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.