  Recent work by the authors led to the development of a mathematical theory
dealing with `second--order hyperbolic Fuchsian systems', as we call them. In
the present paper, we adopt a physical standpoint and discuss the implications
of this theory which provides one with a new tool to tackle the Einstein
equations of general relativity (under certain symmetry assumptions).
Specifically, we formulate the `Fuchsian singular initial value problem' and
apply our general analysis to the broad class of vacuum Gowdy spacetimes with
spatial toroidal topology. Our main focus is on providing a detailed
description of the asymptotic geometry near the initial singularity of these
inhomogeneous cosmological spacetimes and, especially, analyzing the asymptotic
behavior of timelike geodesics ---which represent the trajectories of freely
falling observers --- and null geodesics. In particular, we numerically
construct Gowdy spacetimes which contain a black hole--like region together
with a flat Minkowski--like region. By using the Fuchsian technique, we
investigate the effect of the gravitational interaction between these two
regions and we study the unexpected behavior of geodesic trajectories within
the intermediate part of the spacetime limited by these two regions.
