Closed cosmologies with a perfect fluid and a scalar field

Closed, spatially homogeneous cosmological models with a perfect fluid and a scalar field with an exponential potential are investigated, using dynamical systems methods. First, we consider the closed Friedmann-Robertson-Walker models, discussing the global dynamics in detail. Next, we investigate Kantowski-Sachs models, for which the future and past attractors are determined. The global asymptotic behavior of both the Friedmann-Robertson-Walker and the Kantowski-Sachs models is that they either expand from an initial singularity, 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 ${\kappa }^2<2).\mathrm{}$ As an illustration of the intermediate dynamical behavior of the Kantowski-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.