  Phase space method provides a novel way for deducing qualitative features of
nonlinear differential equations without actually solving them. The method is
applied here for analyzing stability of circular orbits of test particles in
various physically interesting environments. The approach is shown to work in a
revealing way in Schwarzschild spacetime. All relevant conclusions about
circular orbits in the Schwarzschild-de Sitter spacetime are shown to be
remarkably encoded in a single parameter. The analysis in the rotating Kerr
black hole readily exposes information as to how stability depends on the ratio
of source rotation to particle angular momentum. As a wider application, it is
exemplified how the analysis reveals useful information when applied to motion
in a refractive medium, for instance, that of optical black holes.
