Hopf bifurcation, which occurs due to the collision of two excitation modes (see the discussion about this collision in the next section): one mode associated to the oscillations around the minimum of the effective potential of soliton interactions, given by expression (\[eq:freq\_rel\]) and represented in Fig. \[fig:bog\_notrap\] by the solid red curve, and the background spin density mode of lowest energy (represented by the dashed curve for the longer ring), which is given by the analytical expression (in full units) [@Abad2013]: $$\hbar\omega=\sqrt{\frac{\hbar^2 k^2}{2m} \left( \frac{\hbar^2 k^2}{2m} + 
2(g-g_{12})n\right)}.
 \label{eq:spin}$$

The small disagreement between the crossing of these analytical curves and the beginning of instabilities in the numerical results arises from the curvature of the modes near the bifurcation point, and decreases for longer rings. This first instability triggers new collisions between modes and, as a consequence, more instability regions. The longer the ring the higher the number of instability regions in the system, approaching the prediction for the infinite case. It is worth remarking that the bound state mode predicted by Eq. (\[eq:freq\_rel\]), in excellent agreement with the numerics, does not change with the size of the ring.

To analyze the dynamics of dark-dark solitons in the bound state allowed 