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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 761237, 11 pages
http://dx.doi.org/10.1155/2013/761237
Research Article

Exponential Stability and Numerical Methods of Stochastic Recurrent Neural Networks with Delays

1School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China
2Guangdong Electric Power Design Institute, Guangzhou 510663, China

Received 18 April 2013; Accepted 12 July 2013

Academic Editor: Alexander I. Domoshnitsky

Copyright © 2013 Shifang Kuang et al. 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.

Linked References

  1. C. M. Marcus and R. M. Westervelt, “Stability of analog neural networks with delay,” Physical Review A, vol. 39, no. 1, pp. 347–359, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  2. J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, vol. 6, Walter de Gruyter, Berlin, Germany, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  3. X. Li, L. Huang, and J. Wu, “Further results on the stability of delayed cellular neural networks,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 50, no. 9, pp. 1239–1242, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  4. X. M. Li, L. H. Huang, and H. Zhu, “Global stability of cellular neural networks with constant and variable delays,” Nonlinear Analysis. Theory, Methods & Applications, vol. 53, no. 3-4, pp. 319–333, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S. Arik, “Stability analysis of delayed neural networks,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 47, no. 7, pp. 1089–1092, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Arik, “Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays,” IEEE Transactions on Neural Networks, vol. 16, no. 3, pp. 580–586, 2005.
  7. T. Liao and F. Wang, “Global stability for cellular neural networks with time delay,” IEEE Transactions on Neural Networks, vol. 11, no. 6, pp. 1481–1484, 2000.
  8. X. Liao, Q. Liu, and W. Zhang, “Delay-dependent asymptotic stability for neural networks with distributed delays,” Nonlinear Analysis. Real World Applications, vol. 7, no. 5, pp. 1178–1192, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. Kuang, F. Deng, and X. Li, “Stability and hopf bifurcation of a BAM neural network with delayed self-feedback,” in Proceedings of the 7th International Symposium on Neural Networks, vol. 6063 of Lecture Notes in Computer Science, pp. 493–503, 2010.
  10. O. Faydasicok and S. Arik, “Further analysis of global robust stability of neural networks with multiple time delays,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 349, no. 3, pp. 813–825, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. S. Haykin, Neural Networks, Prentice-Hall, NJ, USA, 1994.
  12. S. Blythe, X. Mao, and X. Liao, “Stability of stochastic delay neural networks,” Journal of the Franklin Institute, vol. 338, no. 4, pp. 481–495, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L. Wan and J. Sun, “Mean square exponential stability of stochastic delayed Hopfield neural networks,” Physics Letters A, vol. 343, no. 4, pp. 306–318, 2005.
  14. Q. Zhou and L. Wan, “Exponential stability of stochastic delayed Hopfield neural networks,” Applied Mathematics and Computation, vol. 199, no. 1, pp. 84–89, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Y. Zhang, D. Yue, and E. Tian, “Robust delay-distribution-dependent stability of discrete-time stochastic neural neural networks with time-varying delay,” Neurocomputing, vol. 72, no. 4, pp. 1265–1273, 2009.
  16. Z. Wang, Y. Liu, M. Li, and X. Liu, “Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delay,” IEEE Transactions on Neural Networks, vol. 17, no. 3, pp. 814–820, 2006.
  17. Y. Sun and J. Cao, “pth moment exponential stability of stochastic recurrent neural networks with time-varying delays,” Nonlinear Analysis. Real World Applications, vol. 8, no. 4, pp. 1171–1185, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. C. Huang, Y. He, and H. Wang, “Mean square exponential stability of stochastic recurrent neural networks with time-varying delays,” Computers & Mathematics with Applications, vol. 56, no. 7, pp. 1773–1778, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. U. Küchler and E. Platen, “Strong discrete time approximation of stochastic differential equations with time delay,” Mathematics and Computers in Simulation, vol. 54, no. 1–3, pp. 189–250, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  20. E. Buckwar, “Introduction to the numerical analysis of stochastic delay differential equations,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 297–307, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. D. J. Higham and P. E. Kloeden, “Convergence and stability of implicit methods for jump-diffusion systems,” International Journal of Numerical Analysis and Modeling, vol. 3, no. 2, pp. 125–140, 2006. View at Zentralblatt MATH · View at MathSciNet
  22. X. Mao and S. Sabanis, “Numerical solutions of stochastic differential delay equations under local Lipschitz condition,” Journal of Computational and Applied Mathematics, vol. 151, no. 1, pp. 215–227, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. Liu, W. Cao, and Z. Fan, “Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation,” Journal of Computational and Applied Mathematics, vol. 170, no. 2, pp. 255–268, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. P. Hu and C. Huang, “Stability of stochastic θ-methods for stochastic delay integro-differential equations,” International Journal of Computer Mathematics, vol. 88, no. 7, pp. 1417–1429, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  25. A. Rathinasamy and K. Balachandran, “T-stability of the split-step θ-methods for linear stochastic delay integro-differential equations,” Nonlinear Analysis. Hybrid Systems, vol. 5, no. 4, pp. 639–646, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  26. H. Zhang, S. Gan, and L. Hu, “The split-step backward Euler method for linear stochastic delay differential equations,” Journal of Computational and Applied Mathematics, vol. 225, no. 2, pp. 558–568, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. M. Song and H. Yu, “Numerical solutions of stochastic differential delay equations with Poisson random measure under the generalized Khasminskii-type conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 127397, 24 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  28. Q. Li and S. Gan, “Stability of analytical and numerical solutions for nonlinear stochastic delay differential equations with jumps,” Abstract and Applied Analysis, vol. 2012, Article ID 831082, 13 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. X. Ding, K. Wu, and M. Liu, “Convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations,” International Journal of Computer Mathematics, vol. 83, no. 10, pp. 753–761, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. R. Li, W. Pang, and P. Leung, “Exponential stability of numerical solutions to stochastic delay Hopfield neural networks,” Neurocomputing, vol. 73, no. 4–6, pp. 920–926, 2010.
  31. F. Jiang and Y. Shen, “Stability in the numerical simulation of stochastic delayed Hopfield neural networks,” Neural Coputing and Applications, vol. 22, no. 7-8, pp. 1493–1498, 2013.
  32. A. Rathinasamy, “The split-step θ-methods for stochastic delay Hopfield neural networks,” Applied Mathematical Modelling, vol. 36, no. 8, pp. 3477–3485, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  33. X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 2nd edition, 2008. View at MathSciNet