About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 369725, 14 pages
http://dx.doi.org/10.1155/2012/369725
Research Article

Dynamical Behaviors of Stochastic Reaction-Diffusion Cohen-Grossberg Neural Networks with Delays

1School of Mathematics and Computer Science, Wuhan Textile University, Wuhan 430073, China
2Department of Mathematics, Zhaoqing University, Zhaoqing 526061, China
3School of Management, Wuhan Textile University, Wuhan 430073, China

Received 22 August 2012; Accepted 24 September 2012

Academic Editor: Xiaodi Li

Copyright © 2012 Li Wan 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.

Abstract

This paper investigates dynamical behaviors of stochastic Cohen-Grossberg neural network with delays and reaction diffusion. By employing Lyapunov method, Poincaré inequality and matrix technique, some sufficient criteria on ultimate boundedness, weak attractor, and asymptotic stability are obtained. Finally, a numerical example is given to illustrate the correctness and effectiveness of our theoretical results.

1. Introduction

Cohen and Grossberg proposed and investigated Cohen-Grossberg neural networks in 1983 [1]. Hopfield neural networks, recurrent neural networks, cellular neural networks, and bidirectional associative memory neural networks are special cases of this model. Since then, the Cohen-Grossberg neural networks have been widely studied in the literature, see for example, [212] and references therein.

Strictly speaking, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields. Therefore, we must consider that the activations vary in space as well as in time. In [1319], the authors gave some stability conditions of reaction-diffusion neural networks, but these conditions were independent of diffusion effects.

On the other hand, it has been well recognized that stochastic disturbances are ubiquitous and inevitable in various systems, ranging from electronic implementations to biochemical systems, which are mainly caused by thermal noise, environmental fluctuations, as well as different orders of ongoing events in the overall systems [20, 21]. Therefore, considerable attention has been paid to investigate the dynamics of stochastic neural networks, and many results on stability of stochastic neural networks have been reported in the literature, see for example, [2238] and references therein.

The above references mainly considered the stability of equilibrium point of neural networks. What do we study when the equilibrium point does not exist? Except for stability property, boundedness and attractor are also foundational concepts of dynamical systems, which play an important role in investigating the uniqueness of equilibrium, global asymptotic stability, global exponential stability, the existence of periodic solution, and so on [39, 40]. Recently, ultimate boundedness and attractor of several classes of neural networks with time delays have been reported. In [41], the globally robust ultimate boundedness of integrodifferential neural networks with uncertainties and varying delays was studied. Some sufficient criteria on the ultimate boundedness of deterministic neural networks with both varying and unbounded delays were derived in [42]. In [43, 44], a series of criteria on the boundedness, global exponential stability, and the existence of periodic solution for nonautonomous recurrent neural networks were established. In [45, 46], some criteria on ultimate boundedness and attractor of stochastic neural networks were derived. To the best of our knowledge, there are few results on the ultimate boundedness and attractor of stochastic reaction-diffusion neural networks.

Therefore, the arising questions about the ultimate boundedness, attractor and stability for the stochastic reaction-diffusion Cohen-Grossberg neural networks with time-varying delays are important yet meaningful.

The rest of the paper is organized as follows: some preliminaries are in Section 2, main results are presented in Section 3, a numerical example and conclusions will be drawn in Sections 4 and 5, respectively.

2. Model Description and Assumptions

Consider the following stochastic Cohen-Grossberg neural networks with delays and diffusion terms: for and . In the above model, is the number of neurons in the network; is space variable; is the state variable of the th neuron at time and in space ; and denote the activation functions of the th unit at time and in space ; constant ; presents an amplification function; is an appropriately behavior function; and denote the connection strengths of the th unit on the th unit, respectively; corresponds to the transmission delay and satisfies denotes the external bias on the th unit; is the diffusion function; is a compact set with smooth boundary and measure mes in is the initial boundary value; is -dimensional Brownian motion defined on a complete probability space with a natural filtration generated by , where we associate with the canonical space generated by all and denote by the associated -algebra generated by with the probability measure .

System (2.1) has the following matrix form: where

Let be the space of real Lebesgue measurable functions on and a Banach space for the -norm Note that is -valued function and -measurable -valued random variable, where on , is the space of all continuous -valued functions defined on with a norm .

The following assumptions and lemmas will be used in establishing our main results.(A1) There exist constants , , and such that (A2) There exist constants and such that (A3) is bounded, positive, and continuous, that is, there exist constants , such that , for , .

