Using a completely analytic procedure—based on a suitable extension of a classical method—we discuss an approach to the Poincaré–Mel’nikov theory, which can be conveniently applied also to the case of nonhyperbolic critical points, and even if the critical point is located at the infinity. In this paper, we concentrate our attention on the latter case, and precisely on problems described by Kepler-type potentials in one or two degrees of freedom, in the presence of general time-dependent perturbations. We show that the appearance of chaos (possibly including Arnol’d diffusion) can be proved quite easily and in a direct way, without resorting to singular coordinate transformations, such as the McGehee or blowing-up transformations. Natural examples are provided by the classical Gyldén problem, originally proposed in celestial mechanics, but also of interest in different fields, and by the general three-body problem in classical mechanics.
Cicogna, Giampaolo and Santoprete, Manuele, "An Approach to Mel’nikov Theory in Celestial Mechanics" (2000). Mathematics Faculty Publications. 46.