  Geodesic dynamics is regular in the fields of isolated stationary black
holes. However, due to the presence of unstable periodic orbits, it easily
becomes chaotic under various perturbations. Here we examine what amount of
chaoticity is induced in Schwarzschild space-time by a presence of an
additional source. Following astrophysical motivation, we specifically consider
thin rings or discs lying symmetrically around the hole, and describe the total
field in terms of exact static and axially symmetric solutions of Einstein's
equations. The growth of chaos in time-like geodesic motion is illustrated on
Poincar\'e sections, on time series of position or velocity and their Fourier
spectra, and on time evolution of the orbital `latitudinal action'. The results
are discussed in dependence on the mass and position of the ring/disc and on
geodesic parameters (energy and angular momentum). In the Introduction, we also
add an overview of the literature.
