  To understand the observational properties of cosmological models, in
particular, the temperature of the cosmic microwave background radiation, it is
necessary to study their null geodesics. Dynamical systems theory, in
conjunction with the orthonormal frame approach, has proved to be an invaluable
tool for analyzing spatially homogeneous cosmologies. It is thus natural to use
such techniques to study the geodesics of these models. We therefore augment
the Einstein field equations with the geodesic equations, all written in
dimensionless form, obtaining an extended system of first-order ordinary
differential equations that simultaneously describes the evolution of the
gravitational field and the behavior of the associated geodesics. It is shown
that the extended system is a powerful tool for investigating the effect of
spacetime anisotropies on the temperature of the cosmic microwave background
radiation, and that it can also be used for studying geodesic chaos.
