Resonant terms in a class of systems with stationary bifurcations

Bifurcation control of systems with a single parameter and a single input is addressed in this paper. We focus on systems with quadratic degeneracy, i.e. the quadratic part of the system fully determines local behavior. Three qualitative performances are identified for these systems, namely an unstable isolated equilibrium point, a transcritical bifurcation, and a saddle-node bifurcation. The characterization of these bifurcations are found based on invariant matrices and the coefficients of state feedback. The stability of all equilibrium points in a bifurcation is determined by the invariant matrices and the feedback. It is also proved that the qualitative performance of these bifurcations, such as the type of a bifurcation and the stability of an equilibrium point, is independent of quadratic and higher degree terms in the feedback. The approach in this paper does not require quadratic normal form. However, for systems with cubic degeneracy, the general formula for center manifold is too complicated