Open Access

Exponential Stability for Impulsive BAM Neural Networks with Time-Varying Delays and Reaction-Diffusion Terms

Advances in Difference Equations20072007:078160

DOI: 10.1155/2007/78160

Received: 9 March 2007

Accepted: 16 May 2007

Published: 21 June 2007

Abstract

Impulsive bidirectional associative memory neural network model with time-varying delays and reaction-diffusion terms is considered. Several sufficient conditions ensuring the existence, uniqueness, and global exponential stability of equilibrium point for the addressed neural network are derived by M-matrix theory, analytic methods, and inequality techniques. Moreover, the exponential convergence rate index is estimated, which depends on the system parameters. The obtained results in this paper are less restrictive than previously known criteria. Two examples are given to show the effectiveness of the obtained results.

[1234567891011121314151617181920212223242526272829]

Authors’ Affiliations

(1)
Department of Mathematics, Chongqing Jiaotong University
(2)
Department of Mathematics, Southeast University

References

  1. Kosko B: Bidirectional associative memories. IEEE Transactions on Systems, Man, and Cybernetics 1988,18(1):49–60. 10.1109/21.87054MathSciNetView ArticleGoogle Scholar
  2. Cao J, Song Q: Stability in Cohen-Grossberg-type bidirectional associative memory neural networks with time-varying delays. Nonlinearity 2006,19(7):1601–1617. 10.1088/0951-7715/19/7/008MATHMathSciNetView ArticleGoogle Scholar
  3. Arik S: Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays. IEEE Transactions on Neural Networks 2005,16(3):580–586. 10.1109/TNN.2005.844910View ArticleGoogle Scholar
  4. Gopalsamy K, He X-Z: Delay-independent stability in bidirectional associative memory networks. IEEE Transactions on Neural Networks 1994,5(6):998–1002. 10.1109/72.329700View ArticleGoogle Scholar
  5. Liao X, Wong K-W, Yang S: Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays. Physics Letters A 2003,316(1–2):55–64. 10.1016/S0375-9601(03)01113-7MATHMathSciNetView ArticleGoogle Scholar
  6. Rao VSH, Phaneendra BhRM: Global dynamics of bidirectional associative memory neural networks involving transmission delays and dead zones. Neural Networks 1999,12(3):455–465. 10.1016/S0893-6080(98)00134-8View ArticleGoogle Scholar
  7. Mohamad S: Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks. Physica D 2001,159(3–4):233–251. 10.1016/S0167-2789(01)00344-XMATHMathSciNetView ArticleGoogle Scholar
  8. Cao J, Wang L: Exponential stability and periodic oscillatory solution in BAM networks with delays. IEEE Transactions on Neural Networks 2002,13(2):457–463. 10.1109/72.991431View ArticleGoogle Scholar
  9. Cao J, Liang J, Lam J: Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D 2004,199(3–4):425–436. 10.1016/j.physd.2004.09.012MATHMathSciNetView ArticleGoogle Scholar
  10. Xu S, Lam J: A new approach to exponential stability analysis of neural networks with time-varying delays. Neural Networks 2006,19(1):76–83. 10.1016/j.neunet.2005.05.005MATHView ArticleGoogle Scholar
  11. Guo S, Huang L, Dai B, Zhang Z: Global existence of periodic solutions of BAM neural networks with variable coefficients. Physics Letters A 2003,317(1–2):97–106. 10.1016/j.physleta.2003.08.019MATHMathSciNetView ArticleGoogle Scholar
  12. Park JH: A novel criterion for global asymptotic stability of BAM neural networks with time delays. Chaos, Solitons and Fractals 2006,29(2):446–453. 10.1016/j.chaos.2005.08.018MATHMathSciNetView ArticleGoogle Scholar
  13. Guan Z-H, Chen G: On delayed impulsive Hopfield neural networks. Neural Networks 1999,12(2):273–280. 10.1016/S0893-6080(98)00133-6View ArticleGoogle Scholar
  14. Yang Z, Xu D: Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays. Applied Mathematics and Computation 2006,177(1):63–78. 10.1016/j.amc.2005.10.032MATHMathSciNetView ArticleGoogle Scholar
  15. Ho DWC, Liang J, Lam J: Global exponential stability of impulsive high-order BAM neural networks with time-varying delays. Neural Networks 2006,19(10):1581–1590. 10.1016/j.neunet.2006.02.006MATHView ArticleGoogle Scholar
  16. Akça H, Alassar R, Covachev V, Covacheva Z, Al-Zahrani E: Continuous-time additive Hopfield-type neural networks with impulses. Journal of Mathematical Analysis and Applications 2004,290(2):436–451. 10.1016/j.jmaa.2003.10.005MATHMathSciNetView ArticleGoogle Scholar
  17. Xu D, Yang Z: Impulsive delay differential inequality and stability of neural networks. Journal of Mathematical Analysis and Applications 2005,305(1):107–120. 10.1016/j.jmaa.2004.10.040MATHMathSciNetView ArticleGoogle Scholar
  18. Li Y: Global exponential stability of BAM neural networks with delays and impulses. Chaos, Solitons and Fractals 2005,24(1):279–285.MATHMathSciNetView ArticleGoogle Scholar
  19. Zhang Y, Sun J: Stability of impulsive neural networks with time delays. Physics Letters A 2005,348(1–2):44–50. 10.1016/j.physleta.2005.08.030MATHView ArticleGoogle Scholar
  20. Li Y-T, Yang C-B: Global exponential stability analysis on impulsive BAM neural networks with distributed delays. Journal of Mathematical Analysis and Applications 2006,324(2):1125–1139. 10.1016/j.jmaa.2006.01.016MATHMathSciNetView ArticleGoogle Scholar
  21. Yang F, Zhang C, Wu D: Global stability analysis of impulsive BAM type Cohen-Grossberg neural networks with delays. Applied Mathematics and Computation 2007,186(1):932–940. 10.1016/j.amc.2006.08.016MATHMathSciNetView ArticleGoogle Scholar
  22. Stamov GT, Stamova IM: Almost periodic solutions for impulsive neural networks with delay. Applied Mathematical Modelling 2007,31(7):1263–1270. 10.1016/j.apm.2006.04.008MATHView ArticleGoogle Scholar
  23. Liao XX, Yang SZ, Chen SJ, Fu YL: Stability of general neural networks with reaction-diffusion. Science in China. Series F 2001,44(5):389–395.MATHGoogle Scholar
  24. Wang L, Xu D: Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays. Science in China. Series F 2003,46(6):466–474. 10.1360/02yf0146MATHMathSciNetView ArticleGoogle Scholar
  25. Song Q, Zhao Z, Li Y: Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms. Physics Letters A 2005,335(2–3):213–225. 10.1016/j.physleta.2004.12.007MATHView ArticleGoogle Scholar
  26. Qiu J: Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms. Neurocomputing 2007,70(4–6):1102–1108.View ArticleGoogle Scholar
  27. Cao J, Wang J: Global exponential stability and periodicity of recurrent neural networks with time delays. IEEE Transactions on Circuits and Systems I 2005,52(5):920–931.MathSciNetView ArticleGoogle Scholar
  28. Zhang Q, Wei X, Xu J: New stability conditions for neural networks with constant and variable delays. Chaos, Solitons and Fractals 2005,26(5):1391–1398. 10.1016/j.chaos.2005.04.008MATHMathSciNetView ArticleGoogle Scholar
  29. Song Q, Cao J: Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays. Journal of Computational and Applied Mathematics 2006,197(1):188–203. 10.1016/j.cam.2005.10.029MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Q. Song and J. Cao. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.