  In this work, we present an extensive dynamical analysis of the null
geodesics in the spacetimes proposed by Casadio, Fabbri, and Mazzacurati (CFM),
which model black holes and wormholes in a Randall-Sundrum brane. Geodesic
stability is evaluated by Lyapunov and Jacobi criteria, with coinciding
results. Semistable photon spheres are found in the geometries of interest.
Bifurcations associated with the dynamical system are also investigated,
characterizing a degenerated Bogdanov-Takens bifurcation. Our results suggest
deep connections between the geometric characteristics of the CFM spacetimes
and the dynamics of null geodesics in these backgrounds.
