- 99-426 Cicogna G., Santoprete M.
- An approach to Mel'nikov theory in celestial mechanics
Nov 12, 99
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Abstract. Using a completely analytic procedure -- based on a suitable
extension of a classical method -- we discuss an approach to the
Poincar\'e-Mel'nikov theory, which can be conveniently applied also
to the case of non-hyperbolic 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-like 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 Gyld\'en problem,
originally proposed in celestial mechanics, but also of interest
in different fields, and by the general 3-body problem in