Controllable or uncontrollable? - a bifurcation approach

Given a system with an uncontrollable linearization at the origin, we study the controllability of the system at equilibria around the origin. If the uncontrollable mode is nonzero, we prove that the system always have other equilibria around the origin. We also prove that these equilibria are most likely to be linearly controllable. As a result, the system is qualitatively changed if the equilibrium point is changed. Thus, a bifurcation occurs. The analysis and proof given are based on the normal form of nonlinear control systems.