  The results of the renormalization group are commonly advertised as the
existence of power law singularities near critical points. The classic
predictions are often violated and logarithmic and exponential corrections are
treated on a case-by-case basis. We use the mathematics of normal form theory
to systematically group these into universality families of seemingly unrelated
systems united by common scaling variables. We recover and explain the existing
literature and predict the nonlinear generalization for the universal
homogeneous scaling functions. We show that this procedure leads to a better
handling of the singularity even in classic cases and elaborate our framework
using several examples.
