In spatially homogenous cosmologies with a massive scalar field, oscillations enter the Einstein-matter system via the Klein-Gordon equation. These oscillations are generally nontrivial to control, hindering tasks such as finding the global past and future asymptotic solutions of these systems. At the hand of locally rotationally symmetric Bianchi types I, II and III, I will introduce the idea to view these systems as periodic perturbation problems, and motivate the application of averaging methods.

The standard theory of averaging in nonlinear dynamical systems deals with problems where the perturbation is controlled by a constant parameter. In the presented case the perturbation is controlled by the Hubble scalar, a time dependent quantity. Since for the considered problems the Hubble scalar goes to zero as time goes to infinity, the respective full and averaged solutions converge. We can prove this, and also put forward a conjecture regarding the decay rates of the solutions to the equilibrium, for which I present numerical and analytical support. Under the premise that the conjecture hold one can then find the future stable solution of the full system by finding that of the simpler averaged system.