Fold-Hopf bifurcation in a simplified four-neuron BAM (bidirectional associative memory) neural network with two delays

The bidirectional associative memory (BAM) neural network with four neurons and two delays is considered in the present paper. A linear stability analysis for the trivial equilibrium is firstly employed to provide a possible critical point at which a zero and a pair of pure imaginary eigenvalues occur in the corresponding characteristic equation. A fold-Hopf bifurcation is proved to happen at this critical point by the nonlinear analysis. The coupling strength and the delay are considered as bifurcation parameters to investigate the dynamical behaviors derived from the fold-Hopf bifurcation. Various dynamical behaviours are qualitatively classified in the neighbourhood of the fold-Hopf bifurcation point by using the center manifold reduction (CMR) together with the normal form. The bifurcating periodic solutions are expressed analytically in an approximate form. The validity of the results is shown by their consistency with the numerical simulation.