Abstract
The paper presents a detailed analysis on the dynamics of a two-neuron network with time-delayed connections between the neurons and time-delayed feedback from each neuron to itself. On the basis of characteristic roots method and Hopf bifurcation theorems for functional differential equations, we investigate the existence of local Hopf bifurcation. In addition, the direction of Hopf bifurcation and stability of the periodic solutions bifurcating from the trivial equilibrium are determined based on the normal form theory and center manifold theorem. Moreover, employing the global Hopf bifurcation theory due to [Wu, 1998], we study the global existence of periodic solutions. It is shown that the local Hopf bifurcation indicates the global Hopf bifurcation after the second group critical value of the delay.
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