Control of dynamic bifurcations

We consider differential equations (x) over dot = f(x, lambda) where the parameter lambda = epsilon t moves slowly through a bifurcation point of f. Such a dynamic bifurcation is often accompanied by a potentially dangerous jump transition. We construct smooth scalar feedback controls which avoid these jumps. For transcritical and pitchfork bifurcations, a small constant additive control is usually sufficient. For Hopf bifurcations, we have to construct a more elaborate control creating a suitable bifurcation with double zero eigenvalue.