# Delayed Hopf bifurcation and space-time buffer curves in the Complex
Ginzburg-Landau equation

Ryan Goh  Department of Mathematics and Statistics, Boston University, Boston,
MA 02215, USA Tasso J. Kaper  Department of Mathematics and Statistics, Boston
University, Boston, MA 02215, USA Theodore Vo  School of Mathematics, Monash
University, Clayton, Victoria 3800, Australia

###### Abstract

In this article, the recently-discovered phenomenon of delayed Hopf
bifurcations (DHB) in reaction-diffusion PDEs is analyzed in the cubic Complex
Ginzburg-Landau equation, as an equation in its own right, with a slowly-
varying parameter. We begin by using the classical asymptotic methods of
stationary phase and steepest descents to show that solutions which have
approached the attracting quasi-steady state (QSS) before the Hopf bifurcation
remain near that state for long times after the instantaneous 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 curve in the space-time plane
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. This curve is
called the space-time buffer curve. The homogeneous solution also stops being
exponentially small in a spatially dependent manner, as determined also by the
initial time. Hence, a competition arises between these two solutions, as to
which one ceases to be exponentially small first, and this competition governs
spatial dependence of the DHB. We find four different cases of DHB, depending
on the outcomes of the competition, and we quantify to leading order how these
depend on the main system parameters, including the Hopf frequency, initial
time, initial data, source terms, and diffusivity. Examples are presented for
each case, with source terms that are uni-modal, spatially-periodic, smooth
step function, and algebraically-growing. 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.

Keywords Slow passage through Hopf bifurcation, dynamic bifurcation in PDEs,
spatially-inhomogeneous onset of oscillations, hard onset of oscillations,
stationary phase method, steepest descents method, complex Ginzburg-Landau
equation, reaction-diffusion equations

##  1 Introduction

In applied mathematics, physics, biology, and in many other areas of science
and engineering, the phenomenon of delayed Hopf bifurcation (DHB) is a central
feature of analytic ordinary differential equations (ODEs) in which a
parameter passes slowly through a Hopf point. ODE models and experimental
examples arise in chemistry and pattern formation [9, 22, 23, 38, 39, 61],
nonlinear mechanical oscillators and generalised Rayleigh oscillators [13, 51,
52], electrical engineering [29, 67], fluid dynamics and geophysics [2, 21,
32, 44], neuroscience [7, 10, 11, 12, 16, 25, 26, 35, 55, 56, 59, 62, 63],
cardiac models [40], and the Kaldor model in business [27].

In DHB for ODEs, the key system parameter passes slowly in time through a Hopf
bifurcation value at which the stable equilibrium becomes unstable, yet the
solutions remain near the repelling equilibrium for long times, of length
$\mathcal{O}(1)$ in the slow time, after the Hopf point. As a result, in the
super-critical case, the attendant (post-DHB) onset of oscillations is a hard
onset, with solutions jumping rapidly away from the unstable equilibrium to
the stable limit cycle, which by that time already has a large amplitude. DHB
has been studied in analytic ODEs for more than 50 years, going back at least
to the seminal work [58]. The theory is further developed in [5, 22, 31, 33,
47, 48, 62], and is also presented in recent monographs [41, 66]. Moreover,
many of the above applications have been inspired by [5, 47, 48].

DHB has also been studied [11] in large systems of ODEs. There, the FitzHugh-
Nagumo and Hodgkin-Huxley cable equations are studied with a slowly varying
Neumann boundary condition at one end of the spatial domain and a zero-flux
condition at the other end. The spatial variable in these partial differential
equations (PDEs) is discretised (with centered finite differences for the
Laplacian), and the WKB method is used to analyze DHB in the large system of
ODEs.

Recently, it was discovered that the phenomenon of DHB also occurs in
reaction-diffusion equations [36]. In that work, it was shown using numerics,
physical considerations, and some Fourier analysis that DHB is important for a
variety of reaction-diffusion equations in which there is slow passage through
super-critical Hopf bifurcations. The reaction-diffusion examples in which DHB
has been found [36] include the Complex Ginzburg-Landau equation, the
Brusselator model of the Belousov-Zhabotinsky reaction, the Hodgkin-Huxley
PDE, the FitzHugh-Nagumo PDE, and a spatially-extended pituitary lactotroph
cell model.

There has been rigorous analysis in [3] of spatio-temporal canards and delayed
bifurcations in a class of infinite-dimensional systems on bounded domains,
which includes slow passage through Hopf bifurcation, slow passage through a
Turing bifurcation, and some bifurcations in delay-differential equations. In
that article, under the assumption that a spectral gap exists in the fast sub-
system, a center manifold analysis is performed using the infinite-dimensional
invariant manifold theory of Haragus and Iooss [30]. See also the references
in [3, 65] for more on spatio-temporal canards.

In this article, we use the methods of stationary phase and steepest descents
to analyze the DHB created when the bifurcation parameter $\mu$ slowly
increases through a supercritical Hopf bifurcation (at zero) in the Complex
Ginzburg Landau equation on the real line,

$\begin{split}A_{t}&=(\mu+i\omega_{0})A-(1+i\alpha)|A|^{2}A+\operatorname{\varepsilon}^{\beta}I_{a}(x)+\operatorname{\varepsilon}^{\gamma}dA_{xx}\\\
\mu_{t}&=\operatorname{\varepsilon}.\end{split}$ (1.1)

Here, $x$ is real, $t\geq 0$, $A=A(x,t)$ is complex-valued, and
$0<\operatorname{\varepsilon}\ll 1$ is a small parameter. The linear growth
rate $\mu=\mu(t)$ is real for the main phenomena we study; however, for the
mathematical analysis, it will be advantageous to consider complex values of
$\mu$ in a horizontal strip with mid-line on the real axis and of sufficient
height. The system parameters satisfy $\omega_{0}>0$ and $\mathcal{O}(1)$
independent of $\operatorname{\varepsilon}$, $\alpha$ is real, $\beta>0$,
$\gamma\geq 0$, $d$ may be complex-valued ($d=d_{R}+id_{I}$) with $d_{R}>0$,
and they are independent of $\operatorname{\varepsilon}$. For real values of
$x$, the source term $I_{a}(x)$, which breaks the symmetry $A\to Ae^{i\theta}$
for any real $\theta$ of the CGL equation, is typically taken to be bounded
and positive, with uniformly bounded derivatives. The initial data at
$\mu(0)=\mu_{0}<0$ is $A(x,0)=A_{0}(x)$, and typically taken to be bounded and
continuous for all real $x$. Also, it will be useful to distinguish between
initial data given at $\mu_{0}\leq-\omega_{0}$ and data given at
$-\omega_{0}<\mu_{0}<0$.

