Abstract The recently discovered ring around the dwarf planet (136108) Haumea is located near the 1:3 resonance between the orbital motion of the ring particles and the spin of Haumea. In the current work, we study the dynamics of individual particles in the region where the ring is located. Using the Poincaré surface of section technique, the islands of stability associated with the 1:3 resonance are identified and studied. Throughout its existence, this resonance is shown to be doubled, producing pairs of periodic and quasi-periodic orbits. The fact of being doubled introduces a separatrix, which generates a chaotic layer that reduces the size of the stable regions of the 1:3 resonance significantly. The results also show that there is a minimum equivalent eccentricity (e1$\colon$3) required for the existence of such a resonance. This value seems to be too high to keep a particle within the borders of the ring. On the other hand, the Poincaré surface of sections shows the existence of much larger stable regions, but associated with a family of first-kind periodic orbits. They exist with equivalent eccentricity values lower than e1$\colon$3, covering a large radial distance, which encompasses the region of Haumea’s ring. Therefore, this analysis suggests that Haumea’s ring is in a stable region associated with a first-kind periodic orbit instead of the 1:3 resonance.