Hopf bifurcation in bidirectional associative memory neural networks with delays: Analysis and computation
This paper is concerned with the stability behaviour of some delay differential equations that model bidirectional associative memory neural networks. The system under consideration has the form \(x_i'(t)=-x_i(t)+\sum_{j=1}^na_{ij}f_j(y_j(t-\tau_1))\), \(y_i'(t)=-y_i(t)+\sum_{j=1}^nb_{ij}g_j(y_j(t-\tau_2))\), \(i=1,\dots,N\), where \(a_{ij},b_{ij}\) are the constant connection weights through the neurons in two layers, \(f_i\) and \(g_i\) given activation functions and \(\tau_1,\tau_2\) constant delays. In this problem several sets of sufficient conditions on the data problem \(a_{ij},b_{ij},f_i,g_i\) that imply the asymptotic stability of the zero solution are given. Also the existence of Hopf bifurcations in terms of \(\tau=\tau_1+\tau_2\) is studied proving that for some \(\tau=\tau^*\) the above system has a branch of periodic solutions bifurcating from the zero solution near \(\tau=\tau^*\). Finally, a particular system with \(N=2\) neurons is considered to test the above theoretical results using the delay differential equations solver of \textit{L. F. Shampine} and \textit{S. Thompson} [Appl. Numer. Math. 37, 441--458 (2001; Zbl 0983.65079)].
scientific article; zbMATH DE number 2091371
scientific article