hopf bifurcation control with pd controller

In this paper, we investigate the problem of bifurcation control for a delayed logistic growth model. by choosing the timedelay as the bifurcation parameter, we present a proportional - derivative (pd) controller to control hopf bifurcation. we show that the onset of hopf bifurcation can be delayed or advanced via a pd controller by setting proper controlling parameter. under consideration model as operator equation, apply orthogonal decomposition, compute the center manifold and normal form we determined the direction and stability of bifurcating periodic solutions. therefore the hopf bifurcation of the model became controllable to achieve desirable behaviour which are applicable in certain circumstances.