Periodic Solutions of Recurrent Neural Networks with Distributed Delays and Impulses on Time Scales
In this paper, by using the continuation theorem of coincidence degree theory, M-matrix theory and constructing some suitable Lyapunov functions, some sufficient conditions are obtained for the existence and global exponential stability of periodic solutions of recurrent neural networks with distributed delays and impulses on time scales. Without assuming the boundedness of the activation functions gj, hj , these results are less restrictive than those given in the earlier references.
[1] J. Can, J. Wang, Global asymptotic and robust stability of recurrent neural
networks with time delays, IEEE Trans. Circuits Syst. I 52 (5) (2005)
417-425.
[2] H. Huang, J. Cao and J. Wang, Global exponential stability and periodic
solutions of recurrent neural networks with delays, Phys. Lett. A 298
(5-6) (2002) 393-404.
[3] Y.Y. Wu, Y.Q. Wu, Stability Analysis for Recurrent Neural Networks with
Time-varying Delay, International Journal of Automation and Computing
6 (3) (2009) 223-227.
[4] Y.Y. Wu, T. Li, Y.Q. Wu, Improved Exponential Stability Criteria for
Recurrent Neural Networks with Time-varying Discrete and Distributed
Delays, International Journal of Automation and Computing 7 (2) (2010)
199-204.
[5] C. Xing, Z. Gui, Global Asymptotic Stability of Recurrent Neural
Networks with Time-Varying Delays and Impulses, Fourth International
Conference on Natural Computation 2 (2008) 394-398.
[6] H.G. Zhang, Z.S. Wang, New delay-dependent criterion for the stability
of recurrent neural networks with time-varying delay, Sci. China. Ser.
F-Inf. Sci. 52 (6) (2009) 942-948.
[7] J. Can, D.S. Huang, Y.Z. Qu, Global robust stability of delayed recurrent
neural networks, Chaos, Solitons and Fractals 23 (2005) 221-229.
[8] J. Can, J. Wang, Global exponential stability and periodicity of recurrent
neural networks with time delays, IEEE Trans. Circuits Syst. I 52 (5)
(2005) 920-931.
[9] H.A. Jalab, R.W. Ibrahim, Stability of recurrent neural networks, Int. J.
Comp. Sci. Net. Sec. 12 (6) (2006) 159-164.
[10] J. Cao, J. Wang, Absolute exponential stability of recurrent neural
networks with lipschitz-continuous activation functions and time-varying
delays, Neural Networks 17 (2004) 379-390.
[11] Y. Li, L. Lu, Global exponential stability and existence of periodic
solutions of Hopfield-type neural networks with impulses, Phys. Lett.
A 333 (2004) 62-71.
[12] Y. Li, S. Gao, Global Exponential Stability for Impulsive BAM Neural
Networks with Distributed Delays on Time Scales, Neural Processing
Letters 31 (1) (2010) 65-91.
[13] Y. Li, T. Zhang, Global Exponential Stability of Fuzzy Interval Delayed
Neural Networks with impulses on Time Scales, Int. J. Neural Syst. 19
(6) (2009) 449-456.
[14] Y. Li, X. Chen, L. Zhao, Stability and existence of periodic solutions to
delayed Cohen-Grossberg BAM neural networks with impulses on time
scales, Neurocomputing 72 (7-9) 1621-1630 (2009).
[15] Y. Li, L. Zhao, P. Liu, Existence and exponential stability of periodic
solution of high-order Hopfield neural network with delays on time scales,
Volume 2009 (2009), Article ID 573534, 18 pages.
[16] J. Dai, Y. Li, T. Zhang, Positive periodic solutions for nonautonomous
impulsive neutral functional differential systems with time-varying delays
on time scales, 34 (2010) 1-10.
[17] Y. Li, H. Zhang, Existence of periodic solutions for a periodic mutualism
model on time scales, J. Math. Anal. Appl. 343 (2) (2008) 818-825.
