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Stability and bifurcation in a discrete system of two neurons with delays. (English) Zbl 1154.37381
Summary: We consider a simple discrete two-neuron network model with three delays. The characteristic equation of the linearized system at the zero solution is a polynomial equation involving very high order terms. We derive some sufficient and necessary conditions on the asymptotic stability of the zero solution. Regarding the eigenvalues of connection matrix as the bifurcation parameters, we also consider the existence of three types of bifurcations: Fold bifurcations, Flip bifurcations, and Neimark-Sacker bifurcations. The stability and direction of these three kinds of bifurcations are studied by applying the normal form theory and the center manifold theorem. Our results are a very important generalization to the previous works in this field.

MSC:
37N25Dynamical systems in biology
39A30Stability theory (difference equations)
39A28Bifurcation theory (difference equations)
92C20Neural biology
39A12Discrete version of topics in analysis
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References:
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