Bifurcation analysis in active control system with time delay feedback

In this study, a model of active control system with time delay feedback is investigated, and time delays were set in control loop. An efficient method was used to analyze the stability of the system. By solving the characteristic equation, the local stability and the existence of codimension one bifurcation (Hopf bifurcation and fold bifurcation) were obtained. Regarding the time delay as bifurcation parameters, the direction of the Hopf bifurcation and the stability of bifurcation periodic solution by applying the normal form theory and center manifold theorem. Numerical simulations were also given to illustrate the results obtained.