Analytic continuation in the case of non-regular dependency on a small parameter with an application to celestial mechanics

We consider a non-autonomous system of ordinary differential equations. Assume that the time dependence is periodic with a very high frequency 1/ɛ, where ɛ is a small parameter and differentiability with respect to the parameter is lost when ɛ equals zero. We derive from Arenstorf's implicit function theorem a set of conditions to show the existence of periodic solutions. These conditions look formally like the standard analytic continuation method, namely, checking that a certain minor does not vanish. We apply this result to show the existence of a new class of periodic orbits of very large radii in the three-dimensional elliptic restricted three-body problem for arbitrary values of the masses of the primaries.