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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.

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
Full Text: DOI
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