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Zero singularities of codimension two and three in delay differential equations. (English) Zbl 1149.92002
Summary: We give conditions under which a general class of delay differential equations has a point of Bogdanov-Takens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to two- or three-dimensional systems of ordinary differential equations. We put these equations in normal form and determine how the coefficients of the normal forms depend on the original parameters in the model. Finally, we apply our results to two neural models and compare the predictions of the theory with numerical bifurcation analysis of the full equations. One model involves a transcritical bifurcation, hence we derive and analyse the appropriate unfoldings for this case.
MSC:
92B20General theory of neural networks (mathematical biology)
68T05Learning and adaptive systems
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
34C23Bifurcation (ODE)
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