Estimate of Multiple Attracing Domains for Cohen-Grossberg Neural Network with Distributed Delays

ABSTRACT

In this paper, we present multiplicity results of exponential stability and attracting domains for Cohen- Grossberg neural network (CGNN) with distributed delays. We establish new criteria for the coexistence of equilibrium points and estimate their attracting domains. Moreover, we base our criteria on co-efficients of the networks and the derivative of activation functions within the attracting domains. It is shown that our results are new and complement corresponding results existing in the previous literature.

In this paper, we present multiplicity results of exponential stability and attracting domains for Cohen- Grossberg neural network (CGNN) with distributed delays. We establish new criteria for the coexistence of equilibrium points and estimate their attracting domains. Moreover, we base our criteria on co-efficients of the networks and the derivative of activation functions within the attracting domains. It is shown that our results are new and complement corresponding results existing in the previous literature.

Cite this paper

nullZ. Huang and Z. Wang, "Estimate of Multiple Attracing Domains for Cohen-Grossberg Neural Network with Distributed Delays,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 533-540. doi: 10.4236/am.2011.25070.

nullZ. Huang and Z. Wang, "Estimate of Multiple Attracing Domains for Cohen-Grossberg Neural Network with Distributed Delays,"

References

[1] M. Cohen and S. Grossberg, “Absolute Stability and Global Pattern Formation and Parallel Memory Storage by Competitive Neural Networks,” IEEE Transactions on Systems, Man and Cybernetics, Vol. 13, No. 5, 1983, pp. 815-825.

[2] S. Grossberg, “Nonlinear Neural Networks: Principles, Mechanisms and Architectures,” Neural Networks, Vol. 1, No. 1, 1988, pp. 17-61. doi:10.1016/0893-6080(88)90021-4

[3] Z. K. Huang and Y. H. Xia, “Exponential p-Stability of Second Order Cohen-Grossberg Neural Networks with Transmission Delays and Learning Behavior,” Simulation Modelling Practice and Theory, Vol. 15, No. 6, 2007, pp. 622-634. doi:10.1016/j.simpat.2006.12.003

[4] L. Wang and X. Zou, “Exponential Stability of Cohen- Grossberg Neural Networks,” Neural Networks, Vol. 15, No. 3, 2002, pp. 415-422. doi:10.1016/S0893-6080(02)00025-4

[5] T. Chen and L. Rong, “Robust Global Exponential Stability of Cohen-Grossberg Neural Networks with Time Delays,” IEEE Transactions on Neural Networks, Vol. 15, No. 1, 2004, pp. 203-205. doi:10.1109/TNN.2003.822974

[6] J. D. Cao and J. L. Liang, “Boundedness and Stability for Cohen-Grossberg Neural Network with Time-Varying Delays,” Journal of Mathematical Analysis and Applications, Vol. 296, No. 2, 2004, pp. 665-685. doi:10.1016/j.jmaa.2004.04.039

[7] X. F. Liao, C. G. Li and K.-W. Wong, “Criteria for Exponential Stability of Cohen-Grossberg Neural Networks,” Neural Networks, Vol. 17, No. 10, 2004, pp. 1401-1414. doi:10.1016/j.neunet.2004.08.007

[8] C. X. Huang and L. H. Huang, “Dynamics of a Class of Cohen-Grossberg Neural Networks with Time-Varying Delays,” Nonlinear Analysis: Real World Applications, Vol. 8, No. 1, 2007, pp. 40-52. doi:10.1016/j.nonrwa.2005.04.008

[9] L. Wang, “Stability of Cohen-Grossberg Neural Networks with Distributed Delays,” Applied Mathematics and Computation, Vol. 160, No. 1, 2005, pp. 93-110. doi:10.1016/j.amc.2003.09.014

[10] X. F. Liao and C. D. Li, “Global Attractivity of Cohen- Grossberg Model with Finite and Infinite Delays,” Journal of Mathematical Analysis and Applications, Vol. 315, No. 1, 2006, pp. 244-262. doi:10.1016/j.jmaa.2005.04.076

[11] J. H. Sun and L. Wan, “Global Exponential Stability and Periodic Solutions of Cohen-Grossberg Neural Networks with Continuously Distributed Delays,” Physica D, Vol. 208, No. 1-2, 2005, pp. 1-20. doi:10.1016/j.physd.2005.05.009

[12] L. Wang and X. F. Zou, “Harmless Delays in Cohen- Grossberg Neural Networks,” Physica D: Nonlinear Phenomena, Vol. 170, No. 2, 2002, pp. 162-173. doi:10.1016/S0167-2789(02)00544-4

