a range. Automatica 44(12), 3215–3218 (2008)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ahmad, S., Majeed, R., Hong, K.S., Rehan. M.: Observer design for one-sided Lipschitz nonlinear systems subject to measurement delays. Math. Probl. Eng. Article ID 879492 (2015)Google Scholar
  31. 31.
    Majeed, R., Ahmad, S., Rehan, M.: Delay-range-dependent observer-based control of nonlinear systems under input and output time-delays. Appl. Math. Comput. 262, 145–159 (2015)MathSciNetGoogle Scholar
  32. 32.
    Lee, S.H., Park, M.J., Kwon, O.M., Sakthivel, R.: Master–slave synchronization for nonlinear systems via reliable control with gaussian stochastic process. Appl. Math. Comput. 290, 439–459 (2016)MathSciNetGoogle Scholar
  33. 33.
    Kaviarasan, B., Sakthivel, R., Lim, Y.: Synchronization of complex dynamical networks with uncertain inner coupling and successive delays based on passivity theory. Neurocomputing 186, 127–138 (2016)CrossRefGoogle Scholar
  34. 34.
    Mathiyalagan, K., Anbuvithya, R., Sakthivel, R., Park, J.H., Prakash, P.: Non-fragile \(H_{\infty }\) synchronization of memristor-based neural networks using passivity theory. Neural Netw. 74, 85–100 (2016)CrossRefGoogle Scholar
  35. 35.
    Ma, J., Qin, H.X., Song, X.L., Chu, R.T.: Pattern selection in neuronal network driven by electric autapses with diversity in time delays. Int. J. Mod. Phys. B. 29(01). Article ID 1450239 (2015)Google Scholar
  36. 36.
    Ahn, C.K.: Adaptive \(H_\infty \) anti-synchronization for time-delayed chaotic neural networks. Prog. Theor. Phys. 122(6), 1391–1403 (2009)CrossRefMATHGoogle Scholar
  37. 37.
    Hussain, M., Rehan, M.: Nonlinear time-delay anti-windup compensator synthesis for nonlinear time-delay systems: a delay-range-dependent approach. Neurocomputing 186, 54–65 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • Muhammad Siddique
    • 1
  • Muhammad Rehan
    • 1
  • M. K. L. Bhatti
    • 2
  • Shakeel Ahmed
    • 1
  1. 1.Department of Electrical EngineeringPakistan Institute of Engineering and Applied Sciences (PIEAS)IslamabadPakistan
  2. 2.Department of Electrical EngineeringNFC Institute of Engineering and TechnologyMultanPakistan

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