Hopf bifurcation control for affine systems

We establish conditions to control the Hopf bifurcation of nonlinear systems with two uncontrollable modes on the imaginary axis. We use the center manifold to reduce the system dynamics to dimension two, and find expressions in terms of the original vector fields. The control law designed has a constant term, which establishes the stability of the equilibrium point, and a quadratic term, which determines the orientation and stability of the periodic solution near the origin.