  A mathematically and physically sound three-degree-of-freedom dynamical model
that emulates low- to high-confinement mode (L--H) transitions is elicited from
a singularity theory critique of earlier fragile models. We construct a smooth
map of the parameter space that is consistent both with the requirements of
singularity theory and with the physics of the process. The model is found to
contain two codimension 2 organizing centers and two Hopf bifurcations, which
underlie dynamical behavior that has been observed around L-H transitions but
not mirrored in previous models. The smooth traversal of parameter space
provided by this analysis gives qualitative guidelines for controlling access
to H-mode and oscillatory regimes.
