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International Journal of Mathematics and Mathematical Sciences
Volume 2007, Article ID 84260, 8 pages
Research Article

Bifurcation Analysis for a Two-Dimensional Discrete-Time Hopfield Neural Network with Delays

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, 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.


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.