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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)].

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:

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\textit{L. Wang} and \textit{X. Zou}, J. Comput. Appl. Math. 167, No. 1, 73--90 (2004; Zbl 1054.65076)

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### References:

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