International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 84260, 8 pages
doi:10.1155/2007/84260
Research Article
Bifurcation Analysis for a Two-Dimensional Discrete-Time Hopfield Neural Network with Delays
Department of Mathematics, Yunnan University, Kunming 650091, Yunnan, China
Received 21 September 2006; Revised 18 January 2007; Accepted 27 February 2007
Academic Editor: Virginia Kiryakova
Copyright © 2007 Yaping Ren and Yongkun 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
A bifurcation analysis is undertaken for a discrete-time Hopfield neural
network with four delays. Conditions ensuring the asymptotic stability of the null
solution are obtained with respect to two parameters of the system. Using techniques
developed by Kuznetsov to a discrete-time system, we study the Neimark-Sacker bifurcation (also called Hopf bifurcation for maps) of the system. The direction and the
stability of the Neimark-Sacker bifurcation are investigated by applying the normal form
theory and the center manifold theorem.