  We investigate test particle trajectories in warped spacetimes with a thick
brane warp factor, a cosmological on--brane line element and a time dependent
extra dimension. The geodesic equations are reduced to a first order autonomous
dynamical system. Using analytical methods, we arrive at some useful general
conclusions regarding possible trajectories. Oscillatory motion, suggesting
confinement about the location of the thick brane, arises for a growing warp
factor. On the other hand, we find runaway trajectories (exponential-like) for
a decaying warp factor. Variations of the extra dimensional scale factor yield
certain quantitative differences. Results obtained from explicit numerical
evaluations match well with the qualitative conclusions obtained from the
dynamical systems analysis.
