Invariants and stability of control systems with transcritical and saddle-node bifurcations

State feedback control of bifurcations with quadratic or cubic degeneracy is addressed. Based on normal forms and invariants of nonlinear control systems, the classification of bifurcations for systems with a single uncontrollable mode is obtained. Using invariants, characterizations of stability are derived for a family of bifurcations, including saddle-node bifurcation and transcritical bifurcation in this paper. Bifurcations in systems under feedbacks which are not zero at the critical point are also addressed. In the case of a saddle-node bifurcation, continuous but not differential feedbacks are introduced to remove the bifurcation and to achieve the stability