A procedure has previously been developed for the iterative
construction of invariants associated with magnetic field-line Hamiltonians
that consist of an axisymmetric zeroth-order term plus a non-axisymmetric
perturbation. Approximate
field-line invariants obtained with this procedure are used to examine
the
topological properties of magnetic field lines in a parabolic-current
MHD equilibrium that
was slightly perturbed from axisymmetry in the limit of a periodic cylindrical
configuration. Excellent agreement between Poincaré maps and the
level curves of the
first-order invariant is found for small perturbations. A means of circumventing
the
‘small-divisor problem’ in some cases is identified and
implemented with outstanding results. These results indicate that this
perturbation method can have valuable
consequences for future investigations of magnetic field-line topology.