Feedback stabilization of steady-state and Hopf bifurcations

We consider nonlinear systems with steady-state or Hopf bifurcations for which the bifurcated modes are linearly unstabilizable. The goal is to find sufficiently smooth state feedbacks such that the bifurcation for the closed loop system is supercritical, and at the same time, the linearly controllable subsystem is asymptotically stable. We solve this problem by giving necessary and sufficient conditions under the assumption that certain nondegeneracy conditions are satisfied. We also give explicit construction of stabilizing feedbacks and illustrate the theory using simple examples.