  An idea which has been around in general relativity for more than forty years
is that in the approach to a big bang singularity solutions of the Einstein
equations can be approximated by the Kasner map, which describes a succession
of Kasner epochs. This is already a highly non-trivial statement in the
spatially homogeneous case. There the Einstein equations reduce to ordinary
differential equations and it becomes a statement that the solutions of the
Einstein equations can be approximated by heteroclinic chains of the
corresponding dynamical system. For a long time progress on proving a statement
of this kind rigorously was very slow but recently there has been new progress
in this area, particularly in the case of the vacuum Einstein equations. In
this paper we generalize some of these results to the Einstein-Maxwell
equations. It turns out that this requires new techniques since certain
eigenvalues are in a less favourable configuration in the case with a magnetic
field. The difficulties which arise in that case are overcome by using the fact
that the dynamical system of interest is of geometrical origin and thus has
useful invariant manifolds.
