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Pattern formation and continuation in a trineuron ring with delays. (English) Zbl 1132.34051

A single-directional ring of three neurons with delays is considered. This is modelled by the following system of delay differential equations \[ u^{\prime}_i(t) = - \mu u_i(t) + f(u_{i+1}(t-\tau)), \] where \(i=0,1,2\), \(u_3(t)=u_0(t)\), \(\mu\) is a positive constant, and \(f\in C^1(\mathbb R;\mathbb R)\), \(f(0)=0\). First, linear stability of the model is investigated. Next, Hopf bifurcating periodic orbits are studied in details.

MSC:

34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
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