On the control of Hopf bifurcations

Linear and quadratic normal forms of nonlinear systems with a pair of imaginary uncontrollable modes are derived. Based on the normal form, formulae of feedbacks are found to control the bifurcation of the system. The Hopf bifurcation cannot be removed from the closed-loop system, because the imaginary eigenvalues are uncontrollable. However, is it proved that both the orientation and the stability of the periodic solution can be controlled by state feedback. It is proved that a linear feedback determines the orientation of the periodic solution around the bifurcation point, and the quadratic feedback controls the stability of the periodic solution. The explicit relation between the feedback and the performance of the periodic solution, such as the orientation and stability, is derived