[18] Y. Li, M. Hu, Three Positive Periodic Solutions for a Class
of Higher-Dimensional Functional Differential Equations with Impulses
on Time Scales, Advances in Difference Equations (2009)
doi:10.1155/2009/698463.
[19] J. Liu, Y. Li, L. Zhao, On a periodic predator-prey system with time
delays on time scales, Commun. Nonlinear Sci. Numer. Simulat. 8 (14)
(2009) 3432-3438.
[20] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time
Scales, Birkhauser, Boston, 2003.
[21] L. Bi, M. Bohner, M. Fan, Periodic solutions of functional dynamic
equations with infinite delay, Nonlinear Anal. 68 (2008) 1226-1245.
[22] V. Lakshmikantham, A.S. Vatsala, Hybrid systems on time scales, J.
Comput. Appl. Math. 141 (2002) 227-235.
[23] E.R. Kaufmann, Y.N. Raffoul, Periodic solutions for a neutral nonlinear
dynamic equation on a time scale, J. Math. Anal. Appl. 319 (2006) 315-
25.
[24] M. Bohner, M. Fan, J.M. Zhang, Existence of periodic solutions in
predator-prey and competetion dynamic systems, Nonlinear Anal. Real
Word Appl. 330 (1) (2007) 1-9.
[25] F.H. Wang, C.C. Yeh, S.L. Yu, C.H. Hong, Youngs inequality and related
results on time scales, Appl. Math. Lett. 18 (2005) 983-988.
[26] J.L. Mawhin, Topological degree methods in noninear boundary value
problems, CBMS Regional Conference Series in Mathematics, No. 40,
American Mathematical Society, Provedence, RI, 1979.
[27] D. O,regan, Y.J. Chao, Y.Q. Chen, Topological degree theory and
application, Taylor and Francis Group, Boca Raton, London, New York,
2006.
[28] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical
Science, Academic Press, New York, 1979.
[1] J. Can, J. Wang, Global asymptotic and robust stability of recurrent neural
networks with time delays, IEEE Trans. Circuits Syst. I 52 (5) (2005)
417-425.
[2] H. Huang, J. Cao and J. Wang, Global exponential stability and periodic
solutions of recurrent neural networks with delays, Phys. Lett. A 298
(5-6) (2002) 393-404.
[3] Y.Y. Wu, Y.Q. Wu, Stability Analysis for Recurrent Neural Networks with
Time-varying Delay, International Journal of Automation and Computing
6 (3) (2009) 223-227.
[4] Y.Y. Wu, T. Li, Y.Q. Wu, Improved Exponential Stability Criteria for
Recurrent Neural Networks with Time-varying Discrete and Distributed
Delays, International Journal of Automation and Computing 7 (2) (2010)
199-204.
[5] C. Xing, Z. Gui, Global Asymptotic Stability of Recurrent Neural
Networks with Time-Varying Delays and Impulses, Fourth International
Conference on Natural Computation 2 (2008) 394-398.
[6] H.G. Zhang, Z.S. Wang, New delay-dependent criterion for the stability
of recurrent neural networks with time-varying delay, Sci. China. Ser.
F-Inf. Sci. 52 (6) (2009) 942-948.
[7] J. Can, D.S. Huang, Y.Z. Qu, Global robust stability of delayed recurrent
neural networks, Chaos, Solitons and Fractals 23 (2005) 221-229.
[8] J. Can, J. Wang, Global exponential stability and periodicity of recurrent
neural networks with time delays, IEEE Trans. Circuits Syst. I 52 (5)
(2005) 920-931.
[9] H.A. Jalab, R.W. Ibrahim, Stability of recurrent neural networks, Int. J.
Comp. Sci. Net. Sec. 12 (6) (2006) 159-164.
[10] J. Cao, J. Wang, Absolute exponential stability of recurrent neural
networks with lipschitz-continuous activation functions and time-varying
delays, Neural Networks 17 (2004) 379-390.
[11] Y. Li, L. Lu, Global exponential stability and existence of periodic
solutions of Hopfield-type neural networks with impulses, Phys. Lett.
