  A systematic asymptotic expansion is developed for the gravitational wave
degrees of freedom of a class of expanding, vacuum Gowdy cosmological
spacetimes. In the wave map description of these models, the evolution of the
gravitational wave amplitudes defines an orbit in the target space. The
circumference of this orbit decays to zero indicating that the asymptotic
spacetime is spatially homogeneous. A prescription is given to identify the
asymptotic cosmological model for the gravitational wave degrees of freedom
using the asymptotic point in the target space. The remaining metric function
of the asymptotic cosmological model, found by solving the constraints, is
determined by an effective energy of the gravitational waves rather than from
the asymptotic point in the target space.
