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Projective synchronization of neural networks with mixed time-varying delays and parameter mismatch

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Abstract

In this paper, the projective synchronization of neural networks with mixed time-varying delays and parameter mismatch is discussed. Due to parameter mismatch and projective factor, complete projective synchronization cannot be achieved. Therefore, a new weak projective synchronization scheme is proposed to ensure that coupled neural networks are in a state of synchronization with an error level. Several criteria are derived and the error level is estimated by applying a generalized Halanay inequality and matrix measure. Finally, a numerical example is given to verify the efficiencies of theoretical results.

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References

  1. Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: a universal concept in nonlinear sciences. In: Academic, Physics, Cambridge Nonlinear Science Series. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  2. Pecora, L.M., Carroll, T.L.: Synchronizing chaotic circuits. IEEE Trans. Circuits Syst. 38(4), 453–456 (1991)

    Article  Google Scholar 

  3. Lu, J., Cao, J., Ho, D.W.C.: Adaptive stabilization and synchronization for chaotic Lur’e systems with time-varying delay. IEEE Trans. Circuits Syst. 55(5), 1347–1356 (2008)

    Article  MathSciNet  Google Scholar 

  4. He, W., Cao, J.: Exponential synchronization of chaotic neural networks: a matrix measure approach. Nonlinear Dyn. 55(1), 55–65 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. He, W., Cao, J.: Adaptive synchronization of a class of chaotic neural networks with known or unknown parameters. Phys. Lett. A 372(4), 408–416 (2008)

    Article  MATH  Google Scholar 

  6. Kocarev, L., Parlitz, U.: Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett. 76(11), 1816–1819 (1996)

    Article  Google Scholar 

  7. He, W., Cao, J.: Generalized synchronization of chaotic systems: an auxiliary system approach via matrix measure. Chaos 19, 013118 (2009)

    Article  MathSciNet  Google Scholar 

  8. Li, C., Liao, X., Wong, K.: Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication. Physica D 194(3–4), 187–202 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Sun, Y., Cao, J.: Adaptive lag synchronization of unknown chaotic delayed neural networks with noise perturbation. Phys. Lett. A 364(3–4), 277–285 (2007)

    Article  MATH  Google Scholar 

  10. Corron, N.J., Blakely, J.N., Pethel, S.D.: Lag and anticipating synchronization without time-delay coupling. Chaos 15, 023110 (2005)

    Article  Google Scholar 

  11. Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78(22), 4193–4196 (1997)

    Article  Google Scholar 

  12. Mainieri, R., Rehacek, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82(15), 3042–3045 (1999)

    Article  Google Scholar 

  13. Ghosh, D.: Generalized projective synchronization in time-delayed systems: nonlinear observer approach. Chaos 19, 013102 (2009)

    Article  MathSciNet  Google Scholar 

  14. Yan, J., Li, C.: Generalized projective synchronization of a unified chaotic system. Chaos Solitons Fractals 26(4), 1119–1124 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Li, G.: Generalized projective synchronization of two chaotic systems by using active control. Chaos Solitons Fractals 30(1), 77–82 (2006)

    Article  MATH  Google Scholar 

  16. Du, H., Zeng, Q., Wang, C., Ling, M.: Function projective synchronization in coupled chaotic systems. Nonlinear Anal., Real World Appl. 11(2), 705–712 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Liu, Y., Wang, Z., Liu, X.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Netw. 19(5), 667–675 (2006)

    Article  MATH  Google Scholar 

  18. Song, Q., Cao, J., Zhao, Z.: Periodic solutions and its exponential stability of reaction-diffusion recurrent neural networks with continuously distributed delays. Nonlinear Anal., Real World Appl. 7(1), 65–80 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Liang, J., Cao, J.: Global asymptotic stability of bidirectional associative memory networks with distributed delays. Appl. Math. Comput. 152(2), 415–424 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhao, H.: Global asymptotic stability of Hopfield neural network involving distributed delays. Neural Netw. 17(1), 47–53 (2004)

    Article  MATH  Google Scholar 

  21. Zhao, H.: Existence and global attractivity of almost periodic solution for cellular neural network with distributed delays. Appl. Math. Comput. 154(3), 683–695 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ruan, S., Filfil, R.S.: Dynamics of a two-neuron system with discrete and distributed delays. Physica D 191(3–4), 323–342 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wang, Z., Liu, Y., Liu, X.: On global asymptotic stability of neural networks with discrete and distributed delays. Phys. Lett. A 345(4–6), 299–308 (2005)

    Article  MATH  Google Scholar 

  24. Wang, Z., Liu, Y., Fraser, K., Liu, X.: Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays. Phys. Lett. A 354(4), 288–297 (2006)

    Article  MATH  Google Scholar 

  25. Johnson, G.A., Mar, D.J., Carroll, T.L., Pecora, L.M.: Synchronization and imposed bifurcations in the presence of large parameter mismatch. Phys. Rev. Lett. 80(18), 3956–3959 (1998)

    Article  Google Scholar 

  26. Li, X., Pan, W., Luo, B., Ma, D.: Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization. IEEE J. Quantum Electron. 42(9), 953–960 (2006)

    Article  Google Scholar 

  27. Chen, Y., Cao, L., Sun, M.: Robust modified function projective synchronization in network with unknown parameters and mismatch parameters. Int. J. Nonlinear Sci. 10(1), 17–23 (2010)

    MATH  MathSciNet  Google Scholar 

  28. Shen, L., Liu, W., Ma, J.: Robust function projective synchronization of a class of uncertain chaotic systems. Chaos Solitons Fractals 42(2), 1292–1296 (2009)

    Article  MATH  Google Scholar 

  29. Vidyasagar, M.: Nonlinear Systems Analysis, 2nd edn. Prentice Hall, Upper Saddle River (1992)

    MATH  Google Scholar 

  30. Hom, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge UP, New York (1991)

    Google Scholar 

  31. Wen, L., Yu, Y., Wang, W.: Generalized Halanay inequalities for dissipativity of Volterra functional differential equations. J. Math. Anal. Appl. 347(1), 169–178 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zhang, W., Huang, J., Wei, P.: Weak synchronization of chaotic neural networks with parameter mismatch via periodically intermittent control. Appl. Math. Model. 35(2), 612–620 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. Wu, X., Lu, H.: Generalized projective synchronization between two different general complex dynamical networks with delayed coupling. Phys. Lett. A 374(38), 3932–3941 (2010)

    Article  Google Scholar 

  34. Wang, K., Teng, Z., Jiang, H.: Adaptive synchronization of neural networks with time-varying delay and distributed delay. Physica A 387(2–3), 631–642 (2008)

    Article  Google Scholar 

  35. Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, vol. 3, pp. 2805–2810 (2000)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Shun Chen & Jinde Cao

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  1. Shun Chen
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  2. Jinde Cao
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Correspondence to Jinde Cao.

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Chen, S., Cao, J. Projective synchronization of neural networks with mixed time-varying delays and parameter mismatch. Nonlinear Dyn 67, 1397–1406 (2012). https://doi.org/10.1007/s11071-011-0076-5

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  • DOI: https://doi.org/10.1007/s11071-011-0076-5

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