I moti quasi periodici e la stabilità del sistema solare. 1, Dagli epicicli al punto omoclino di Poincaré

The problem of stability of planetary motion is revisited with the aim of illustrating some emerging aspects from the historical development of our knowledge. The note is divided in two parts. The first one is concerned with the classical methods and ends up with the work of Poincar\'e. The second one deals with the discoveries of the last 50 years. The first part of the note starts with the attempts to represent the motions of the planets as being quasiperiodic, actually by means of epicicles as in the classical theories. In this framework the Lindststedt's expansion method is illustrated by applying it to Duffing's equation. This introduces the main problem of classical astronomy, namely the role of resonances that shows up in either form of secular terms or of small divisors in the series expansions of the solutions of the equation. Then the discovery of the chaotic behaviour of orbits by Poincar\'e is recalled by illustrating in some detail the phenomenon of homoclinic intersections.