Periodic delay effects on cutting tool dynamics

We consider a delay equation whose delay is perturbed by a small periodic fluctuation. In Particular, it is assumed that the delay equation exhibits a Hopf bifurcation when its delay Is unperturbed. The periodically perturbed system exhibits more delicate bifurcations than a Hopf bifurcation. We show that these bifurcations are well explained by the Bogdanov-Takens bifurcations when the ration between the frequencies of the periodic solution of the Unperturbed system (Hopf bifurcation) and the external periodic perturbation is 1:2. Our anaylsis is based on center manifold reduction theory.