Discrete Dynamics in Nature and Society
Volume 2006 (2006), Article ID 32529, 29 pages
Stability and bifurcation in a simplified four-neuron BAM neural network with multiple delays
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.
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.