Consider the system of two differential-delay equations with two delays \[ dx_1 /dt =-a_0 x_1 (t) +a_1 F_1 (x_1 (t - \tau_1),\;x_2(t-\tau_2)),\tag{*} \]
\[ dx_2 /dt =- b_0 x_2 (t) + b_1 F_2 (x_1 (t- \tau_1), \;x_2 (t - \tau_2)) \] under the following assumptions:
(i) \(F_j \in C^3 (\mathbb{R}^2 , \mathbb{R}^2), F_j (0,0)=0, \;\partial F_j / \partial x_j (0,0)=0\) for \(j=1,2\).
(ii) \( \partial F_1 / \partial x_2 (0,0) \neq 0, \;\partial F_2 / \partial x_1 (0,0) \neq 0, \;x_2 F_1 (x_1 ,x_2) \neq 0\) for \( x_2 \neq 0, \;x_1 F_2 (x_1 ,x_2) \neq 0\) for \(x_1 \neq 0\).
(iii) For \( j=0,1, a_j\) and \(b_j\) are constants where \(a_0 > 0, \;b_0 > 0.\)
The authors use the method of degree theory to study the global existence of periodic solutions to \((*)\). To this purpose, they apply a global Hopf bifurcation result to \((*)\) with \(\alpha =-a_1 b_1 \;\partial F_1 / \partial x_2 (0,0) \;\partial F_2 / \partial x_1 (0,0)\) as bifurcation parameter. As an example they consider a neural network without self-connection.