Lemma 2.1 (Poincaré inequality, [47]). Assume that a real-valued function satisfies , where is a bounded domain of with a smooth boundary . Then, which is the lowest positive eigenvalue of the Neumann boundary problem: is the gradient operator, is the Laplace operator.

Remark 2.2. Assumption (A1) is less conservative than that in [26, 28], since the constants , , , and are allowed to be positive, negative, or zero, that is to say, the activation function in (A1) is assumed to be neither monotonic, differentiable, nor bounded. Assumption (A2) is weaker than those given in [23, 27, 30] since is not required to be zero or smaller than 1 and is allowed to take any value.

Remark 2.3. According to the eigenvalue theory of elliptic operators, the lowest eigenvalue is only determined by [47]. For example, if , then ; if , then .

The notation (resp., ) means that matrix is symmetric-positive definite (resp., positive semidefinite). denotes the transpose of the matrix . represents the minimum eigenvalue of matrix . .

3. Main Results

Theorem 3.1. Suppose that assumptions (A1)–(A3) hold and there exist some matrices , , , , , and such that the following linear matrix inequality hold:(A4)where means the symmetric term,
Then system (2.1) is stochastically ultimately bounded, that is, if for any , there is a positive constant such that the solution of system (2.1) satisfies

Proof. If , then it follows from (A4) that there exists a sufficiently small such that where
If , then it follows from (A4) that there exists a sufficiently small such that where , , and are the same as in (3.4),
Consider the following Lyapunov functional:
Applying Itô formula in [48] to along (2.2), one obtains
From assumptions (A1)–(A4), one obtains
From the boundary condition and Lemma 2.1, one obtains where “·” is inner product, ,
Combining (3.10) and (3.11) into (3.9), we have where or .
In addition, it follows from (A1) that Similarly, one obtains
From (3.13)–(3.15), one derives or where , Thus, one obtains
For any , set . By Chebyshev’s inequality and (3.20), we obtain which implies The proof is completed.

Theorem 3.1 shows that there exists such that for any , . Let be denoted by Clearly, is closed, bounded, and invariant. Moreover, with no less than probability , which means that attracts the solutions infinitely many times with no less than probability , so we may say that is a weak attractor for the solutions.

Theorem 3.2. Suppose that all conditions of Theorem 3.1 hold. Then there exists a weak attractor for the solutions of system (2.1).

Theorem 3.3. Suppose that all conditions of Theorem 3.1 hold and . Then zero solution of system (2.1) is mean square exponential stability.

Remark 3.4. Assumption (A4) depends on and , so the criteria on the stability, ultimate boundedness, and weak attractor depend on diffusion effects and the derivative of the delays and are independent of the magnitude of the delays.

4. An Example

In this section, a numerical example is presented to demonstrate the validity and effectiveness of our theoretical results.

Example 4.1. Consider the following system where , , , , , , , , is one-dimensional Brownian motion. Then we compute that , , , , , , , , and . By using the Matlab LMI Toolbox, for , based on Theorem 3.1, such system is stochastically ultimately bounded when

5. Conclusion

In this paper, new results and sufficient criteria on the ultimate boundedness, weak attractor, and stability are established for stochastic reaction-diffusion Cohen-Grossberg neural networks with delays by using Lyapunov method, Poincaré inequality and matrix technique. The criteria depend on diffusion effect and derivative of the delays and are independent of the magnitude of the delays.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11271295, 10926128, 11047114, and 71171152), Science and Technology Research Projects of Hubei Provincial Department of Education (nos. Q20111607 and Q20111611) and Young Talent Cultivation Projects of Guangdong (LYM09134).

