  We discuss the structure of beta functions as determined by the recursive
nature of Dyson--Schwinger equations turned into an analysis of ordinary
differential equations, with particular emphasis given to quantum
electrodynamics. In particular we determine when a separatrix for solutions to
such ODEs exists and clarify the existence of Landau poles beyond perturbation
theory. Both are determined in terms of explicit conditions on the asymptotics
for the growth of skeleton graphs.
