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Zbl 1149.34046
Huang, Chuangxia; He, Yigang; Huang, Lihong; Zhaohui, Yuan
Hopf bifurcation analysis of two neurons with three delays. (English)
[J] Nonlinear Anal., Real World Appl. 8, No. 3, 903-921 (2007). ISSN 1468-1218

The authors study linear stability and give conditions and direction for Hopf bifurcations in the following system of coupled delay differential equations (two neurons with three delays) $$\align \dot{x}(t) & = - x(t) +a_{11}f(x(t-\tau)) +a_{12}f(y(t-\tau_1)) ,\\ \dot{y}(t) & = - y(t) +a_{21}f(x(t-\tau_2)) +a_{22}f(y(t-\tau)). \endalign$$ Here $x$ and $y$ are scalar variables corresponding to two neurons, $\tau_j$ denote the transmission delays, $a_{ij}$ are synaptic weights. Additionally, $f(0)=0$ so that the zero solution is the equilibrium, which produces Hopf bifurcations studied.
[Sergiy Yanchuk (Berlin)]

MSC 2000:: *34K18 Bifurcation theory of functional differential equations
34K20 Stability theory of functional-differential equations
34K13 Periodic solutions of functional differential equations

Keywords: Hopf bifurcation; neural network; linear stability

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