Stellar dynamics in razor-thin discs with massive nuclear black holes

The bifurcations of orbit-averaged dynamics are studied in a class of razor-thin discs with central black holes. The model used here consists of a perturbed harmonic oscillator Hamiltonian augmented with a GM r potential. Through a sequence of conformal and canonical transformations, we reduce the phase-space flows of the system to a set of non-linear differential equations on a sphere. Based on the critical points of the averaged system, we classify orbit families and reveal the existence of six types of periodic motions: circular , long - and short-axis elliptical , long - and short-axis radial and inclined radial orbits. Long-axis elliptical orbits and their surrounding tubes have significant features: whilst they keep stars away from the centre, they elongate in the same direction as the density profile. These properties are helpful in the construction of self-consistent equilibria.