  We continue the study of time-like geodesic dynamics in exact static, axially
and reflection symmetric space-times describing the fields of a Schwarzschild
black hole surrounded by thin discs or rings. In the first paper of this
series, the rise (and decline) of geodesic chaos with ring/disc mass and
position and with test particle energy was revealed on Poincar\'e sections and
on time series of position or velocity and their power spectra. In the second
paper we compared these results with those obtained by two recurrence methods,
focusing on "sticky" orbits whose different parts show different degrees of
chaoticity. Here we complement the analysis by using several Lyapunov-type
coefficients which quantify the rate of orbital divergence. After comparing the
results with those obtained by the previous methods, we specifically consider a
system involving a black hole surrounded by a small thin disc or a large ring,
having in mind the configuration which probably occurs in galactic nuclei.
Within the range of parameters which roughly corresponds to our Galactic
center, we found that the black-hole accretion disc does not have a significant
gravitational effect on the dynamics of free motion at larger radii, while the
inner circumnuclear molecular ring (concentrated above 1 parsec radius) can
only induce some irregularity in motion of stars ("particles") on smaller radii
if its mass reaches 10 to 30% of the central black hole (which is the upper
estimate given in the literature), if it is sufficiently compact (which does
not hold but maybe for its inner rim) and if the stars can get to its close
vicinity. The outer dust ring between 60 and 100 parsecs appears to be less
important for the geodesic dynamics in its interior.
