  In a 2D conservative Hamiltonian system there is a formal integral $\Phi$
besides the energy H. This is not convergent near a stable periodic orbit, but
it is convergent near an unstable periodic orbit. We explain this difference
and we find the convergence radius along the asymptotic curves. In simple
mappings this radius is infinite. This allows the theoretical calculation of
the asymptotic curves and their intersections at homoclinic points. However in
more complex mappings and in Hamiltonian systems the radius of convergence is
in general finite and does not allow the theoretical calculation of any
homoclinic point. Then we develop a method similar to analytic continuation,
applicable in systems expressed in action-angle variables, that allows the
calculation of the asymptotic curves to an arbitrary length. In this way we can
study analytically the chaotic regions near the unstable periodic orbit and
near its homoclinic points.