[13] K. N. Lu, D. Y. Xu and Z. C. Yang, “Global Attraction and Stability for Cohen-Grossberg Neural Networks with Delays,” Neural Networks, Vol. 19, No. 10, 2006, pp. 1538-1549. doi:10.1016/j.neunet.2006.07.006

[14] Z. K. Huang, X. H. Wang and Y. H. Xia, “Exponential Stability of Impulsive Cohen-Grossberg Networks with Distributed Delays,” International Journal of Circuit Theory and Applications, Vol. 36, No. 3, 2008, pp. 345-365. doi:10.1002/cta.424

[15] J. Hopfield, “Neurons with Graded Response have Collective Computational Properties like Those of Two State Neurons,” Proceedings of the National Academy of Sciences, USA, Vol. 81, No. 10, 1984, pp. 3088-3092. doi:10.1073/pnas.81.10.3088

[16] C. Y. Cheng, K. H. Lin and C. W. Shih, “Multistability in Recurrent Neural Networks,” SIAM Journal on Applied Mathematics, Vol. 66, No. 4, 2006, pp. 1301-1320. doi:10.1137/050632440

[17] Z. Zeng, D. S. Huang and Z. Wang, “Memory Pattern Analysis of Cellular Neural Networks,” Physics Letters A, Vol. 342, No. 1-2, 2005, pp. 114-128. doi:10.1016/j.physleta.2005.05.017

[18] C. Y. Cheng, K. H. Lin and C. W. Shih, “Multistability and Convergence in Delayed Neural Networks,” Physica D, Vol. 225, No. 1, 2007, pp. 61-74. doi:10.1016/j.physd.2006.10.003

[19] L. P. Shayer and S. A. Campbell, “Stability, Bifurcation and Multistability in a System of Two Coupled Neurons with Multiple Time Delays,” SIAM Journal on Applied Mathematics, Vol. 61, No. 2, 2000, pp. 673-700. doi:10.1137/S0036139998344015

[20] Z. K. Huang, Q. K. Song and C. H. Feng, “Multistability in Networks with Self-Excitation and High-Order Synaptic Connectivity,” IEEE Transactions on Circuits and Systems-I: Regular Papers, Vol. 57, No. 8, 2010, pp. 2144-215. doi:10.1109/TCSI.2009.2037401

[21] Z. K. Huang, X. H. Wang and C. H. Feng, “Multiperiodicity of Periodically Oscillated Discrete-Time Neural Networks with Transient Excitatory Self-Connections and Sigmoidal Nonlinearities,” IEEE Transactions on Neural Networks, Vol. 21, No. 10, 2010, pp. 1643-1655. doi:10.1109/TNN.2010.2067225

[22] Y. M. Chen, “Global Stability of Neural Networks with Distributed Delays,” Neural Networks, Vol. 15, No. 7, 2002, pp. 867-871. doi:10.1016/S0893-6080(02)00039-4

[23] H. Y. Zhao, “Global Asymptotic Stability of Hopfield Neural Network involving Distributed Delays,” Neural Networks, Vol. 17, No. 1, 2004, pp. 47-53. doi:10.1016/S0893-6080(03)00077-7

[1] M. Cohen and S. Grossberg, “Absolute Stability and Global Pattern Formation and Parallel Memory Storage by Competitive Neural Networks,” IEEE Transactions on Systems, Man and Cybernetics, Vol. 13, No. 5, 1983, pp. 815-825.

[2] S. Grossberg, “Nonlinear Neural Networks: Principles, Mechanisms and Architectures,” Neural Networks, Vol. 1, No. 1, 1988, pp. 17-61. doi:10.1016/0893-6080(88)90021-4

[3] Z. K. Huang and Y. H. Xia, “Exponential p-Stability of Second Order Cohen-Grossberg Neural Networks with Transmission Delays and Learning Behavior,” Simulation Modelling Practice and Theory, Vol. 15, No. 6, 2007, pp. 622-634. doi:10.1016/j.simpat.2006.12.003

[4] L. Wang and X. Zou, “Exponential Stability of Cohen- Grossberg Neural Networks,” Neural Networks, Vol. 15, No. 3, 2002, pp. 415-422. doi:10.1016/S0893-6080(02)00025-4

[5] T. Chen and L. Rong, “Robust Global Exponential Stability of Cohen-Grossberg Neural Networks with Time Delays,” IEEE Transactions on Neural Networks, Vol. 15, No. 1, 2004, pp. 203-205. doi:10.1109/TNN.2003.822974

[6] J. D. Cao and J. L. Liang, “Boundedness and Stability for Cohen-Grossberg Neural Network with Time-Varying Delays,” Journal of Mathematical Analysis and Applications, Vol. 296, No. 2, 2004, pp. 665-685. doi:10.1016/j.jmaa.2004.04.039

