  In this article, the phenomenon of delayed Hopf bifurcations (DHB) in
reaction-diffusion PDEs is analyzed in the cubic Complex Ginzburg-Landau
equation with a slowly-varying parameter. We use the classical asymptotic
methods of stationary phase and steepest descents to show that solutions which
approach the attracting quasi-steady state (QSS) before the Hopf bifurcation
remain near that state for long times after the Hopf bifurcation and the QSS
has become repelling. In the complex time plane, the phase function of the
linear PDE has a saddle point, and the Stokes and anti-Stokes lines are central
to the asymptotics. The nonlinear terms are treated by applying an iterative
method to the mild form of the PDE given by perturbations about the linear
particular solution. This tracks the closeness of solutions near the attracting
and repelling QSS. Next, we show that beyond a key Stokes line through the
saddle there is a space-time buffer curve along which the particular solution
of the linear PDE ceases to be exponentially small, causing the solution of the
nonlinear PDE to diverge from the repelling QSS and exhibit large-amplitude
oscillations. The homogeneous solution also stops being exponentially small in
a spatially dependent manner, as determined also by the initial time. We find
four different cases of DHB, depending on the competition between the
homogeneous and particular solutions, and we quantify how these depend on
system parameters. Examples are presented for each case, with uni-modal,
spatially-periodic, smooth step, and algebraically-growing source terms. Also,
rich spatio-temporal dynamics are observed in the post-DHB oscillations.
Finally, it is shown that large-amplitude source terms can be designed so that
solutions spend substantially longer times near the repelling QSS, and hence
region-specific control over the delayed onset of oscillations can be achieved.
