The stability and invariants of control systems with pitchfork or cusp bifurcations

The control of bifurcation by state feedback for nonlinear systems with cubic degeneracy is addressed in this paper. Normal forms and invariants are employed to classify the bifurcations of the closed-loop systems. It is proved that, for systems with uncontrollable linearization, the closed-loop system under state feedback usually has either pitchfork or cusp bifurcations if the quadratic degeneracy condition does not hold. Cubic invariants are found to characterize the bifurcations of the closed-loop systems. It is also proved that the cusp or pitchfork bifurcations addressed in this paper can be rendered supercritical by state feedbacks under some weak conditions