Dynamical system methods are used to analyse the qualitative behavior of solutions to the semiclassical Einstein equations in Friedmann-Lemai tre-Robertson-Walker (FLRW) spacetimes. We specialise to the case of the conformally coupled, massless quantum scalar field and the electromagnetic field. If the quantum fieelds are in the conformal vacuum state we obtain a two-dimensional dynamical system. For this system we can idendify the structurally stable cases, obtain bifurcation diagrams and investigate the (Lyapunov-) stability of the Minkowski and deSitter equilibrium points. Finally we are able to compare the qualitative behavior of solutions to other matter models and in particular to the CDM model of cosmology.