References

  1. M. A. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 13, no. 5, pp. 815–826, 1983. View at Zentralblatt MATH
  2. Z. Chen and J. Ruan, “Global dynamic analysis of general Cohen-Grossberg neural networks with impulse,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1830–1837, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. T. Huang, A. Chan, Y. Huang, and J. Cao, “Stability of Cohen-Grossberg neural networks with time-varying delays,” Neural Networks, vol. 20, no. 8, pp. 868–873, 2007. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Huang, C. Li, and G. Chen, “Stability of Cohen-Grossberg neural networks with unbounded distributed delays,” Chaos, Solitons & Fractals, vol. 34, no. 3, pp. 992–996, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. Z. W. Ping and J. G. Lu, “Global exponential stability of impulsive Cohen-Grossberg neural networks with continuously distributed delays,” Chaos, Solitons & Fractals, vol. 41, no. 1, pp. 164–174, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. Li and J. Yan, “Dynamical analysis of Cohen-Grossberg neural networks with time-delays and impulses,” Neurocomputing, vol. 72, no. 10–12, pp. 2303–2309, 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. M. Tan and Y. Zhang, “New sufficient conditions for global asymptotic stability of Cohen-Grossberg neural networks with time-varying delays,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2139–2145, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. M. Gao and B. Cui, “Robust exponential stability of interval Cohen-Grossberg neural networks with time-varying delays,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 1914–1928, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. C. Li, Y. K. Li, and Y. Ye, “Exponential stability of fuzzy Cohen-Grossberg neural networks with time delays and impulsive effects,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3599–3606, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Y. K. Li and L. Yang, “Anti-periodic solutions for Cohen-Grossberg neural networks with bounded and unbounded delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 7, pp. 3134–3140, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. X. D. Li, “Exponential stability of Cohen-Grossberg-type BAM neural networks with time-varying delays via impulsive control,” Neurocomputing, vol. 73, no. 1–3, pp. 525–530, 2009. View at Publisher · View at Google Scholar · View at Scopus
  12. J. Yu, C. Hu, H. Jiang, and Z. Teng, “Exponential synchronization of Cohen-Grossberg neural networks via periodically intermittent control,” Neurocomputing, vol. 74, no. 10, pp. 1776–1782, 2011. View at Publisher · View at Google Scholar · View at Scopus
  13. J. Liang and J. Cao, “Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays,” Physics Letters A, vol. 314, no. 5-6, pp. 434–442, 2003. View at Publisher · View at Google Scholar · View at Scopus
  14. Z. J. Zhao, Q. K. Song, and J. Y. Zhang, “Exponential periodicity and stability of neural networks with reaction-diffusion terms and both variable and unbounded delays,” Computers & Mathematics with Applications, vol. 51, no. 3-4, pp. 475–486, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. X. Lou and B. Cui, “Boundedness and exponential stability for nonautonomous cellular neural networks with reaction-diffusion terms,” Chaos, Solitons & Fractals, vol. 33, no. 2, pp. 653–662, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. K. Li, Z. Li, and X. Zhang, “Exponential stability of reaction-diffusion generalized Cohen-Grossberg neural networks with both variable and distributed delays,” International Mathematical Forum, vol. 2, no. 29–32, pp. 1399–1414, 2007. View at Zentralblatt MATH
  17. R. Wu and W. Zhang, “Global exponential stability of delayed reaction-diffusion neural networks with time-varying coefficients,” Expert Systems with Applications, vol. 36, no. 6, pp. 9834–9838, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. Z. A. Li and K. L. Li, “Stability analysis of impulsive Cohen-Grossberg neural networks with distributed delays and reaction-diffusion terms,” Applied Mathematical Modelling, vol. 33, no. 3, pp. 1337–1348, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. J. Pan and S. M. Zhong, “Dynamical behaviors of impulsive reaction-diffusion Cohen-Grossberg neural network with delays,” Neurocomputing, vol. 73, no. 7–9, pp. 1344–1351, 2010. View at Publisher · View at Google Scholar · View at Scopus
  20. M. Kærn, T. C. Elston, W. J. Blake, and J. J. Collins, “Stochasticity in gene expression: from theories to phenotypes,” Nature Reviews Genetics, vol. 6, no. 6, pp. 451–464, 2005. View at Publisher · View at Google Scholar · View at Scopus
  21. K. Sriram, S. Soliman, and F. Fages, “Dynamics of the interlocked positive feedback loops explaining the robust epigenetic switching in Candida albicans,” Journal of Theoretical Biology, vol. 258, no. 1, pp. 71–88, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. C. Huang and J. D. Cao, “On pth moment exponential stability of stochastic Cohen-Grossberg neural networks with time-varying delays,” Neurocomputing, vol. 73, no. 4–6, pp. 986–990, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. M. Dong, H. Zhang, and Y. Wang, “Dynamics analysis of impulsive stochastic Cohen-Grossberg neural networks with Markovian jumping and mixed time delays,” Neurocomputing, vol. 72, no. 7–9, pp. 1999–2004, 2009. View at Publisher · View at Google Scholar · View at Scopus
