  We study singularity formation in spherically symmetric solutions of the
charge-one and charge-two sector of the (2+1)-dimensional S^2 sigma-model and
the (4+1)-dimensional Yang-Mills model, near the adiabatic limit. These
equations are non-integrable, and so studies are performed numerically on
rotationally symmetric solutions using an iterative finite differencing scheme
that is numerically stable. We evaluate the accuracy of predictions made with
the geodesic approximation. We find that the geodesic approximation is
extremely accurate for the charge-two sigma-model and the Yang-Mills model,
both of which exhibit fast blowup. The charge-one sigma-model exhibits slow
blowup. There the geodesic approximation must be modified by applying an
infrared cutoff that depends on initial conditions.
