Skip to main content
SpringerLink
Log in

Weak, modified and function projective synchronization of Cohen–Grossberg neural networks with mixed time-varying delays and parameter mismatch via matrix measure approach

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

This paper is concerned with the modified function projective synchronization of Cohen–Grossberg neural networks systems with parameter mismatch and mixed time-varying delays. Due to the existence of parameter mismatch between the drive and slave systems, complete modified function projective synchronization is not possible to achieve. So a new concept, viz., weak modified function projective synchronization, is discussed up to a small error bound. Several generic criteria are derived to show weak modified function projective synchronization between the systems. The estimation of error bound is done using matrix measure and Halanay inequality. Simulation results are proposed graphically for different particular cases to show the synchronization between parameter-mismatched systems, which validate the effectiveness of our proposed theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. An H, Chen Y (2008) The function cascade synchronization method and applications. Commun Nonlinear Sci Numer Simul 13(10):2246–2255. https://doi.org/10.1016/j.cnsns.2007.05.029

    MathSciNet  MATH  Google Scholar 

  2. Ashwin P, Buescu J, Stewart I (1994) Bubbling of attractors and synchronisation of chaotic oscillators. Phys Lett A 193(2):126–139. https://doi.org/10.1016/0375-9601(94)90947-4

    MathSciNet  MATH  Google Scholar 

  3. Bao H, Cao J (2016) Finite-time generalized synchronization of nonidentical delayed chaotic systems. Nonlinear Anal Model Control 21(3):306–324

    MathSciNet  MATH  Google Scholar 

  4. Cao J, Tao Q (2001) Estimation on domain of attraction and convergence rate of hopfield continuous feedback neural networks. J Comput Syst Sci 62(3):528–534. https://doi.org/10.1006/jcss.2000.1722

    MathSciNet  MATH  Google Scholar 

  5. Chee CY, Xu D (2005) Secure digital communication using controlled projective synchronisation of chaos. Chaos Solitons Fractals 23(3):1063–1070

    MATH  Google Scholar 

  6. Chen S, Cao J (2012) Projective synchronization of neural networks with mixed time-varying delays and parameter mismatch. Nonlinear Dyn 67(2):1397–1406

    MathSciNet  MATH  Google Scholar 

  7. Chen Y, An H, Li Z (2008) The function cascade synchronization approach with uncertain parameters or not for hyperchaotic systems. Appl Math Comput 197(1):96–110. https://doi.org/10.1016/j.amc.2007.07.036

    MathSciNet  MATH  Google Scholar 

  8. Chen L, Chai Y, Wu R (2011) Modified function projective synchronization of chaotic neural networks with delays based on observer. IntSoliton J Mod Phys C 22(02):169–180

    MATH  Google Scholar 

  9. Cheng CJ, Liao TL, Yan JJ, Hwang CC (2005) Synchronization of neural networks by decentralized feedback control. Phys Lett A 338(1):28–35

    MATH  Google Scholar 

  10. Cheng CJ, Liao TL, Hwang CC (2005) Exponential synchronization of a class of chaotic neural networks. Chaos Solitons Fractals 24(1):197–206

    MathSciNet  MATH  Google Scholar 

  11. Cohen MA, Grossberg S (1983) Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans Syst Man Cybern SMC–13(5):815–826

    MathSciNet  MATH  Google Scholar 

  12. Du H, Zeng Q, Wang C (2009) Modified function projective synchronization of chaotic system. Chaos Solitons Fractals 42(4):2399–2404. https://doi.org/10.1016/j.chaos.2009.03.120

    MATH  Google Scholar 

  13. Du H, Zeng Q, Lü N (2010) A general method for modified function projective lag synchronization in chaotic systems. Phys Lett A 374(13):1493–1496. https://doi.org/10.1016/j.physleta.2010.01.058

    MATH  Google Scholar 

  14. Feng Yl, Wang Jc (2010) Modified projective synchronization of the unified chaotic system. J Zhangzhou Norm Univ (Nat Sci) 3:005

    Google Scholar 

  15. Gan Q (2012) Adaptive synchronization of Cohen–Grossberg neural networks with unknown parameters and mixed time-varying delays. Commun Nonlinear Sci Numer Simul 17(7):3040–3049

    MathSciNet  MATH  Google Scholar 

  16. Gauthier DJ, Bienfang JC (1996) Intermittent loss of synchronization in coupled chaotic oscillators: toward a new criterion for high-quality synchronization. Phys Rev Lett 77:1751–1754. https://doi.org/10.1103/PhysRevLett.77.1751

    Google Scholar 

  17. González-Miranda JM (1996) Synchronization of symmetric chaotic systems. Phys Rev E 53:5656–5669. https://doi.org/10.1103/PhysRevE.53.5656

    Google Scholar 

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

    MATH  Google Scholar 

  19. He W, Cao J (2009) Exponential synchronization of chaotic neural networks: a matrix measure approach. Nonlinear Dyn 55(1–2):55–65

    MathSciNet  MATH  Google Scholar 

  20. Hongyue D, Qingshuang Z, Mingxiang L (2008) Adaptive modified function projective synchronization with known or unknown parameters. In: IEEE international symposium on knowledge acquisition and modeling workshop, 2008. KAM Workshop 2008

  21. Huang J, Li C, Huang T, Han Q (2013) Lag quasisynchronization of coupled delayed systems with parameter mismatch by periodically intermittent control. Nonlinear Dyn 71(3):469–478

    MathSciNet  MATH  Google Scholar 

  22. Kouomou YC, Colet P, Gastaud N, Larger L (2004) Effect of parameter mismatch on the synchronization of chaotic semiconductor lasers with electro-optical feedback. Phys Rev E 69:056226. https://doi.org/10.1103/PhysRevE.69.056226

