Quasi-periodic Saddle-Node Bifurcations Near a Differentiable Singularity for Forced Oscillations

In the two-parameter unfolding of a Bogdanov-Takens singularity for autonomous differential equations in the plane with reflection symmetry, it is known in one case that there is a curve Γ in parameter space that corresponds to nonhyperbolic periodic orbits, and all one-parameter paths that cross Γ transversally give saddle-node bifurcations of periodic orbits. In the analogous situation for periodically forced systems, the curve Γ is replaced by a Cantor set of parameter values that corresponds to nonhyperbolic quasi-periodic tori, and there is a restricted set of one-parameter paths that give quasi-periodic saddle-node bifurcations of tori. We require only finite differentiability of the system (C2 dependence on parameters, Ck dependence on state variables, k ≥ 29). The proof of this result uses a version of the Nash-Moser implicit function theorem that obtains C2 dependence of the implicitly defined function on parameters.