Stability and bifurcation in the circular restricted $ (N+2) $-body problem in the sphere $ \mathbb{S}^2 $ with logarithmic potential

In this paper we study part of the dynamics of a circular restricted $ (N+2) $-body problem on the sphere $ \mathbb{S}^2 $ and considering the logarithmic potential, where $ N $ primaries remain in a ring type configuration (identical masses placed at the vertices of a regular polygon in a fixed parallel and rotating uniformly with respect to the $ Z $-axis) and a $ (N+1) $-th primary of mass $ M\in \mathbb{R} $ fixed at the south pole of $ \mathbb{S}^2 $. Such a particular configuration will be called ring-pole configuration (RP). An infinitesimal mass particle has an equilibrium position at the north pole for any value of $ M $, any parallel where the ring has been fixed (we use as parameter $ z = \cos\theta $, where $ \theta $ is the polar angle of the ring) and any number $ N\geq 2 $ of masses forming the ring. We study the non-linear stability of the north pole in terms of the parameters $ (z, M, N) $ and some bifurcations near the north pole.