The PDE (1.1) has an attracting Quasi-Steady State (QSS) for all
$\mu<-\delta$, where $\delta>0$, small, and $\mathcal{O}(1)$, which solutions
approach at an exponential rate. Similarly, it has a repelling QSS for all
$\mu>\delta$, from which solutions diverge at an exponential rate. For
example, in the base case of $\beta=\frac{1}{2}$ and $\gamma=1$, the
attracting QSS (for $\mu<-\delta$) and the repelling QSS (for $\mu>\delta$)
are given by

$A_{\rm
QSS}(x,\mu)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{a}(x)}{\mu+i\omega_{0}}+\operatorname{\varepsilon}^{\frac{3}{2}}\left(\frac{I_{a}(x)+d(\mu+i\omega_{0}){I_{a}}^{\prime\prime}(x)}{(\mu+i\omega_{0})^{3}}-\frac{(1+i\alpha)I_{a}^{3}(x)}{(\mu+i\omega_{0})^{2}(\mu^{2}+\omega_{0}^{2})}\right)+\mathcal{O}(\operatorname{\varepsilon}^{\frac{5}{2}}).$
(1.2)

Here, the $\mathcal{O}(\operatorname{\varepsilon}^{\frac{5}{2}})$ terms depend
on $x$ and $\mu$. The QSS may also be derived for other $\beta$ and $\gamma$.

In [36], DHB is observed for solutions which are in a fixed $\mathcal{O}(1)$
neighbourhood of the attracting QSS for any $\mu_{0}$ sufficiently negative.
These solutions all continue to approach the attracting QSS until $\mu=0$,
where the instantaneous Hopf bifurcation occurs. However, rather than
immediately tracking the stable (post-DHB) oscillatory state as it grows in
amplitude, these solutions remain near the repelling QSS for long times into
$\mu>0$. Moreover, the amount of time any such solution $A(x,t)$ spends near
the repelling QSS can, and generally does, depend on $x$. See Figure 1.

Figure 1:  DHB in the PDE (1.1) in the base case of $\beta=\frac{1}{2}$ and
$\gamma=1$, with a Gaussian source term $I_{a}(x)=e^{-\frac{x^{2}}{4}}$ on the
domain $[-\ell,\ell]$ ($\ell=30$). The solution is near the attracting QSS
while $\mu\in[\mu_{0},0)$ (with $\mu_{0}=-1$, not shown). Then, for $\mu>0$,
i.e., after the instantaneous Hopf bifurcation, the solution stays near the
repelling QSS (green state) at least until $\mu=\omega_{0}=0.5$ at all points
$x$. The duration of the DHB (i.e., the length of time the solution stays near
the repelling QSS) is spatially dependent. At $x=0$, $A(x,\mu)$ first leaves a
neighbourhood of the repelling QSS at time $\mu=\omega_{0}$, and the large-
amplitude oscillations first set in there. Then, as one steps outward from the
center, with $|x|>0$, the duration of the DHB grows beyond $\mu=\omega_{0}$.
The hard onset of the large-amplitude oscillations is governed to leading
order by the space-time buffer curve studied in this article. Here,
$\operatorname{\varepsilon}=0.01,\omega_{0}=\tfrac{1}{2},d_{R}=1,d_{I}=0,$ and
$\alpha=0$.

Our first goal in this article is to derive a general formula for the space-
time buffer curve of system (1.1) with general source terms $I_{a}(x)$. The
space-time buffer curve corresponds to the $x$-dependent (post-DHB) time at
which the solution cannot remain near the repelling QSS any longer for each
point $x$, irrespective of how far before the slowly-varying Hopf point the
solution was attracted to the (pre-Hopf) stable QSS. We directly to leading
order apply the classical methods of stationary phase and steepest descents
(see for example [8, 37, 46]) to the linear CGL equation, obtained by
linearising (1.1) about $A=0$. The coefficient $(\mu+i\omega_{0})$ on the
linear homogeneous term vanishes at $\mu=-i\omega_{0}$ in the complex $\mu$
plane. This is a saddle point of the complex phase
$-\frac{1}{2}(\mu+i\omega_{0})^{2}$, since the derivative of the phase
vanishes there. Moreover, the lines of stationary phase of the linear PDE
through this saddle point, along which the real part of the phase vanishes,
are given by $\mu_{I}=\pm\mu_{R}-\omega_{0}$, where $\mu=\mu_{R}+i\mu_{I}$. In
the vicinity of the saddle, analysis along the relevant Stokes and anti-Stokes
lines shows that all solutions with initial data at $\mu_{0}\leq-\omega_{0}$
stay near the attracting QSS while $\mu$ is negative and then, also, that they
stay near the repelling QSS at all points $x$ at least until $\mu$ reaches
$+\omega_{0}$, to leading order. More importantly, application of the methods
of stationary phase and steepest descents yields the formula for the space-
time buffer curve, $(\mu_{\rm stbc}(x),x)$ with $\mu_{\rm
stbc}(x)\geq\omega_{0}$ for all $x$, for the general class of source terms
considered here. This space-time buffer curve represents the $x$-dependent
times at which the particular solution of the linearised PDE ceases to be
exponentially small, and at which the hard onset of oscillations occurs, to
leading order. Similarly, there is a homogeneous exit time curve,
$\mu_{h}(x)$, along which the homogeneous component of the PDE has magnitude
one. Therefore, it is important to determine, at each point $x$, which of the
two times $\mu_{\rm stbc}(x)$ and $\mu_{h}(x)$ is smaller, causing the
solution of the full cubic PDE to stop being exponentially small first. The
smaller of the two times marks the duration of the DHB, to leading order.

After completing this first goal, we study how the properties of DHB depend on
the outcome of this competition between $\mu_{\rm stbc}(x)$ and $\mu_{h}(x)$,
as well as on the properties of $I_{a}(x)$. For solutions with initial data
given at any $\mu_{0}\leq-\omega_{0}$, we identify three cases of DHB: one in
which the particular solution determines the onset of oscillations at all $x$
in the domain; another case in which the time of onset is determined by the
particular solution for some intervals on the domain and by the homogeneous
solution on the complementary intervals; and, a third case in which the
homogeneous solution determines the exit time at all points. We examine
spatially uni-modal, spatially periodic, and smoothed step function source
terms. In these examples, quantitative agreement is found between the leading
order analysis in all cases of DHB and the results of direct numerical
simulations of (1.1) with zero-flux boundary conditions on $[-\ell,\ell]$ and
all values of $\ell$ for which simulations were conducted. Balanced symmetric
Strang operator splitting [60] was used, with centered finite differences for
the Laplacian and fourth-order Runge-Kutta for the time-stepping.