A 333 (2004) 62-71.
[12] Y. Li, S. Gao, Global Exponential Stability for Impulsive BAM Neural
Networks with Distributed Delays on Time Scales, Neural Processing
Letters 31 (1) (2010) 65-91.
[13] Y. Li, T. Zhang, Global Exponential Stability of Fuzzy Interval Delayed
Neural Networks with impulses on Time Scales, Int. J. Neural Syst. 19
(6) (2009) 449-456.
[14] Y. Li, X. Chen, L. Zhao, Stability and existence of periodic solutions to
delayed Cohen-Grossberg BAM neural networks with impulses on time
scales, Neurocomputing 72 (7-9) 1621-1630 (2009).
[15] Y. Li, L. Zhao, P. Liu, Existence and exponential stability of periodic
solution of high-order Hopfield neural network with delays on time scales,
Volume 2009 (2009), Article ID 573534, 18 pages.
[16] J. Dai, Y. Li, T. Zhang, Positive periodic solutions for nonautonomous
impulsive neutral functional differential systems with time-varying delays
on time scales, 34 (2010) 1-10.
[17] Y. Li, H. Zhang, Existence of periodic solutions for a periodic mutualism
model on time scales, J. Math. Anal. Appl. 343 (2) (2008) 818-825.
[18] Y. Li, M. Hu, Three Positive Periodic Solutions for a Class
of Higher-Dimensional Functional Differential Equations with Impulses
on Time Scales, Advances in Difference Equations (2009)
doi:10.1155/2009/698463.
[19] J. Liu, Y. Li, L. Zhao, On a periodic predator-prey system with time
delays on time scales, Commun. Nonlinear Sci. Numer. Simulat. 8 (14)
(2009) 3432-3438.
[20] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time
Scales, Birkhauser, Boston, 2003.
[21] L. Bi, M. Bohner, M. Fan, Periodic solutions of functional dynamic
equations with infinite delay, Nonlinear Anal. 68 (2008) 1226-1245.
[22] V. Lakshmikantham, A.S. Vatsala, Hybrid systems on time scales, J.
Comput. Appl. Math. 141 (2002) 227-235.
[23] E.R. Kaufmann, Y.N. Raffoul, Periodic solutions for a neutral nonlinear
dynamic equation on a time scale, J. Math. Anal. Appl. 319 (2006) 315-
25.
[24] M. Bohner, M. Fan, J.M. Zhang, Existence of periodic solutions in
predator-prey and competetion dynamic systems, Nonlinear Anal. Real
Word Appl. 330 (1) (2007) 1-9.
[25] F.H. Wang, C.C. Yeh, S.L. Yu, C.H. Hong, Youngs inequality and related
results on time scales, Appl. Math. Lett. 18 (2005) 983-988.
[26] J.L. Mawhin, Topological degree methods in noninear boundary value
problems, CBMS Regional Conference Series in Mathematics, No. 40,
American Mathematical Society, Provedence, RI, 1979.
[27] D. O,regan, Y.J. Chao, Y.Q. Chen, Topological degree theory and
application, Taylor and Francis Group, Boca Raton, London, New York,
2006.
[28] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical
Science, Academic Press, New York, 1979.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54661", author = "Yaping Ren and Yongkun Li", title = "Periodic Solutions of Recurrent Neural Networks with Distributed Delays and Impulses on Time Scales", abstract = "In this paper, by using the continuation theorem of coincidence degree theory, M-matrix theory and constructing some suitable Lyapunov functions, some sufficient conditions are obtained for the existence and global exponential stability of periodic solutions of recurrent neural networks with distributed delays and impulses on time scales. Without assuming the boundedness of the activation functions gj, hj , these results are less restrictive than those given in the earlier references.
", keywords = "Recurrent neural networks, global exponential stability, periodic solutions, distributed delays, impulses, time scales.", volume = "5", number = "8", pages = "1235-10", }
