Stability and Neimark–Sacker bifurcation of numerical discretization of delay differential equations

A kind of a discrete delay model obtained by Euler method is investigated. Firstly, the linear stability of the model is studied. It is found that there exist Neimark–Sacker bifurcations when the delay passes a sequence of critical values. Then the explicit algorithm for determining the direction and stability of the Neimark–Sacker bifurcations are derived by using the normal form theory and center manifold theorem. Finally, computer simulations are provided to illustrate the analytical results found.