In addition, we show that solutions of (1.1) can exhibit spatially-dependent
DHB also when the initial data is given at times
$\mu_{0}\in(-\omega_{0},-\delta]$ for some small $\delta>0$. We label this as
Case 4 of DHB. For these solutions with initial data given at times much
closer to the time of the instantaneous Hopf bifurcation, the spatial
dependence of $\mu_{h}(x)$ causes the exit time from a neighborhood of the
repelling QSS to be asymmetric about $\mu=0$ with respect to the time of entry
into a neighborhood of the attracting QSS. This contrasts with the dynamics of
DHB in analytic ODEs, where the entry-exit function (also known as the way-in
way-out function) is symmetric for initial conditions given close to the
instantaneous Hopf bifurcation.

After having carried out the analysis of DHB in the above four cases, we use a
formal analysis to show that solutions of the full, cubic CGL equation (1.1)
with initial data given at time $\mu_{0}\leq-\omega_{0}$ are also close to the
attracting QSS (now of the full nonlinear PDE) on $\mu<0$ and remain close to
the repelling QSS after the instantaneous Hopf bifurcation ($\mu=0$) at least
until $\mu=\omega_{0}$ in Case 1 of DHB. The nonlinear analysis is carried out
using the integral form of the equation governing the difference between the
solution of the full nonlinear PDE and the particular solution of the linear
PDE. Use of an iterative method then establishes the closeness to the
repelling QSS in the cubic PDE, and it reveals how the asymptotic expansion of
the QSS is naturally generated. The main result is that, to leading order,
solutions of the cubic CGL (1.1) stay near the repelling QSS until the same
space-time buffer curve in Case 1 of DHB. Moreover, we note that, the
situation here for DHB in the CGL PDE is similar in this respect to that for
DHB in analytic ODEs, where the linear problem determines the buffer point to
leading order in the ODEs, and the nonlinear terms in the analytic ODE (such
as the cubic term $-|A|^{2}A$) only contribute at higher order to DHB.

Finally, we extend the main DHB results for the base case of the PDE (1.1) in
several directions. The simplest extension is to take into account the higher
order terms in the instantaneous Hopf bifurcation curve for the base case. To
leading order, this curve is given by $\mu=0$ in the space-time plane. The
first non-zero correction occurs at $\mathcal{O}(\operatorname{\varepsilon})$,
and we will study its impact on DHB. As a second extension, we study the DHB
also in the base case but now with source terms which do not satisfy the
hypotheses imposed on $I_{a}(x)$ in the general analysis, namely with an
algebraically-growing and a sign-changing function. Nevertheless, for each of
these sources, we also find good agreement between the analytically calculated
space-time buffer curve and the numerically calculated spatially-dependent
times at which the solutions leave a neighborhood of the repelling QSS and the
oscillations set in for the nonlinear PDE (1.1).

In a third direction, we extend the analysis of the base case to
asymptotically large-amplitude ($\beta=-\frac{1}{2}$) source terms in the PDE
(1.1), while retaining small diffusivity ($\gamma=1$). Here, the QSS are
highly-nontrivial, and the instantaneous Hopf bifurcation times are spatially
dependent, instead of being homogeneous at $\mu=0$, to leading order. We
explore the more complex spatial dependence of the Hopf bifurcation curve and
the space-time buffer curve. An example shows that it is possible to choose
the amplitude and form of the large source term (and hence of the resulting
QSS) to design even more complex spatio-temporal onset of oscillations, giving
region-specific control over the onset of oscillations. In a fourth direction,
we briefly extend the analysis of the base case to an example with
$\mathcal{O}(1)$ diffusivity and $\mathcal{O}(1)$ amplitude source term,
(i.e., $\gamma=0$ and $\beta=0$ in (1.1)). The space-time buffer curve gets
somewhat flattened out compared to the case of
$\mathcal{O}(\operatorname{\varepsilon})$ diffusivity.

While our primary motivation is to carry out this analysis of DHB in the CGL
PDE (1.1) and to derive a method that can be used on other reaction-diffusion
(R-D) systems known to exhibit DHB [36], another motivation for understanding
the phenomenon of DHB in PDEs is that some ODE models exhibiting DHB are
simplified versions or conceptual models of more complex phenomena which
involve diffusion and advection. An example is the Maasch-Saltzman ODE model
of glacial cycles [44], in which DHB is advanced as a possible mechanism for
the mid-Pleistocene transition from 40,000 year glacial cycles to
approximately 100,000 year cycles. See also [21]. Since the Maasch-Saltzman
model is a useful conceptual ODE model, one would also like to know whether or
not the corresponding PDE models, such as a more fully developed PDE model of
the Pleistocene glacial cycles, can also exhibit DHB. Otherwise, for these
problems, the phenomenon of DHB would only be of more limited interest. Along
with [3, 36], this work presents a step in that larger direction, showing that
DHB also occurs in nonlinear spatially-extended systems.

We observe that the analysis presented herein builds naturally on the results
known about DHB in analytic ODEs. In fact, in the case of $d=0$ and a
spatially-homogeneous source term $I_{a}(x)\equiv I_{a}$, system (1.1) reduces
to a prototypical form of DHB in analytic ODEs. In this case, we directly
recover the known DHB results for analytic ODEs, see [5, 31, 47, 48, 58]. The
hard onset of oscillations occurs to leading order at the buffer point
$\mu=\omega_{0}$ of the ODEs, and it is spatially uniform. An example is
provided by the Shishkova equation,
$\operatorname{\varepsilon}z_{\mu}=(\mu+i\omega_{0})z-(1+i\alpha)|z|^{2}z+\operatorname{\varepsilon}h(\mu)$
for $z(\mu;\operatorname{\varepsilon})$, with $h(-i\omega_{0})\neq 0$ and
$0<\operatorname{\varepsilon}\ll 1$. See [58]. This equation has a family of
attracting slow invariant manifolds for $\mu<-\delta$, where $\delta>0$ is
small and independent of $\operatorname{\varepsilon}$, and a family of
repelling slow invariant manifolds for $\mu>\delta$. These families of
manifolds may be extended in to the regions $\mu>0$ and $\mu<0$, respectively,
and they are exponentially close to each other on $(-\omega_{0},\omega_{0})$
to leading order. By the theory of DHB in ODEs, any solution which enters a
fixed, small neighbourhood of an attracting slow invariant manifold at a value
of $\mu<-\omega_{0}$ must exit a neighbourhood of the repelling slow invariant
manifold at $\mu=\omega_{0}$ to leading order, which is the buffer point. This
delayed loss of stability occurs at $\mu=\omega_{0}$ (to leading order) for
these solutions independently of how much time they have spent spiraling in
toward the attracting manifold before $\mu$ reaches $-\omega_{0}$. This is
because the term $\operatorname{\varepsilon}h(\mu)$ breaks the symmetry of
$z\to-z$ in the Shishkova equation and because the intersection at
$\mu=\omega_{0}$ of the Stokes line through the saddle (or nilpotent) point at
$\mu=-i\omega_{0}$ with the real axis acts as a barrier, or buffer point.
Hence, all solutions that have been attracted to that slow manifold must
diverge away from the repelling manifold along with it, irrespective of how
far back in the past they approached the attracting manifold.

