Advances in Differential Equations and Control Processes
Volume 12, Issue 1, Pages 23 - 49
(August 2013)
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NORMAL FORM OF THE EQUIVARIANT HOPF BIFURCATION IN A NEURAL NETWORK INCORPORATING DISTRIBUTED DELAY
Israel Ncube
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Abstract: We consider a network of three identical neurons whose dynamical behaviour is governed by Hopfield’s model incorporating distributed and discrete signal transmission delays to account for the finite switching speed of amplifiers (neurons). The model for such a network is a system of coupled nonlinear delay differential equations. It is established that two cases of a single Hopf bifurcation may occur at the trivial equilibrium of the system, as a consequence of the symmetry of the network. These single Hopf bifurcations are the simple and the double root. Using centre manifold and normal form calculations, the paper establishes the normal form of the equivariant Hopf bifurcation on the centre manifold. |
Keywords and phrases: neural network, delay differential equations, symmetry, discrete delay, distributed delay, characteristic equation, normal form, unfolding, Hopf bifurcation. |
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