Bifurcation analysis in a discrete-time single-directional network with delays☆
In this paper, we consider a simple discrete-time single-directional network of four neurons. The characteristics equation of the linearized system at the zero solution is a polynomial equation involving very high-order terms. We ﬁrst derive some sufﬁcient and necessary conditions ensuring that all the characteristic roots have modulus less than 1. Hence, the zero solution of the model is asymptotically stable. Then, we study the existence of three types of bifurcations, such as fold bifurcations, ﬂip bifurcations, and Neimark-Sacker (NS) bifurcations. Based on the normal form theory and the center manifold theorem, we discuss their bifurcation directions and the stability of bifurcated solutions. In addition, several codimension two bifurcations can be met in the system when curves of codimension one bifurcations intersect or meet tangentially. We proceed through listing smooth normal forms for all the possible codimension 2 bifurcations. © 2007 Elsevier B.V. All rights reserved.
版权说明：以下全部内容由郭上江上传于 2010年01月07日 11时22分05秒，版权归本人所有。