Finally, we remark that, just as is observed here, lines of stationary phase
and lines of steepest ascents and descents play central roles in the
asymptotics of solutions of many linear and nonlinear ODEs and PDEs. For the
general theory of the Stokes phenomenon, see for example [8, 14, 15, 18, 20,
37, 45, 46, 50]. We follow naming conventions in [8, 37, 46]. Moreover, in
many of these equations, there are multiple components which cease to be
exponentially small, by crossing Stokes lines for example, and which
transition through modulus one to becoming large.

This article is organised as follows. In Section 2, the linear CGL equation is
studied in the base case of (1.1) with $\beta=1/2$ and $\gamma=1$,
establishing that solutions with $\mu_{0}<-\omega_{0}$ remain near the
attracting and repelling QSSs at least until $\mu=+\omega_{0}$ for all $x$. In
Section 3, the space-time buffer curve $(\mu_{\rm stbc}(x),x)$ is derived,
where $\mu_{\rm stbc}(x)\geq\omega_{0}$ for bounded, positive source terms,
showing that DHB occurs in the CGL PDE. Also, the examples of different
$I_{a}(x)$ are given to show how their extrema and spatial form (uni-modal,
periodic, smoothed step function) determine the location of the space-time
buffer curve and the dynamics of DHB. In Section 4, we examine the homogeneous
solutions and derive the homogeneous exit time curve, $\mu_{h}(x)$. This
establishes the spatially-dependent exit times caused by the homogeneous
components of the linear solutions, both when the initial data is given at a
time $\mu_{0}\leq-\omega_{0}$, and when it is given at a time
$\mu_{0}\in(-\omega_{0},0)$. In Section 5, the four cases of DHB are
classified, with examples. In Section 6, the analysis of the nonlinear PDE is
presented, establishing the closeness of the solutions to the repelling QSS of
the full cubic CGL equation. In Section 7, it is shown how the DHB of the
solutions of (1.1) is influenced by the
$\mathcal{O}(\operatorname{\varepsilon})$ terms in the time of the
instantaneous Hopf bifurcation. In Section 8, examples are presented with
algebraically-growing and sign-changing source terms, to push beyond the
analysis of the general system. Then, the impact of the large-amplitude source
terms ($\beta=-1/2$) is analyzed in Section 9. Also, the extension of the DHB
results and space-time buffer curve formula to the case of $\mathcal{O}(1)$
diffusivity ($\gamma=0$) and amplitude source term ($\beta=0$) is given in
Section 10. Conclusions and discussion are presented in Section 11.

##  2 Linear analysis for solutions with $\mu_{0}<-\omega_{0}$

In this section, we analyze the linear CGL equation, obtained by linearising
(1.1) about $A=0$, in the base case of moderate-amplitude source terms
($\beta=\frac{1}{2}$) and small-amplitude diffusivity ($\gamma=1$),

$\begin{split}A_{t}&=(\mu+i\omega_{0})A+\sqrt{\operatorname{\varepsilon}}I_{a}(x)+\operatorname{\varepsilon}dA_{xx}\\\
\mu_{t}&=\operatorname{\varepsilon}.\end{split}$

Equivalently, the system may be written as a scalar equation for $A(x,\mu)$,

$\operatorname{\varepsilon}A_{\mu}=(\mu+i\omega_{0})A+\sqrt{\operatorname{\varepsilon}}I_{a}(x)+\operatorname{\varepsilon}dA_{xx}.$
(2.1)

Here, $\mu(t)=\mu_{0}+\operatorname{\varepsilon}t$, and we focus on data for
which $\mu_{0}\leq-\omega_{0}$. The other case, with data given at a time
$\mu_{0}\in(-\omega_{0},0)$, is analyzed in Section 4.

For $0<\operatorname{\varepsilon}\ll 1$, all solutions with
$\mu_{0}\leq-\omega_{0}$ rapidly and exponentially approach an attracting QSS,
see (1.2) (which also contains the terms in the QSS for the full nonlinear
equation), since the real part of the coefficient on the linear term is
negative and stays well bounded away from zero for these $\mu<0$. In this
section, we show that the solutions with $\mu_{0}\leq-\omega_{0}$ not only
remain close to the attracting QSS until the time of the instantaneous Hopf
bifurcation, but after the parameter crosses the instantaneous Hopf
bifurcation at $\mu=0$ they remain close to the repelling QSS as well, at
least until the time $\mu=+\omega_{0}$ at all points $x$ for the functions
$I_{a}(x)$ we study. This will be shown using the classical methods of
stationary phase and steepest descents, see [8, 37, 46].

###  2.1 The homogeneous and particular solutions

Define the following new dependent variable, which is based on an integrating
factor,

$B(x,\mu)=A(x,\mu)e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}.$
(2.2)

Equation (2.1) may then be written as

$\sqrt{\operatorname{\varepsilon}}B_{\mu}=I_{a}(x)e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}+\sqrt{\operatorname{\varepsilon}}dB_{xx}.$
(2.3)

By Duhamel’s Principle, the solution consists of homogeneous and particular
components,

$B(x,\mu)=B_{h}(x,\mu)+B_{p}(x,\mu).$ (2.4)

The homogeneous component satisfies $(B_{h})_{\mu}=d(B_{h})_{xx}$ with
$B_{h}(x,\mu_{0})=A_{0}(x)$,

$B_{h}(x,\mu)=\frac{e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu_{0}+i\omega_{0})^{2}}}{\sqrt{4\pi
d(\mu-\mu_{0})}}\int_{\mathbb{R}}e^{\frac{-(x-y)^{2}}{4d(\mu-\mu_{0})}}A_{0}(y)dy.$
(2.5)

This homogeneous solution is valid (at least) for all real $\mu>\mu_{0}$,
recalling that $\mu_{0}<-\omega_{0}$, and throughout this article, we will
evaluate or estimate it on the ${\rm Re}(\mu)$-axis. Nevertheless, we note
that, with the initial data used in the examples, $B_{h}(x,\mu)$ is actually
analytic in the complex $\mu$ plane, excluding the branch point and cut. Note
that, for general initial data, $A_{0}(x)$ is Gevrey regular of order
$k=\frac{1}{2}$ on an appropriate domain implies that the homogeneous solution
$A_{h}(x,\mu)$ is analytic, by standard theory for homogeneous heat equations.
See for example [54], and recall that a function $f(z)$ is Gevrey regular of
order $k$ on a set $|z|<r$, if there exist positive constants $C_{1},C_{2}$
such that ${\rm max}_{|z|\leq r}|\frac{d^{k}f}{dz^{k}}(z)|\leq
C_{1}C_{2}^{k}(k!)^{\frac{1}{2}}$.

