Discrete Dynamics in Nature and Society
Volume 2006 (2006), Article ID 32529, 29 pages
doi:10.1155/DDNS/2006/32529

Stability and bifurcation in a simplified four-neuron BAM neural network with multiple delays

Xiang-Ping Yan1,2 and Wan-Tong Li2

1School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
2School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China

Received 23 May 2005; Accepted 4 September 2005

Copyright © 2006 Xiang-Ping Yan and Wan-Tong Li. 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

We first study the distribution of the zeros of a fourth-degree exponential polynomial. Then we apply the obtained results to a simplified bidirectional associated memory (BAM) neural network with four neurons and multiple time delays. By taking the sum of the delays as the bifurcation parameter, it is shown that under certain assumptions the steady state is absolutely stable. Under another set of conditions, there are some critical values of the delay, when the delay crosses these critical values, the Hopf bifurcation occurs. Furthermore, some explicit formulae determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form theory and center manifold reduction. Numerical simulations supporting the theoretical analysis are also included.