[7] X. F. Liao, C. G. Li and K.-W. Wong, “Criteria for Exponential Stability of Cohen-Grossberg Neural Networks,” Neural Networks, Vol. 17, No. 10, 2004, pp. 1401-1414. doi:10.1016/j.neunet.2004.08.007

[8] C. X. Huang and L. H. Huang, “Dynamics of a Class of Cohen-Grossberg Neural Networks with Time-Varying Delays,” Nonlinear Analysis: Real World Applications, Vol. 8, No. 1, 2007, pp. 40-52. doi:10.1016/j.nonrwa.2005.04.008

[9] L. Wang, “Stability of Cohen-Grossberg Neural Networks with Distributed Delays,” Applied Mathematics and Computation, Vol. 160, No. 1, 2005, pp. 93-110. doi:10.1016/j.amc.2003.09.014

[10] X. F. Liao and C. D. Li, “Global Attractivity of Cohen- Grossberg Model with Finite and Infinite Delays,” Journal of Mathematical Analysis and Applications, Vol. 315, No. 1, 2006, pp. 244-262. doi:10.1016/j.jmaa.2005.04.076

[11] J. H. Sun and L. Wan, “Global Exponential Stability and Periodic Solutions of Cohen-Grossberg Neural Networks with Continuously Distributed Delays,” Physica D, Vol. 208, No. 1-2, 2005, pp. 1-20. doi:10.1016/j.physd.2005.05.009

[12] L. Wang and X. F. Zou, “Harmless Delays in Cohen- Grossberg Neural Networks,” Physica D: Nonlinear Phenomena, Vol. 170, No. 2, 2002, pp. 162-173. doi:10.1016/S0167-2789(02)00544-4

[13] K. N. Lu, D. Y. Xu and Z. C. Yang, “Global Attraction and Stability for Cohen-Grossberg Neural Networks with Delays,” Neural Networks, Vol. 19, No. 10, 2006, pp. 1538-1549. doi:10.1016/j.neunet.2006.07.006

[14] Z. K. Huang, X. H. Wang and Y. H. Xia, “Exponential Stability of Impulsive Cohen-Grossberg Networks with Distributed Delays,” International Journal of Circuit Theory and Applications, Vol. 36, No. 3, 2008, pp. 345-365. doi:10.1002/cta.424

[15] J. Hopfield, “Neurons with Graded Response have Collective Computational Properties like Those of Two State Neurons,” Proceedings of the National Academy of Sciences, USA, Vol. 81, No. 10, 1984, pp. 3088-3092. doi:10.1073/pnas.81.10.3088

[16] C. Y. Cheng, K. H. Lin and C. W. Shih, “Multistability in Recurrent Neural Networks,” SIAM Journal on Applied Mathematics, Vol. 66, No. 4, 2006, pp. 1301-1320. doi:10.1137/050632440

[17] Z. Zeng, D. S. Huang and Z. Wang, “Memory Pattern Analysis of Cellular Neural Networks,” Physics Letters A, Vol. 342, No. 1-2, 2005, pp. 114-128. doi:10.1016/j.physleta.2005.05.017

[18] C. Y. Cheng, K. H. Lin and C. W. Shih, “Multistability and Convergence in Delayed Neural Networks,” Physica D, Vol. 225, No. 1, 2007, pp. 61-74. doi:10.1016/j.physd.2006.10.003

[19] L. P. Shayer and S. A. Campbell, “Stability, Bifurcation and Multistability in a System of Two Coupled Neurons with Multiple Time Delays,” SIAM Journal on Applied Mathematics, Vol. 61, No. 2, 2000, pp. 673-700. doi:10.1137/S0036139998344015

[20] Z. K. Huang, Q. K. Song and C. H. Feng, “Multistability in Networks with Self-Excitation and High-Order Synaptic Connectivity,” IEEE Transactions on Circuits and Systems-I: Regular Papers, Vol. 57, No. 8, 2010, pp. 2144-215. doi:10.1109/TCSI.2009.2037401

[21] Z. K. Huang, X. H. Wang and C. H. Feng, “Multiperiodicity of Periodically Oscillated Discrete-Time Neural Networks with Transient Excitatory Self-Connections and Sigmoidal Nonlinearities,” IEEE Transactions on Neural Networks, Vol. 21, No. 10, 2010, pp. 1643-1655. doi:10.1109/TNN.2010.2067225

[22] Y. M. Chen, “Global Stability of Neural Networks with Distributed Delays,” Neural Networks, Vol. 15, No. 7, 2002, pp. 867-871. doi:10.1016/S0893-6080(02)00039-4

[23] H. Y. Zhao, “Global Asymptotic Stability of Hopfield Neural Network involving Distributed Delays,” Neural Networks, Vol. 17, No. 1, 2004, pp. 47-53. doi:10.1016/S0893-6080(03)00077-7