The particular solution satisfies the full linear PDE (2.3), but with zero
initial condition at $\mu_{0}$,

$\begin{split}B_{p}(x,\mu)&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{\mu_{0}}^{\mu}g(x,\mu-\tilde{\mu})e^{-\frac{1}{2\operatorname{\varepsilon}}(\tilde{\mu}+i\omega_{0})^{2}}d\tilde{\mu},\\\
g(x,\mu-\tilde{\mu})&=\frac{1}{\sqrt{4\pi
d(\mu-\tilde{\mu})}}\int_{\mathbb{R}}e^{\frac{-(x-y)^{2}}{4d(\mu-\tilde{\mu})}}I_{a}(y)dy.\end{split}$
(2.6)

The source terms $I_{a}(x)$ we study are such that $g(x,\mu-{\tilde{\mu}})$ is
analytic in a region of the complex plane which includes the portion of the
real axis with $\mu>{\tilde{\mu}}$, but which excludes a small neighborhood of
the branch point and cut. Then, in turn, $B_{p}(x,\mu)$ is analytic in an
appropriate region about the segment $(\mu_{0},\omega_{0})$ of the ${\rm
Re}(\mu)$-axis. For more general source terms, one needs to require that
$I_{a}(x)$ is Gevrey regular of order $k=\frac{1}{2}$ in a region containing a
segment of the ${\rm Re}(x)$-axis, sufficiently large to guarantee that
$B_{p}(x,\mu)$ is analytic in $\mu$ for ${\rm Re}(\mu)>{\rm
Re}({\tilde{\mu}})$. This follows from standard theory for the analyticity of
solutions. We refer to [6, 43, 54] for the general theory of analyticity of
solutions and Gevrey regularity of order $k$ for homogeneous and inhomogeneous
heat equations.

In the complex ${\tilde{\mu}}$ plane
(${\tilde{\mu}}={\tilde{\mu}}_{R}+i{\tilde{\mu}}_{I}$), the phase function in
the integrand of $B_{p}$ is

$-\frac{1}{2}({\tilde{\mu}}+i\omega_{0})^{2}=\phi+i\psi,\ \ {\rm where}\ \
\phi=-\frac{1}{2}\left({\tilde{\mu}}_{R}^{2}-({\tilde{\mu}}_{I}+\omega_{0})^{2}\right)\
\ {\rm and}\ \ \psi=-{\tilde{\mu}}_{R}({\tilde{\mu}}_{I}+\omega_{0}).$ (2.7)

This phase has a saddle point at ${\tilde{\mu}}=-i\omega_{0}$, and the
topography induced by this saddle will play a central role in the analysis.
The level sets of $\phi$ are hyperbolas and also known as Stokes lines. The
Stokes lines with $\phi=0$ through the saddle (which are the asymptotes of the
hyperbolas) bound the valleys and hills. They may be parametrised by
${\tilde{\mu}}_{R}$ via ${\tilde{\mu}}_{I}=\pm{\tilde{\mu}}_{R}-\omega_{0}.$
Also, the level sets of $\psi$ are hyperbolas (with asymptotes given by the
axes), and they are referred to as anti-Stokes lines. The geometry is
illustrated in Figure 2. See [8, 18, 37, 46], for example, for the general
theory of Stokes and anti-Stokes lines.

Remark. The integral involving $A_{0}(y)$ in (2.5) is the same as the integral
involving $I_{a}(y)$ in (2.6)(b), in the case when $A_{0}(y)$ is chosen as a
multiple of $I_{a}(y)$ and with $\mu_{0}$ replacing ${\tilde{\mu}}$.

###  2.2 Tracking the particular solutions near the attracting QSS

In this section, we briefly show using steepest descents that solutions with
$\mu_{0}\leq-\omega_{0}$ stay near the attracting QSS before the instantaneous
Hopf bifurcation. Although the result is of course well known, it is useful to
demonstrate briefly how the asymptotic method of steepest descents naturally
finds the asymptotic expansion of the attracting QSS.

Figure 2:  The contour $C_{a}=C_{a1}\bigcup C_{a2}$ in the complex
${\tilde{\mu}}$ plane. Here, and in other figures below, the Stokes lines are
the blue curves, and the anti-Stokes lines are the red curves. The Stokes
lines through the saddle (thick blue lines) separate the hills (unshaded) and
valleys (grey shaded) of the phase function.

Let $\delta>0$ be sufficiently small and independent of
$\operatorname{\varepsilon}$. We fix an arbitrary value of
$\mu\in(-\omega_{0},-\delta]$, and track the particular solution $B_{p}$ from
$\mu_{0}$ to the fixed value $\mu$. Let $C_{a}$ denote the contour
$C_{a1}\bigcup C_{a2}$, where $C_{a1}$ consists of the semi-infinite segment
of the steepest descent curve $\psi\equiv-\omega_{0}\mu_{0}$ from the point
$\mu_{0}$ on the real axis out to infinity ($-\infty-i\omega_{0}$) toward the
horizontal asymptote ${\tilde{\mu}}_{I}=-\omega_{0}$, and $C_{a2}$ consists of
the semi-infinite segment of the steepest ascent curve
$\psi\equiv-\omega_{0}\mu$ from infinity back up to the fixed value $\mu$ on
the real axis. Note that $C_{a2}$ crosses the Stokes line
$\mu_{I}=-\mu_{R}-\omega_{0}$ at the point
$q_{a}=-\sqrt{-\omega_{0}\mu}+i(\sqrt{-\omega_{0}\mu}-\omega_{0})$. See Figure
2.

We track $B_{p}$ along $C_{a}$ from $\mu_{0}$ to the fixed value $\mu$ on
$(-\omega_{0},-\delta]$. By (2.6), the solution is

$\begin{split}B_{p}(x,\mu)&=\mathcal{I}_{a1}+\mathcal{I}_{a2},\quad{\rm
where}\\\
\mathcal{I}_{ai}&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{C_{ai}}g(x,\mu-{\tilde{\mu}})e^{-\frac{1}{2\operatorname{\varepsilon}}({\tilde{\mu}}+i\omega_{0})^{2}}d{\tilde{\mu}},\
\ i=1,2,\qquad\mu\in(-\omega_{0},-\delta].\end{split}$ (2.8)

Directly from $\phi$, the real part of the complex phase (2.7), and the
analyticity of $g$, one finds

$|\mathcal{I}_{a1}|\leq
Ce^{-\frac{1}{2\operatorname{\varepsilon}}(\mu_{0}^{2}-\omega_{0}^{2})},\quad{\rm
for}\ {\rm some}\ C>0.$ (2.9)