  24. Q. Song and Z. Wang, “Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays,” Physica A, vol. 387, no. 13, pp. 3314–3326, 2008. View at Publisher · View at Google Scholar · View at Scopus
  25. C. H. Wang, Y. G. Kao, and G. W. Yang, “Exponential stability of impulsive stochastic fuzzy reaction–diffusion Cohen–Grossberg neural networks with mixed delays,” Neurocomputing, vol. 89, pp. 55–63, 2012. View at Publisher · View at Google Scholar · View at Scopus
  26. H. Huang and G. Feng, “Delay-dependent stability for uncertain stochastic neural networks with time-varying delay,” Physica A, vol. 381, no. 1-2, pp. 93–103, 2007. View at Publisher · View at Google Scholar · View at Scopus
  27. H. Y. Zhao, N. Ding, and L. Chen, “Almost sure exponential stability of stochastic fuzzy cellular neural networks with delays,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 1653–1659, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. W. H. Chen and X. M. Lu, “Mean square exponential stability of uncertain stochastic delayed neural networks,” Physics Letters A, vol. 372, no. 7, pp. 1061–1069, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. C. Huang and J. D. Cao, “Almost sure exponential stability of stochastic cellular neural networks with unbounded distributed delays,” Neurocomputing, vol. 72, no. 13–15, pp. 3352–3356, 2009. View at Publisher · View at Google Scholar · View at Scopus
  30. C. Huang, P. Chen, Y. He, L. Huang, and W. Tan, “Almost sure exponential stability of delayed Hopfield neural networks,” Applied Mathematics Letters, vol. 21, no. 7, pp. 701–705, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. 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
  32. R. Rakkiyappan and P. Balasubramaniam, “Delay-dependent asymptotic stability for stochastic delayed recurrent neural networks with time varying delays,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 526–533, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. Y. Sun and J. D. 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
  34. Z. Wang, J. Fang, and X. Liu, “Global stability of stochastic high-order neural networks with discrete and distributed delays,” Chaos, Solitons & Fractals, vol. 36, no. 2, pp. 388–396, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. X. D. Li, “Existence and global exponential stability of periodic solution for delayed neural networks with impulsive and stochastic effects,” Neurocomputing, vol. 73, no. 4–6, pp. 749–758, 2010. View at Publisher · View at Google Scholar · View at Scopus
  36. Y. Ou, H. Y. Liu, Y. L. Si, and Z. G. Feng, “Stability analysis of discrete-time stochastic neural networks with time-varying delays,” Neurocomputing, vol. 73, no. 4–6, pp. 740–748, 2010. View at Publisher · View at Google Scholar · View at Scopus
  37. Q. Zhu and J. Cao, “Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 41, no. 2, pp. 341–353, 2011. View at Publisher · View at Google Scholar · View at Scopus
  38. Q. Zhu, C. Huang, and X. Yang, “Exponential stability for stochastic jumping BAM neural networks with time-varying and distributed delays,” Nonlinear Analysis: Hybrid Systems, vol. 5, no. 1, pp. 52–77, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. P. Wang, D. Li, and Q. Hu, “Bounds of the hyper-chaotic Lorenz-Stenflo system,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2514–2520, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  40. P. Wang, D. Li, X. Wu, J. Lü, and X. Yu, “Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems,” International Journal of Bifurcation and Chaos, vol. 21, no. 9, pp. 2679–2694, 2011. View at Publisher · View at Google Scholar
  41. X. Y. Lou and B. Cui, “Global robust dissipativity for integro-differential systems modeling neural networks with delays,” Chaos, Solitons & Fractals, vol. 36, no. 2, pp. 469–478, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  42. Q. Song and Z. Zhao, “Global dissipativity of neural networks with both variable and unbounded delays,” Chaos, Solitons & Fractals, vol. 25, no. 2, pp. 393–401, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. H. Jiang and Z. Teng, “Global eponential stability of cellular neural networks with time-varying coefficients and delays,” Neural Networks, vol. 17, no. 10, pp. 1415–1425, 2004. View at Publisher · View at Google Scholar · View at Scopus
  44. H. Jiang and Z. Teng, “Boundedness, periodic solutions and global stability for cellular neural networks with variable coefficients and infinite delays,” Neurocomputing, vol. 72, no. 10–12, pp. 2455–2463, 2009. View at Publisher · View at Google Scholar · View at Scopus
  45. L. Wan and Q. H. Zhou, “Attractor and ultimate boundedness for stochastic cellular neural networks with delays,” Nonlinear Analysis: Real World Applications, vol. 12, no. 5, pp. 2561–2566, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  46. L. Wan, Q. H. Zhou, P. Wang, and J. Li, “Ultimate boundedness and an attractor for stochastic Hopfield neural networks with time-varying delays,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 953–958, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  47. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, NY, USA, 1998. View at Publisher · View at Google Scholar
  48. X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, 1997.