  It is known that the asymptotic invariant manifolds around an unstable
periodic orbit in conservative systems can be represented by convergent series
(Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable
manifolds intersect at an infinity of homoclinic points, generating a
complicated homoclinic tangle. In the case of simple mappings it was found (Da
Silva Ritter et al. 1987) that the domain of convergence of the formal series
extends to infinity along the invariant manifolds. This allows in practice to
study the homoclinic tangle using only series. However in the case of
Hamiltonian systems, or mappings with a finite analyticity domain,the
convergence of the series along the asymptotic manifolds is also finite. Here,
we provide numerical indications that the convergence does not reach any
homoclinic points. We discuss in detail the convergence problem in various
cases and we find the degree of approximation of the analytical invariant
manifolds to the real (numerical) manifolds as i) the order of truncation of
the series increases, and ii) we use higher numerical precision in computing
the coefficients of the series. Then we introduce a new method of series
composition, by using action-angle variables, that allows the calculation of
the asymptotic manifolds up to an a arbitrarily large extent. This is the first
case of an analytic development that allows the computation of the invariant
manifolds and their intersections in a Hamiltonian system for an extent long
enough to allow the study of homoclinic chaos by analytical means.
