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

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)].

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

Citations:

Zbl 0983.65079

Software:

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References:

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