Neural Computing and Applications

, Volume 29, Issue 9, pp 565–584 | Cite as

Existence, uniqueness, and global asymptotic stability analysis for delayed complex-valued Cohen–Grossberg BAM neural networks

  • K. Subramanian
  • P. Muthukumar
Original Article


This paper investigates the existence, uniqueness, and global asymptotic stability of equilibrium point for a complex-valued Cohen–Grossberg delayed bidirectional associative memory neural networks. The two types of complex-valued behaved functions, amplification functions and activation functions, are considered. By using homeomorphism theory and inequality technique, the sufficient conditions for the existence of unique equilibrium point are obtained. Then, by constructing a suitable Lyapunov–Krasovskii functional, the global asymptotic stability condition of the proposed neural networks is derived in terms of linear matrix inequalities. This linear matrix inequality can be efficiently solved via the standard numerical packages. Finally, the numerical examples are given to validate the effectiveness of theoretical results.


Complex-valued Cohen–Grossberg BAM neural networks Existence and uniqueness of equilibrium point Global asymptotic stability Linear matrix inequalities Time delays 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


  1. 1.
    Chua LO, Yang L (1988) Cellular neural networks: applications. IEEE Trans Circuits Syst 35(10):1273–1290MathSciNetCrossRefGoogle Scholar
  2. 2.
    Sakthivel R, Anbuvithya R, Mathiyalagan K, Ma YK, Prakash P (2016) Reliable anti-synchronization conditions for BAM memristive neural networks with different memductance functions. Appl Math Comput 275:213–228MathSciNetGoogle Scholar
  3. 3.
    Mathiyalagan K, Sakthivel R, Marshal Anthoni S (2011) New stability and stabilization criteria for fuzzy neural networks with various activation functions. Phys Scr 84(1):015007CrossRefzbMATHGoogle Scholar
  4. 4.
    Zhu Q, Cao J (2014) Mean-square exponential input-to-state stability of stochastic delayed neural networks. Neurocomputing 131:157–163CrossRefGoogle Scholar
  5. 5.
    Liu L, Zhu Q (2015) Almost sure exponential stability of numerical solutions to stochastic delay Hopfield neural networks. Appl Math Comput 266:698–712MathSciNetGoogle Scholar
  6. 6.
    Rakkiyappan R, Chandrasekar A, Lakshmanan S, Park JH (2014) Exponential stability of Markovian jumping stochastic Cohen–Grossberg neural networks with mode-dependent probabilistic time-varying delays and impulses. Neurocomputing 131:265–277CrossRefGoogle Scholar
  7. 7.
    Huang Z (2016) Almost periodic solutions for fuzzy cellular neural networks with time-varying delays. Neural Comput Appl, 1–8Google Scholar
  8. 8.
    Aizenberg I (2011) Complex-valued neural networks with multi-valued neurons. Springer, New YorkCrossRefzbMATHGoogle Scholar
  9. 9.
    Hirose A (ed) Complex-valued neural networks: advances and applications, vol 18. Wiley 2013. Springer 2012Google Scholar
  10. 10.
    Hu J, Wang J (2012) Global stability of complex-valued recurrent neural networks with time-delays. IEEE Trans Neural Netw Learn Syst 23(6):853–865CrossRefGoogle Scholar
  11. 11.
    Rakkiyappan R, Velmurugan G, Li X (2015) Complete stability analysis of complex-valued neural networks with time delays and impulses. Neural Process Lett 41(3):435–468CrossRefzbMATHGoogle Scholar
  12. 12.
    Mathews JH, Howell RW (2012) Complex analysis for mathematics and engineering. Jones & Bartlett PublishersGoogle Scholar
  13. 13.
    Kuroe Y, Yoshid M, Mori T (2003) On activation functions for complex-valued neural networks-existence of energy functions. In: Artificial neural networks and neural information processing, New York: Springer, pp 174–175Google Scholar
