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Two-parameter bifurcations in a network of two neurons with multiple delays. (English) Zbl 1136.34058
The following system of delay-differential equations is considered
$\dot x_1(t) = -x_1(t) + \beta f(x_1(t-\tau)) + a_{12} f(x_2(t-\tau_1)),$
$\dot x_2(t) = -x_2(t) + \beta f(x_2(t-\tau)) + a_{21} f(x_1(t-\tau_2)).$ Here $$\tau$$, $$\tau_1$$ and $$\tau_2$$ are positive time delays, which satisfy $$\tau_1+\tau_2=2\tau$$, and $$f:\mathbb R\to\mathbb R$$ ia a $$C^1$$-smooth function with $$f(0)=0$$.
Considering the corresponding characteristic equation for the equilibrium $$x_1=x_2=0$$, the authors obtain conditions for various codimension-1 and codimension-2 bifurcations, give formulas for the normal form coefficients, and give information about the bifurcating solutions.

MSC:
 34K18 Bifurcation theory of functional-differential equations 34K20 Stability theory of functional-differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics 34K17 Transformation and reduction of functional-differential equations and systems, normal forms
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