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 is studied the dynamics of individual particles in the region where is located the ring. Using the Poincare Surface of Section technique, the islands of stability associated with the 1:3 resonance are identified and studied. Along all its existence this resonance showed 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 significantly reduces the size of the stable regions of the 1:3 resonance. The results also show that there is a minimum equivalent eccentricity ($e_{1:3}$) for the existence of such resonance. This value seems to be too high to keep a particle within the borders of the ring. On the other hand, the Poincare Surface of Sections show 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 $e_{1:3}$, and covering a large radial distance, which encompasses the region of the Haumeas ring. Therefore, this analysis suggests the Haumeas ring is in a stable region associated with a first kind periodic orbit instead of the 1:3 resonance.
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