  14. 14.
    Liu X, Fang K, Liu B (2009) A synthesis method based on stability analysis for complex-valued Hopfield neural network. In: Proceedings of 7th Asian control conference, pp 1245–1250Google Scholar
  15. 15.
    Velmurugan G, Rakkiyappan R, Cao J (2015) Further analysis of global \(\mu \)-stability of complex-valued neural networks with unbounded time-varying delays. Neural Netw 67:14–27CrossRefGoogle Scholar
  16. 16.
    Sakthivel R, Raja R, Marshal Anthoni S (2010) Asymptotic stability of delayed stochastic genetic regulatory networks with impulses. Phys Scr 82(5):055009CrossRefzbMATHGoogle Scholar
  17. 17.
    Mathiyalagan K, Su H, Shi P, Sakthivel R (2015) Exponential filtering for discrete-time switched neural networks with random delays. IEEE Trans Cybern 45(4):676–687CrossRefGoogle Scholar
  18. 18.
    Dong T, Liao X, Wang A (2015) Stability and Hopf bifurcation of a complex-valued neural network with two time delays. Nonlinear Dyn 82(1–2):173–184MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhou B, Song Q (2013) Boundedness and complete stability of complex-valued neural networks with time delay. IEEE Trans Neural Netw Learn Syst 24(8):1227–1238CrossRefGoogle Scholar
  20. 20.
    Cohen MA, Grossberg S (1983) Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans Syst Man Cybern 5:815–826MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wu H, Zhang X, Li R, Yao R (2015) Adaptive exponential synchronization of delayed Cohen–Grossberg neural networks with discontinuous activations. Int J Mach Learn Cybern. doi: 10.1007/s13042-014-0258-9
  22. 22.
    Xie W, Zhu Q (2015) Mean square exponential stability of stochastic fuzzy delayed Cohen–Grossberg neural networks with expectations in the coefficients. Neurocomputing 166:133–139CrossRefGoogle Scholar
  23. 23.
    Zhang Z, Yu S (2016) Global asymptotic stability for a class of complex-valued Cohen–Grossberg neural networks with time delays. Neurocomputing 171:1158–1166CrossRefGoogle Scholar
  24. 24.
    Kosko B (1987) Adaptive bidirectional associative memories. Appl Opt 26:4947–4960CrossRefGoogle Scholar
  25. 25.
    Zhu Q, Rakkiyappan R, Chandrasekar A (2014) Stochastic stability of Markovian jump BAM neural networks with leakage delays and impulse control. Neurocomputing 136:136–151CrossRefGoogle Scholar
  26. 26.
    Zhu Q, Cao J (2012) Stability analysis of Markovian jump stochastic BAM neural networks with impulse control and mixed time delays. IEEE Trans Neural Netw Learn Syst 23(3):467–479MathSciNetCrossRefGoogle Scholar
  27. 27.
    Li X (2009) Existence and global exponential stability of periodic solution for impulsive Cohen–Grossberg-type BAM neural networks with continuously distributed delays. Appl Math Comput 215(1):292–307MathSciNetzbMATHGoogle Scholar
  28. 28.
    Zhang Z, Liu W, Zhou D (2012) Global asymptotic stability to a generalized Cohen–Grossberg BAM neural networks of neutral type delays. Neural networks 25:94–105CrossRefzbMATHGoogle Scholar
  29. 29.
    Xiong W, Shi Y, Cao J (2015) Stability analysis of two-dimensional neutral-type Cohen–Grossberg BAM neural networks. Neural Comput Appl, pp 1–14Google Scholar
  30. 30.
    Wang Z, Huang L (2016) Global stability analysis for delayed complex-valued BAM neural networks. Neurocomputing 173:2083–2089CrossRefGoogle Scholar
  31. 31.
    Zhang Z, Lin C, Chen B (2014) Global stability criterion for delayed complex-valued recurrent neural networks. IEEE Trans Neural Netw Learn Syst 25(9):1704–1708CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  1. 1.Department of MathematicsGandhigram Rural Institute - Deemed UniversityGandhigramIndia

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