In this paper, we consider a planar system with two delays:
ẋ1(t) = −a0x1(t) + a1F1 (x1(t − τ1), x2(τ−t2)).
ẋ2(t) = −b0x2(t) + b1F2 (x1(t − τ1), x2(t − τxs2)).
Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.