Stability and bifurcation in the circular restricted (n+2)-body problem in the sphere s2 with logarithmic potential

In this paper we study part of the dynamics of a circular restricted (N +2)-body problem on the sphere S2 and considering the logarithmic poten-tial, 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 uni-formly with respect to the Z-axis) and a (N + 1)-th primary of mass M is an element of R fixed at the south pole of S2. Such a particular configuration will be called ring-pole configuration (RP). An infinitesimal mass particle has an equilib-rium 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 >= 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.