EXISTENCE AND GLOBAL EXPONENTIAL STABILITY OF A PERIODIC SOLUTION TO DISCRETE-TIME COHEN-GROSSBERG BAM NEURAL NETWORKS WITH DELAYS

- Journal title : Journal of the Korean Mathematical Society
- Volume 48, Issue 4, 2011, pp.727-747
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2011.48.4.727

Title & Authors

EXISTENCE AND GLOBAL EXPONENTIAL STABILITY OF A PERIODIC SOLUTION TO DISCRETE-TIME COHEN-GROSSBERG BAM NEURAL NETWORKS WITH DELAYS

Zhang, Zhengqiu; Wang, Liping;

Zhang, Zhengqiu; Wang, Liping;

Abstract

By employing coincidence degree theory and using Halanay-type inequality technique, a sufficient condition is given to guarantee the existence and global exponential stability of periodic solutions for the two-dimensional discrete-time Cohen-Grossberg BAM neural networks. Compared with the results in existing papers, in our result on the existence of periodic solution, the boundedness conditions on the activation are replaced with global Lipschitz conditions. In our result on the existence and global exponential stability of periodic solution, the assumptions in existing papers that the value of activation functions at zero is zero are removed.

Keywords

periodic solutions;two-dimensional discrete-time Cohen-Grossberg BAM neural networks;global exponential stability;coincidence degree theory;Lyapunov functional;

Language

English

Cited by

References

1.

S. Arik and Z. Orman, Global stability analysis of Cohen-Grossberg neural networks with time varying delays, Phys. Lett. A 341 (2005), no. 5-6, 410-421.

2.

J. Cao and J. Liang, Boundedness and stability of Cohen-Grossberg neural networks with time-varying delays, J. Math. Anal. Appl. 296 (2004), no. 2, 665-685.

3.

A. P. Chen and J. D. Cao, Periodic bi-directional Cohen-Grossberg neural networks with distributed delays, Nonlinear Anal. 66 (2007), no. 12, 2947-2961.

4.

A. P. Chen and F. L. Chen, Periodic solution to BAM neural network with delays on time scalesNeurocomputing 73 (2009), no. 1-3, 274-282.

5.

A. P. Chen, L. H. Huang, Z. G. Liu, and J. D.Cao, Periodic bidirectional associative memory neural networks with distributed delaysJ. Math. Anal. Appl. 317 (2006), no. 1, 80-102.

6.

T. Chen and L. Rong, Delay-independent stability analysis of Cohen-Grossberg neural networks, Phys. Lett. A 317 (2003), no. 5-6, 436-449.

7.

Z. Chen, D. H. Zhao, and X. L. Fu, Discrete analogue of high-order periodic Cohen- Grossberg neural networks with delays, Appl. Math. Comput. 214 (2009), no. 1, 210-217.

8.

M. Cohen and S. Grossberg, Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Systems Man Cybernet. 13 (1983), no. 5, 815-826.

9.

B. T. Cui and W. Wu, Global exponential stability of Cohen-Grossberg neural networks with distributed delays, Neurocomputing 72 (2008), no. 1-3, 386-391.

10.

J. Feng and S. Xu, New criteria on global robust stability of Cohen-Grossberg neural networks with time varying delays, Neurocomputing 72 (2008), no. 1-3, 445-457.

11.

J. Feng, S. Y. Xu, and Y. Zou, Delay-dependent stability of neutral type neural networks with distributed delays, Neurocomputing 72 (2009), 2576-2580.

12.

R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Springer Berlin, 1977.

13.

Y. Li, Existence and stability of periodic solutions for Cohen-Grossberg neural networks with multiple delays, Chaos Solitons Fractals 20 (2004), no. 3, 459-466.

14.

Y. K. Li, Global stability and existence of periodic solutions of discrete delayed cellular neural networks, Phys. Lett. A 333 (2004), no. 1-2, 51-61.

15.

X. Liao, C. Li, and K. Wong, Criteria for exponential stability of Cohen-Grossberg neural networks, Neural Networks 17 (2004), no. 10, 1401-1414.

16.

J. Liang and J. Cao, Exponential stability of continuous-time and discrete-time bidirectional associative memory networks with delays, Chaos Solitons Fractals 22 (2004), no. 4, 773-785.

17.

L. Liang, J. Cao, and D. Ho, Discrete-time bidirectional associative memory neural networks with variable delays, Phys. Lett. A 335 (2005), 226-234.

18.

J. A. Liu and G. D. Zong, New delay-dependent asymptotic stability conditions concern- ing BAM neural networks of neutral type, Neurocomputing 72 (2009), 2549-2555.

19.

W. L. Lu and T. P. Chen, $R^n_+$ global stability of a Cohen-Grossberg network system with nonnegative equilibria, Neural Networks 20 (2007), no. 6, 714-722.

20.

