  The propagation and Poincar\'e mapping of perturbed Keplerian motion is a key
topic in celestial mechanics and astrodynamics, e.g. to study the stability of
orbits or design bounded relative trajectories. The high-order transfer map
(HOTM) method enables efficient mapping of perturbed Keplerian orbits over many
revolutions. For this, the method uses the high-order Taylor expansion of a
Poincar\'e or stroboscopic map, which is accurate close to the expansion point.
In this paper, we investigate the performance of the HOTM method using
different element sets for building the high-order map. The element sets
investigated are the classical orbital elements, modified equinoctial elements,
Hill variables, cylindrical coordinates and Deprit's ideal elements. The
performances of the different coordinate sets are tested by comparing the
accuracy and efficiency of mapping low-Earth and highly-elliptical orbits
perturbed by $J_2$ with numerical propagation. The accuracy of HOTM depends
strongly on the choice of elements and type of orbit. A new set of elements is
introduced that enables extremely accurate mapping of the state, even for high
eccentricities and higher-order zonal perturbations. Finally, the high-order
map is shown to be very useful for the determination and study of fixed points
and centre manifolds of Poincar\'e maps.