The main work then is to derive the result for $\mathcal{I}_{a2}$, which we do
using two different parametrisations of $C_{2}$, explicitly here using
${\tilde{\mu}}_{R}$, and implicitly in Appendix A,

$C_{a2}:\quad{\tilde{\mu}}={\tilde{\mu}}_{R}+i{\tilde{\mu}}_{I}({\tilde{\mu}}_{R}),\quad{\rm
with}\ \
{\tilde{\mu}}_{I}({\tilde{\mu}}_{R})=-\omega_{0}\left(1-\frac{\mu}{{\tilde{\mu}}_{R}}\right).$
(2.10)

Along $C_{a2}$, ${\tilde{\mu}}_{R}$ increases from $-\infty$ to $\mu$, and
${\tilde{\mu}}_{I}$ increases from $-\omega_{0}$ to zero. Also, we observe
that, by (2.10),
$\frac{d{\tilde{\mu}}}{d{\tilde{\mu}}_{R}}=1-\frac{i\omega_{0}\mu}{{\tilde{\mu}}_{R}^{2}}$.
Hence,

$\mathcal{I}_{a2}=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{-\infty}^{\mu}g\left(x,\mu-{\tilde{\mu}}_{R}+i\omega_{0}\left(1-\frac{\mu}{{\tilde{\mu}}_{R}}\right)\right)e^{-\frac{1}{2\operatorname{\varepsilon}}\left({\tilde{\mu}}_{R}+i\frac{\omega_{0}\mu}{{\tilde{\mu}}_{R}}\right)^{2}}\left(1-\frac{i\omega_{0}\mu}{{\tilde{\mu}}_{R}^{2}}\right)d{\tilde{\mu}}_{R}.$

Now, it is useful to define

$u({\tilde{\mu}}_{R})=-\operatorname{\varepsilon}\frac{g\left(x,\mu-{\tilde{\mu}}_{R}+i\omega_{0}\left(1-\frac{\mu}{{\tilde{\mu}}_{R}}\right)\right)}{\left({\tilde{\mu}}_{R}+\frac{i\omega_{0}\mu}{{\tilde{\mu}}_{R}}\right)}\qquad
v({\tilde{\mu}}_{R})=e^{-\frac{1}{2\operatorname{\varepsilon}}\left({\tilde{\mu}}_{R}+i\frac{\omega_{0}\mu}{{\tilde{\mu}}_{R}}\right)^{2}}.$
(2.11)

Hence, using integration by parts, formula (2.6)(b) for $g$, and
$\lim_{\chi\to 0^{+}}g(x,\chi)=I_{a}(x)$, we find

$\begin{split}\mathcal{I}_{a2}&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{-\infty}^{\mu}u\frac{dv}{d{\tilde{\mu}}_{R}}d{\tilde{\mu}}_{R}\\\
&=-\sqrt{\operatorname{\varepsilon}}\frac{I_{a}(x)e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}}{(\mu+i\omega_{0})}-\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{-\infty}^{\mu}v\frac{du}{d{\tilde{\mu}}_{R}}d{\tilde{\mu}}_{R}\end{split}$
(2.12)

Proceeding to higher order, using $g_{\mu}=dg_{xx}$ and $\lim_{\chi\to
0^{+}}g_{xx}(x,\chi)={I_{a}}^{\prime\prime}(x)$, we find

$\mathcal{I}_{a2}=\left\\{-\sqrt{\operatorname{\varepsilon}}\frac{I_{a}(x)}{\mu+i\omega_{0}}+\operatorname{\varepsilon}^{\frac{3}{2}}\frac{\left[I_{a}(x)+d{I_{a}}^{\prime\prime}(x)(\mu+i\omega_{0})\right]}{(\mu+i\omega_{0})^{3}}+\mathcal{O}\left(\frac{\operatorname{\varepsilon}^{\frac{5}{2}}}{(\mu+i\omega_{0})^{5}}\right)\right\\}e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}.$
(2.13)

Summing (2.9) and (2.13) and recalling
$\mu_{0}\leq-\omega_{0}<\mu\leq-\delta$, we have

$B_{p}(x,\mu)=\mathcal{I}_{a1}+\mathcal{I}_{a2}=\left\\{-\frac{\sqrt{\operatorname{\varepsilon}}I_{a}(x)}{\mu+i\omega_{0}}+\frac{\operatorname{\varepsilon}^{\frac{3}{2}}\left[I_{a}(x)+d{I_{a}}^{\prime\prime}(x)(\mu+i\omega_{0})\right]}{(\mu+i\omega_{0})^{3}}+\mathcal{O}\left(\frac{\operatorname{\varepsilon}^{\frac{5}{2}}}{(\mu+i\omega_{0})^{5}}\right)\right\\}e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}.$
(2.14)

Finally, we translate the formula back to the $A$ equation using (2.2),

$A_{p}(x,\mu)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{a}(x)}{\mu+i\omega_{0}}+{\operatorname{\varepsilon}}^{\frac{3}{2}}\frac{\left[I_{a}(x)+d{I_{a}}^{\prime\prime}(x)(\mu+i\omega_{0})\right]}{(\mu+i\omega_{0})^{3}}+\mathcal{O}\left(\frac{\operatorname{\varepsilon}^{\frac{5}{2}}}{(\mu+i\omega_{0})^{5}}\right),\quad-\omega_{0}<\mu\leq-\delta.$
(2.15)

The first and second terms here are exactly the first and second order terms
in the asymptotic expansion of the attracting QSS for the linear CGL; cf.
(1.2), where the expansion is given for the cubic CGL. (Note that the cubic
term $-|A|^{2}A$ in (1.1) gives rise to an additional term at
$\mathcal{O}(\operatorname{\varepsilon}^{\frac{3}{2}})$ given by
$-\frac{(1+i\alpha)I_{a}^{3}(x)}{(\mu+i\omega_{0})^{2}(\mu^{2}+\omega_{0}^{2})}$
in the asymptotics of the QSS, see (1.2).) The remainder, which is uniform in
$x$, contains $\mathcal{I}_{a1}$ and the higher order terms in the asymptotic
expansion of the attracting QSS, and one may continue using integration by
parts on $\mathcal{I}_{a2}$ to derive them. The remainder terms also include
an exponentially small term coming from the attraction of the initial data to
the QSS. Therefore, we have shown that, with $\mu_{0}<-\omega_{0}$, the
particular solution $A_{p}$ is close to the attracting QSS for all
$\mu\in(-\omega_{0},-\delta]$.

###  2.3 Tracking the particular solutions near the repelling QSS

In this section, we track the solutions with initial data given at
$\mu_{0}\leq-\omega_{0}$ beyond the instantaneous Hopf bifurcation point at
$\mu=0$ into the regime where $\mu>0$. We show that for any
$\mu\in[\delta,\omega_{0}]$ these solutions are close to the repelling QSS at
all points $x$. We use the method of stationary phase, as well as steepest
descents, taking advantage of the saddle point at ${\tilde{\mu}}=-i\omega_{0}$
in (2.7).

