Stability and bifurcation analysis in an amplitude equation with delayed feedback

An amplitude equation is considered. The linear stability of the equation with direct control is investigated, and hence a bifurcation set is provided in the appropriate parameter plane. It is found that there exist stability switches when delay varies, and the Hopf bifurcation occurs when delay passes through a sequence of critical values. Furthermore, the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, some numerical simulations are performed to illustrate the obtained results.