Evolution of the Bianchi type I, Bianchi type III, and the Kantowski-Sachs universe: Isotropization and inflation

We study the Einstein-Klein-Gordon equations for a convex positive potential in a Bianchi type I, a Bianchi type III, and a Kantowski-Sachs universe. After analyzing the inherent properties of the system of differential equations, the study of the asymptotic behaviors of the solutions and their stability is done for an exponential potential. The results are compared with those of Burd and Barrow. In contrast with their results, we show that for the Bianchi type I case isotropy can be reached without inflation and we find new critical points which lead to new exact solutions. On the other hand, we recover the result of Burd and Barrow, that if inflation occurs, then isotropy is always reached. The numerical integration is also done and all the asymptotical behaviors are confirmed.