  A novel technique to determine invariant curves in nonlinear beam dynamics
based on the method of formal series has been developed. It is first shown how
the solution of the Hamilton equations of motion describing nonlinear betatron
oscillations in the presence of a single sextupole can be represented in a
nonperturbative form. Further, the solution of the Hamilton-Jacobi equation is
obtained in a closed symbolic form as a ratio of two series in the perturbation
parameter (and the nonlinear action invariant), rather than a conventional
power series according to canonical perturbation theory. It is well behaved
even for large values of the perturbation parameter close to strong structural
resonances. The relationship between existing invariant curves and the
so-called scattering orbits in classical scattering theory has been revealed.