Figure 3:  The contour $C_{r}=C_{r1}\bigcup C_{r2}\bigcup C_{r3}\bigcup
C_{r4}$ in the complex ${\tilde{\mu}}$ plane.

We fix an arbitrary value of $\mu\in[\delta,\omega_{0}]$. We consider the
contour $C_{r}=C_{r1}\bigcup C_{r2}\bigcup C_{r3}\bigcup C_{r4}$, where
$C_{r1}=[\mu_{0},-\omega_{0}]$; $C_{r2}$ is the segment of the Stokes line
${\tilde{\mu}}_{I}=-{\tilde{\mu}}_{R}-\omega_{0}$ from $-\omega_{0}$ down to
the saddle point at $-i\omega_{0}$; $C_{r3}$ consists of the segment of the
Stokes line ${\tilde{\mu}}_{I}={\tilde{\mu}}_{R}-\omega_{0}$ from the saddle
point at $-i\omega_{0}$ up to the point
$q_{r}=\sqrt{\omega_{0}\mu}+i(\sqrt{\omega_{0}\mu}-\omega_{0})$, for this
fixed value of $\mu$; and, $C_{r4}$ consists of the segment of the steepest
ascent curve $\psi=-\omega_{0}\mu$ from $q_{r}$ up to the point $\mu$. See
Figure 3.

We take any solution $B_{p}$ with $\mu_{0}\leq-\omega_{0}$ on or near the
attracting QSS, and we track it along $C_{r}$ to the fixed value $\mu$. At
that point, the solution is

$\begin{split}B_{p}(x,\mu)&=\mathcal{I}_{r1}+\mathcal{I}_{r2}+\mathcal{I}_{r3}+\mathcal{I}_{r4},\quad{\rm
where}\\\
\mathcal{I}_{ri}&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{C_{ri}}g(x,\mu-{\tilde{\mu}})e^{-\frac{1}{2\operatorname{\varepsilon}}({\tilde{\mu}}+i\omega_{0})^{2}}d{\tilde{\mu}},\
\ i=1,2,3,4,\quad\delta\leq\mu\leq\omega_{0}.\end{split}$ (2.16)

The integral $\mathcal{I}_{r1}$ along the segment
$C_{r1}=[\mu_{0},-\omega_{0}]$ may be evaluated in the same manner as used for
the integral (2.13). However, here $\mu\in[\delta,\omega_{0}]$ and the Taylor
expansion is about ${\tilde{\mu}}=-\omega_{0}$,

$\mathcal{I}_{r1}=\left\\{\frac{\sqrt{\operatorname{\varepsilon}}}{2\omega_{0}}(1+i)g(x,\mu+\omega_{0})+\mathcal{O}\left(\operatorname{\varepsilon}^{\frac{3}{2}}\right)\right\\}e^{\frac{i\omega_{0}^{2}}{\operatorname{\varepsilon}}}.$
(2.17)

Next, we parametrise $C_{r2}$ by ${\tilde{\mu}}_{R}$, with
${\tilde{\mu}}_{I}({\tilde{\mu}}_{R})=-({\tilde{\mu}}_{R}+\omega_{0})$, and
${\tilde{\mu}}_{R}:-\omega_{0}\to 0$. Hence,
$-\frac{1}{2}(\tilde{\mu}+i\omega_{0})^{2}=i{\tilde{\mu}}_{R}^{2}$ for all
${\tilde{\mu}}$ on $C_{r2}$. It is purely imaginary, corresponding to the fact
that $C_{r2}$ lies on a Stokes line $\phi=0$. Hence, for each ${\tilde{\mu}}$
on $C_{r2}$, the integrand is of the form to which the method of stationary
phase applies, namely $g\cdot
e^{\frac{i}{\operatorname{\varepsilon}}h({\tilde{\mu}}_{R})}$ with
$h({\tilde{\mu}}_{R})={{\tilde{\mu}}_{R}}^{2}$. Moreover, the end point
$\tilde{\mu}=-i\omega_{0}$ of $C_{r2}$ (${\tilde{\mu}}_{R}=0$) is a point of
stationary phase, since $h^{\prime}(0)=0$ and $h^{\prime\prime}(0)=2\neq 0$,
and it is the only such point along $C_{r2}$. [For the general method in which
an end point is a saddle (or turning) point, see for example Section 4.1 of
[46], especially formula (4.14).]

Applying the method of stationary phase, we insert the parametrisation of
$C_{r2}$, use ${\tilde{\nu}}=-{\tilde{\mu}}_{R}$, Taylor expand about
${\tilde{\nu}}=0$ (i.e., ${\tilde{\mu}}=-i\omega_{0}$), and observe that the
dominant contribution asymptotically comes from the point of stationary phase
at the saddle,

$\begin{split}\mathcal{I}_{r2}&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{-\omega_{0}}^{0}g(x,\mu-[{\tilde{\mu}}_{R}-i({\tilde{\mu}}_{R}+\omega_{0})])e^{\frac{i}{\operatorname{\varepsilon}}{\tilde{\mu}}_{R}^{2}}(1-i)d{\tilde{\mu}}_{R}\\\
&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}(1-i)\int_{0}^{\omega_{0}}g(x,\mu+[{\tilde{\nu}}+i(-{\tilde{\nu}}+\omega_{0})])e^{\frac{i}{\operatorname{\varepsilon}}{\tilde{\nu}}^{2}}d{\tilde{\nu}}\\\
&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}(1-i)g(x,\mu+i\omega_{0})\int_{0}^{\omega_{0}}(1+\mathcal{O}({\tilde{\nu}}))e^{\frac{i}{\operatorname{\varepsilon}}{\tilde{\nu}}^{2}}d{\tilde{\nu}}\\\
&=\sqrt{\frac{\pi}{2h^{\prime\prime}(0)}}e^{\frac{i\pi}{4}}(1-i)g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}})\\\
&=\sqrt{\frac{\pi}{2}}g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}}),\quad{\rm
for}\ {\rm any}\ \mu\in[\delta,\omega_{0}].\end{split}$ (2.18)

This leading order term in $\mathcal{I}_{r2}$ will turn out to be half of the
leading order term in the total integral for $B_{p}(x,\mu)$ for each
$\mu\in[\delta,\omega_{0}]$.

