  In this paper we have used the dynamical systems analysis to study the
dynamics of a five-dimensional universe in the form of a warped product
spacetime with a spacelike dynamic extra dimension. We have decomposed the
geodesic equations to get the motion along the extra dimension and have studied
the associated dynamical system when the cross-diagonal element of the Einstein
tensor vanishes, and also when it is non-vanishing. In the first case,
introducing the concept of an energy function along the phase path in terms of
the extra-dimensional coordinate, we have examined how the energy function
depends on the warp factor. The energy function has been used as a measure of
the amount of perturbation caused by a brane displacement. Geometrically the
effect of brane displacement is manifested in terms of a coordinate translation
along the extra dimension, thereby producing a change in the geodesic motion
along the extra dimension in the region close to the brane. Then we studied the
geodesic motion under a conventional metric perturbation in the form of
homothetic motion and conformal motion and examined the nature of critical
points for a Mashhoon-Wesson-type metric. Finally we investigated the motion
for null and timelike geodesics under the condition when the cross-diagonal
element of the Einstein tensor is non-vanishing and examined the effects of
perturbation on the critical points of the dynamical system.
