Summary: We consider a simplified bidirectional associated memory (BAM) neural network model with four neurons and multiple time delays. The global existence of periodic solutions bifurcating from Hopf bifurcations is investigated by applying the global Hopf bifurcation theorem due to J. Wu
[Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034
)] and Bendixson’s criterion for high-dimensional ordinary differential equations due to Y. Li
and J. S. Muldowney
[ J. Differ. Equations 106, No. 1, 27–39 (1993; Zbl 0786.34033
)]. It is shown that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the sum of two delays. Numerical simulations supporting the theoretical analysis are also included.