Bautin bifurcation analysis for synchronous solution of a coupled FHN neural system with delay

In this paper, the Bautin bifurcation of synchronous solution of a coupled FHN neural system with delay is investigated. Firstly, the method of Lyapunov functional is used to obtain the synchronization conditions of the neural system, and then distributions of the roots of the characteristic equation associated with the linearization of the synchrosystem are discussed. Center manifold and normal form are employed to calculate its Lyapunov coefficients. A group of sufficient conditions are given to present Bautin bifurcation of the synchrosystem by applying the Bautin bifurcation theorem of delay differential equations developed by Anca-Veronica Ion. The Bautin bifurcation diagram in the physical parameter space is provided to illustrate the correctness of our theoretical analysis.