Chaos in motion around black holes

The geodesic motion around Kerr black holes is regular, but this may change when an additional source is present, even if the space-time symmetry were not lowered. In this thesis we study the eect of a simple (static and axially symmetric) additional source on geodesic dynamics in the eld of a Schwarzschild black hole. In the static case the complete space-time can be described by an exact solution of Einstein equations thanks to a relatively simple superposition of the central-black-hole and the external-source metrics. Following the astrophysical motivation, we will specically consider, as the external source, the thin ring (linear source) of the Bach-Weyl type and innite thin discs (planar sources) of several types (inverted counter-rotating discs of Morgan and Morgan and the discs with power-law radial shape of density). The results may be relevant e.g. for a long-term behaviour of discrete sources (stars) in the eld of a very massive black hole in a galactic nucleus, surrounded by an accretion disc and/or by a massive toroid