Closed, spatially homogeneous cosmological models with a perfect fluid and
a scalar field with an exponential potential are investigated, using dynami
cal systems methods. First, we consider the closed Friedmann-Robertson-Walk
er models, discussing the global dynamics in detail. Next, we investigate K
antowski-Sachs models, for which the future and past attractors are determi
ned. The global asymptotic behavior of both the Friedmann-Robertson-Walker
and the Kantowski-Sachs models is that they either expand from an initial s
ingularity, reach a maximum expansion and thereafter recollapse to a final
singularity (for all values of the potential parameter kappa), or else they
expand forever towards a flat power-law inflationary solution (when kappa(
2) <2). As an illustration of the intermediate dynamical behavior of the Ka
ntowski-Sachs models, we examine the cases of no baryotropic fluid, and of
a massless scalar field in detail. We also briefly discuss Bianchi type IX
models.