Free motion around black holes with discs or rings: between integrability and chaos - II

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 previous paper, the rise (and decline) of geodesic chaos with ring/disc mass and position and with test particle energy was revealed on Poincaré sections, on time series of position or velocity and their power spectra, and on time evolution of the orbital `latitudinal action'. In agreement with the KAM theory of nearly integrable dynamical systems and with the results observed in similar gravitational systems in the literature, we found orbits of very different degrees of chaoticity in the phase space of perturbed fields. Here we compare selected orbits in more detail and try to classify them according to the characteristics of the corresponding phase-variable time series, mainly according to the shape of the time-series power spectra, and also applying two recurrence methods: the method of `average directional vectors', which traces the directions in which the trajectory (recurrently) passes through a chosen phase-space cell, and the `recurrence-matrix' method, which consists of statistics over the recurrences themselves. All the methods proved simple and powerful, while it is interesting to observe how they differ in sensitivity to certain types of behaviour.