Motion around a monopole + ring system - I. Stability of equatorial circular orbits versus regularity of three-dimensional motion

We study the motion of test particles around a centre of attraction represented by a monopole (with and without spheroidal deformation) surrounded by a ring, given as a superposition of Morgan and Morgan discs. We deal with two kinds of bounded orbits: (i) equatorial circular orbits and (ii) general three-dimensional orbits. The first case provides a method to perform a linear stability analysis of these structures by studying the behaviour of vertical and epicyclic frequencies as functions of the mass ratio, the size of the ring and/or the quadrupolar deformation. In the second case, we study the influence of these parameters in the regularity or chaoticity of motion. We find that there is a close connection between linear stability (or instability) of equatorial circular orbits and regularity (or chaoticity) of the three-dimensional motion.