Bifurcation control of complex networks model via PD controller

In this paper, we investigate the problem of bifurcation control for a complex networks in a small-world networks model with time delay. By choosing the nonlinear interactive parameter as the bifurcation parameter, we present a Proportional-Derivative (PD) feedback controller to control Hopf bifurcation which inherently happens due to the networks topology. Stability analysis shows that the onset of Hopf bifurcation can be delayed or advanced via a PD controller by setting proper control parameters. Therefore the Hopf bifurcation of the model became controllable to achieve desirable behaviors which is applicable in certain circumstances. Meanwhile, the direction and stability of bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Finally, numerical simulations confirm the effectiveness of the control strategy in controlling the Hopf bifurcation for the complex network model.