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The complex classification of trans-Neptunian objects (TNOs) that arecaptured in mean-motion resonances (MMRs) and the constraint of their multipleorigins are two significant open problems concerning the Solar System. Thecase-by-case study of the different MMRs and their characteristics provideinformation about their origin and dynamics, which helps us to understand theearly stages of the Solar System evolution. In this paper, we study thedynamics of the detected TNOs close to a 3/1 MMR with Neptune. We initially usea semi-analytic three-body model to investigate the coplanar secular dynamicsof these objects and find the stationary points. We then use surface sectionsand stability maps to analyse the non-averaged dynamics. These methods allow usto isolate the different stability regions and determine the extent of thechaotic regions. We show that stability maps are an extremely powerful tool forstudying the resonant dynamics when they are computed in terms of the resonantangle. We then use these maps to study the non-planar three-body problem andthe full dynamics in the presence of planetary perturbations. We confirm thatTNOs near the 3/1 MMR regions can exist at very high inclinations. In theframework of the three-body problem, many of these objects can also be stableoutside the 3/1 MMR owing to a Kozai secular resonance. However, when we takeinto account the perturbations of the four giant planets, the Kozai regionsdisappear and only the 3/1 MMR region remains, with eccentricities $e \lesssim0.5$.

Dynamics of trans-Neptunian objects near the 3/1 mean-motion resonance with Neptune