Control of dynamic bifurcations

Abstract

We consider differential equations xẋ = f (x, λ) where the parameter λ = et 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.