    Google Scholar 

  23. Li GH (2007) Generalized projective synchronization between lorenz system and chen’s system. Chaos Solitons Fractals 32(4):1454–1458. https://doi.org/10.1016/j.chaos.2005.11.073

    MATH  Google Scholar 

  24. Li GH (2007) Modified projective synchronization of chaotic system. Chaos Solitons Fractals 32(5):1786–1790. https://doi.org/10.1016/j.chaos.2005.12.009

    MathSciNet  MATH  Google Scholar 

  25. Li K, Song Q (2008) Exponential stability of impulsive Cohen–Grossberg neural networks with time-varying delays and reaction–diffusion terms. Neurocomputing 72(1–3):231–240

    Google Scholar 

  26. Li T, Song AG, Fei SM (2009) Robust stability of stochastic Cohen–Grossberg neural networks with mixed time-varying delays. Neurocomputing 73(1–3):542–551

    Google Scholar 

  27. Liang J, Cao J (2007) Global output convergence of recurrent neural networks with distributed delays. Nonlinear Anal Real World Appl 8(1):187–197

    MathSciNet  MATH  Google Scholar 

  28. Liao X, Li C, Wong KW (2004) Criteria for exponential stability of Cohen–Grossberg neural networks. Neural Netw 17(10):1401–1414

    MATH  Google Scholar 

  29. Liu C, Li C, Li C (2011) Quasi-synchronization of delayed chaotic systems with parameters mismatch and stochastic perturbation. Commun Nonlinear Sci Numer Simul 16(10):4108–4119. https://doi.org/10.1016/j.cnsns.2011.02.033

    MathSciNet  MATH  Google Scholar 

  30. Mainieri R, Rehacek J (1999) Projective synchronization in three-dimensional chaotic systems. Phys Rev Lett 82:3042–3045. https://doi.org/10.1103/PhysRevLett.82.3042

    Google Scholar 

  31. Milanović V, Zaghloul ME (1996) Synchronization of chaotic neural networks and applications to communications. Int J Bifurc Chaos 6(12b):2571–2585

    MATH  Google Scholar 

  32. Pan J, Zhong S (2010) Dynamical behaviors of impulsive reaction–diffusion Cohen–Grossberg neural network with delays. Neurocomputing 73(7–9):1344–1351

    MATH  Google Scholar 

  33. Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64:821–824. https://doi.org/10.1103/PhysRevLett.64.821

    MathSciNet  MATH  Google Scholar 

  34. Pradeepa C, Cao Y, Murugesuc R, Rakkiyappand R (2019) An event-triggered synchronization of semi-Markov jump neural networks with time-varying delays based on generalized free-weighting-matrix approach. Math Comput Simul 155:41–56

    MathSciNet  Google Scholar 

  35. Rosenblum MG, Pikovsky AS, Kurths J (1996) Phase synchronization of chaotic oscillators. Phys Rev Lett 76(11):1804

    MATH  Google Scholar 

  36. Rulkov NF, Sushchik MM (1997) Robustness of synchronized chaotic oscillations. Int J Bifurc Chaos 07(03):625–643. https://doi.org/10.1142/S0218127497000431

    MATH  Google Scholar 

  37. Rulkov NF, Sushchik MM, Tsimring LS, Abarbanel HD (1995) Generalized synchronization of chaos in directionally coupled chaotic systems. Phys Rev E 51(2):980

    Google Scholar 

  38. Tang Z, Park JH, Feng J (2017) Impulsive effects on quasi-synchronization of neural networks with parameter mismatches and time-varying delay. IEEE Trans Neural Net Learn Syst 29:908–919

    Google Scholar 

  39. Wang L, Zou X (2002) Exponential stability of Cohen–Grossberg neural networks. Neural Netw 15(3):415–422

    Google Scholar 

  40. Wang Z, Cao J, Guo Z (2018) Dissipativity analysis and stabilization for discontinuous delayed complex-valued networks via matrix measure method. Adv Differ Equ 2018:340

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  43. Yanchuk S, Maistrenko Y, Lading B, Mosekilde E (2000) Effects of a parameter mismatch on the synchronization of two coupled chaotic oscillators. Int J Bifurc Chaos 10(11):2629–2648. https://doi.org/10.1142/S0218127400001584

    MATH  Google Scholar 

  44. Yu J, Hu C, Jiang H, Teng Z (2011) Exponential synchronization of Cohen–Grossberg neural networks via periodically intermittent control. Neurocomputing 74(10):1776–1782

    Google Scholar 

  45. Zahreddine Z (2003) Matrix measure and application to stability of matrices and interval dynamical systems. Int J Math Math Sci 2003(2):75–85

    MathSciNet  MATH  Google Scholar 

  46. Zhu Q, Cao J (2010) Adaptive synchronization of chaotic Cohen–Crossberg neural networks with mixed time delays. Nonlinear Dyn 61(3):517–534

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are extending their heartfelt thanks to the revered reviewers for their constructive suggestions toward up-gradation of the article.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi, 221005, India

    Rakesh Kumar & Subir Das

Authors
  1. Rakesh Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Subir Das
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rakesh Kumar.

Ethics declarations

Conflict of interest

The authors Subir Das, Professor, Department of Mathematical Sciences, IIT (BHU), Varanasi, India, and Mr. Rakesh Kumar, who is pursuing his Ph.D. degree under the supervision of Prof. S. Das, declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, R., Das, S. Weak, modified and function projective synchronization of Cohen–Grossberg neural networks with mixed time-varying delays and parameter mismatch via matrix measure approach. Neural Comput & Applic 32, 7321–7332 (2020). https://doi.org/10.1007/s00521-019-04227-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-019-04227-4

Keywords

Search

Navigation