  The complex classification of trans-Neptunian objects (TNOs) that are
captured in mean-motion resonances (MMRs) and the constraint of their multiple
origins are two significant open problems concerning the Solar System. The
case-by-case study of the different MMRs and their characteristics provide
information about their origin and dynamics, which helps us to understand the
early stages of the Solar System evolution. In this paper, we study the
dynamics of the detected TNOs close to a 3/1 MMR with Neptune. We initially use
a semi-analytic three-body model to investigate the coplanar secular dynamics
of these objects and find the stationary points. We then use surface sections
and stability maps to analyse the non-averaged dynamics. These methods allow us
to isolate the different stability regions and determine the extent of the
chaotic regions. We show that stability maps are an extremely powerful tool for
studying the resonant dynamics when they are computed in terms of the resonant
angle. We then use these maps to study the non-planar three-body problem and
the full dynamics in the presence of planetary perturbations. We confirm that
TNOs near the 3/1 MMR regions can exist at very high inclinations. In the
framework of the three-body problem, many of these objects can also be stable
outside the 3/1 MMR owing to a Kozai secular resonance. However, when we take
into account the perturbations of the four giant planets, the Kozai regions
disappear and only the 3/1 MMR region remains, with eccentricities $e \lesssim
0.5$.
