  Singularity Theory is used to comprehensively investigate the bifurcations of
the steady-states of the traveling wave ODEs of the cubic-quintic
Ginzburg-Landau equation (CGLE). These correspond to plane waves of the PDE. In
addition to the most general situation, we also derive the degeneracy
conditions on the eight coefficients of the CGLE under which the equation for
the steady states assumes each of the possible quartic (the quartic fold and an
unnamed form), cubic (the pitchfork and the winged cusp), and quadratic (four
possible cases) normal forms for singularities of codimension up to three.
Since the actual governing equations are employed, all results are globally
valid, and not just of local applicability. In each case, the recognition
problem for the unfolded singularity is treated. The transition varieties, i.e.
the hysteresis, isola, and double limit curves are presented for each normal
form. For both the most general case, as well as for various combinations of
coefficients relevant to the particular cases, the bifurcations curves are
mapped out in the various regions of parameter space delimited by these
varieties. The multiplicities and interactions of the plane wave solutions are
then comprehensively deduced from the bifurcation plots in each regime, and
include features such as regimes of hysteresis among co-existing states,
domains featuring more than one interval of hysteresis, and isola behavior
featuring dynamics unrelated to the primary solution branch in limited ranges
of parameter space.