K. N. Lu, D. Y. Xu, and Z. C. Yang, Global attraction and stability for Cohen-Grossberg neural networks with delays, Neural Networks 19 (2006), no. 10 1538-1549.

21.

S. Mohamad and K. Gopalsamy, Dynamics of a class of discrete time neural networks and their continuous time counterparts, Math. Comput. Simulation 53 (2000), no. 1-2, 1-39.

22.

S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput. 135 (2003), no. 1, 17-38.

23.

S. Mohamad and A. G. Naim, Discrete-time analogues of integrodifferential equations modelling bidirectional neural networks, J. Comput. Appl. Math. 138 (2002), no. 1, 1-20.

24.

Ju H. Park, Robust stability of bidirectional associative memory neural networks with time delays, Phys. Lett. A 349 (2006), no. 6, 494-499.

25.

Ju H. Park and O. M. Kwon, Delay-dependent stability criterion for bidirectional as- sociative memory neural networks with interval time-varying delays, Modern Physics Letters B 23 (2009), no. 1, 35-46.

26.

Ju H. Park, S. M. Lee, and O. M. Kwon, On exponential stability of associative memory neural networks with time-varying delays, Chaos Solitons Fractals 39 (2009), no. 3, 1083-1091.

27.

Ju H. Park, C. H. Park, O. M. Kwon, and S. M. Lee, A new stability criterion for bidirectional associative memory neural networks of neutral type, Appl. Math. Comput. 199 (2008), no. 2, 716-722.

28.

L. B. Rong and T. P. Chen, New results on the robust stability of Cohen-Grossberg neural networks with delays, Neural Process Lett. 24 (2006), no. 3, 193-202.

29.

Q. K. Song and J. D. Cao, Robust stability in Cohen-Grossberg neural networks with both time-varying and distributed delays, Neural Process Lett. 27 (2008), no. 2, 179-196.

30.

C. Sun and C. B. Feng, Discrete-time analogues of intergrodifferential equations mod- elling neural networks, Phys. Lett. A 334 (2005), 180-191.

31.

J. Sun and L. Wang, Global exponential stability and periodic solutions of Cohen- Grossberg neural networks with continuously distributed delays, Phys. D 208 (2005), no. 1-2, 1-20.

32.

L. Wang, Stability of Cohen-Grossberg neural networks with distributed delays, Appl. Math. Comput. 160 (2005), no. 1, 93-110.

33.

Z. D. Wang, Y. R. Liu, and M. Li, Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays, IEEE Trans. Neural Networks 17 (2006), no. 3, 814-820.

34.

Z. D. Wang, Y. R. Liu, and X. H. Liu, On global asymptotic stability of neural networks with discrete and distributed delays, Phys Lett. A 345 (2005), no. 4-6, 299-308.

35.

W. Wu, B. T. Cui, and X. Y. Lou, Some criteria for asymptotic stability of Cohen- Grossberg neural networks with time varying delays, Neurocomputing 70 (2007), no. 4-6, 1085-1088.

36.

W. Xiong and J. Cao, Global exponential stability of discrete-time Cohen-Grossberg neural networks, Neurocomputing 64 (2005), 433-446.

37.

H. J. Xiang and J. D. Cao, Exponential stability of periodic solution to Cohen-Grossberg type BAM neural networks with time-varying delays, Neurocomputing 72 (2009), no. 7-8, 1702-1711.

38.

W. W. Yu, J. D. Cao, and J. Wang, An LMI approach to global asymptotic stability of the delayed Cohen-Grossberg neural networks via nonsmooth analysis, Neural Networks 20 (2007), no. 7, 810-818.

39.

Z. G. Zeng, J.W, Global exponential stability of recurrent networks with time-varying delays in the presence of strong external stimuli, Neural Networks 19 (2006), no. 10, 1528-1537.

40.

J. Zhang, Y. Suda, and H. Komine, Global exponential stability of Cohen-Grossberg neural networks with variable delays, Phys. Lett. A 338 (2005), no. 1, 44-50.

41.

Z. Q. Zhang and D. M. Zhou, Existence and global exponential stability of a periodic solution for a discrete-time interval general BAM neural networks, J. Franklin Inst. 347 (2010), no. 5, 763-780.

42.

H. Y. Zhao and L. Wang, Hopf bifucation in Cohen-Grossberg neural network with discrete delays, Nonlinear Anal.: Real World Appli. 8 (2007), no. 1, 73-89.

43.

H. Zhao, L. Sun, and G. Wang, Periodic oscillation of discrete-time bidirectional asso- ciative memory neural networks, Neurocomputing 70 (2007), 2924-2930.

44.

T. Zhou, Y. Liu, and Y. C. Liu, Existence and global exponential stability of periodic solution for discrete time BAM neural networks, Appl. Math. Comput. 182 (2006), no. 2, 1341-1354.

45.

Z. Orman and S. Arik, New results for global stability of Cohen-Grossberg neural net- works with multiple time delays, Neurocomputing 71 (2008), no. 16-18, 3053-3063.