Hopf bifurcation in a prototype delayed system

A simple model described by an autonomous continuous delayed differential equation with one variable, proposed by Ucar, is considered in this paper. We will show that Hopf bifurcations occur in this simple system. The stability of bifurcating periodic solutions and the direction of Hopf bifurcations are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical example is given to demonstrate the effectiveness of the theoretical analysis