Hopf bifurcation in bidirectional associative memory neural networks with delays: Analysis and computation. (English) Zbl 1054.65076

Wang, Lin; Zou, Xingfu

J. Comput. Appl. Math. 167, No. 1, 73-90 (2004).

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 L. F. Shampine and S. Thompson [Appl. Numer. Math. 37, 441–458 (2001; Zbl 0983.65079)].

Reviewer: Manuel Calvo (Zaragoza)

MSC:

65L07	Numerical investigation of stability of solutions to ordinary differential equations
34K20	Stability theory of functional-differential equations
34K18	Bifurcation theory of functional-differential equations
92B20	Neural networks for/in biological studies, artificial life and related topics
34K25	Asymptotic theory of functional-differential equations

Keywords:

delay differential equations; bidirectional neural networks; stability; periodic solutions; Hopf bifurcation

Citations:

Zbl 0983.65079

Software:

dde23

References:

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