Next, we show that $\mathcal{I}_{r3}$ gives the other half of the leading
order term in $B_{p}$. By the definition of $C_{r3}$, for any
$\mu\in[\delta,\omega_{0}]$, we have
$\mathcal{I}_{r3}=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{-i\omega_{0}}^{q_{r}}g(x,\mu-{\tilde{\mu}})e^{-\frac{1}{2\operatorname{\varepsilon}}({\tilde{\mu}}+i\omega_{0})^{2}}d{\tilde{\mu}}.$
We also use ${\tilde{\mu}}_{R}$ to parametrise the segment $C_{r3}$ as
${\tilde{\mu}}_{I}={\tilde{\mu}}_{R}-\omega_{0}$, now with
${\tilde{\mu}}_{R}:0\to\sqrt{\omega_{0}\mu}$. Hence,
$-\frac{1}{2}(\tilde{\mu}+i\omega_{0})^{2}=-i{\tilde{\mu}}_{R}^{2}$ along
$C_{r3}$; and, for each ${\tilde{\mu}}$ on $C_{r3}$, the integral is also of
the form to which the method of stationary phase applies, namely $g\cdot
e^{\frac{i}{\operatorname{\varepsilon}}{\tilde{h}}(\tilde{\mu}_{R})}$, with
${\tilde{h}}({\tilde{\mu}_{R}})=-{\tilde{\mu}}_{R}^{2}$. Moreover, the initial
point $\tilde{\mu}=-i\omega_{0}$ of $C_{r3}$ (${\tilde{\mu}}_{R}=0$) is a
point of stationary phase, since ${\tilde{h}}^{\prime}(0)=0$ and
${\tilde{h}}^{\prime\prime}(0)=-2\neq 0$, and it is the only such point along
$C_{r3}$. We find

$\begin{split}\mathcal{I}_{r3}&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{0}^{\sqrt{\omega_{0}\mu}}g(x,\mu-[{\tilde{\mu}}_{R}+i({\tilde{\mu}}_{R}-\omega_{0})])e^{-\frac{i}{\operatorname{\varepsilon}}{\tilde{\mu}}_{R}^{2}}(1+i)d{\tilde{\mu}}_{R}\\\
&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}g(x,\mu+i\omega_{0})\int_{0}^{\sqrt{\omega_{0}\mu}}e^{-\frac{i}{\operatorname{\varepsilon}}{\tilde{\mu}}_{R}^{2}}(1+\mathcal{O}({\tilde{\mu}}_{R}))(1+i)d{\tilde{\mu}}_{R}\\\
&=\sqrt{\frac{\pi}{2|{\tilde{h}}^{\prime\prime}(0)|}}g(x,\mu+i\omega_{0})e^{-\frac{i\pi}{4}}(1+i)+\mathcal{O}(\sqrt{\operatorname{\varepsilon}})\\\
&=\sqrt{\frac{\pi}{2}}g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}}),\quad{\rm
for}\ {\rm any}\ \mu\in[\delta,\omega_{0}].\end{split}$ (2.19)

Finally, we calculate $\mathcal{I}_{r4}$. Implicitly parametrise $C_{r4}$
using $\sigma$,

$C_{r4}:\quad-\frac{1}{2}({\tilde{\mu}}+i\omega_{0})^{2}=-\frac{1}{2}(\mu+i\omega_{0})^{2}+\sigma.$

The parameter $\sigma$ starts from $-\frac{1}{2}(\omega_{0}^{2}-\mu^{2})$ at
the point $q_{r}$ and increases monotonically along $C_{r4}$ to zero at the
point $\mu$. The explicit representation is

${\tilde{\mu}}(\sigma)=-i\omega_{0}+\left[(\mu+i\omega_{0})^{2}-2\sigma\right]^{\frac{1}{2}}.$

The integration along $C_{r4}$ follows in a manner similar to that along
$C_{a2}$ in Appendix A, except that here one starts at $q_{r}$ and also here
$\mu>0$,

$\mathcal{I}_{r4}=\sqrt{\operatorname{\varepsilon}}e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\int_{-\frac{1}{2}(\omega_{0}^{2}-\mu^{2})}^{0}g(x,\mu+i\omega_{0}-[(\mu+i\omega_{0})^{2}-2\sigma]^{\frac{1}{2}})e^{\frac{\sigma}{\operatorname{\varepsilon}}}\left[(\mu+i\omega_{0})^{2}-2\sigma\right]^{-\frac{1}{2}}d\sigma.$

Then, with a similar Taylor expansion, one finds

$\mathcal{I}_{r4}=\left[-\frac{\sqrt{\operatorname{\varepsilon}}I_{a}(x)}{\mu+i\omega_{0}}+\operatorname{\varepsilon}^{\frac{3}{2}}\left(\frac{I_{a}(x)+d(\mu+i\omega_{0}){I_{a}}^{\prime\prime}(x)}{(\mu+i\omega_{0})^{3}}\right)+\mathcal{O}\left(\frac{\operatorname{\varepsilon}^{\frac{5}{2}}}{(\mu+i\omega_{0})^{5}}\right)\right]e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}.$
(2.20)

Summing (2.17), (2.18), (2.19), and (2.20), we have

$\begin{split}B_{p}(x,\mu)&=\mathcal{I}_{r1}+\mathcal{I}_{r2}+\mathcal{I}_{r3}+\mathcal{I}_{r4}\\\
&=\left[-\frac{\sqrt{\operatorname{\varepsilon}}I_{a}(x)}{\mu+i\omega_{0}}+\operatorname{\varepsilon}^{\frac{3}{2}}\left(\frac{I_{a}(x)+d(\mu+i\omega_{0}){I_{a}}^{\prime\prime}(x)}{(\mu+i\omega_{0})^{3}}\right)\right]e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\\\
&+\mathcal{O}\left(\frac{\operatorname{\varepsilon}^{\frac{5}{2}}e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}}{(\mu+i\omega_{0})^{5}}\right)+\sqrt{2\pi}g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}}),\quad{\rm
for}\ \delta\leq\mu\leq\omega_{0}.\end{split}$ (2.21)

Finally, we translate the formula back to the $A$ equation using (2.2),

$\begin{split}A_{p}(x,\mu)&=-\sqrt{\operatorname{\varepsilon}}\frac{I_{a}(x)}{\mu+i\omega_{0}}+\operatorname{\varepsilon}^{\frac{3}{2}}\left(\frac{I_{a}(x)+d(\mu+i\omega_{0}){I_{a}}^{\prime\prime}(x)}{(\mu+i\omega_{0})^{3}}\right)+\mathcal{O}\left(\frac{\operatorname{\varepsilon}^{\frac{5}{2}}}{(\mu+i\omega_{0})^{5}}\right)\\\
&+\left(\sqrt{2\pi}g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}})\right)e^{\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}},\quad{\rm
for}\ \delta\leq\mu\leq\omega_{0}.\end{split}$ (2.22)

The first and second terms are precisely the leading order terms in the
expansion of the repelling QSS for the linear CGL (cf. (1.2), where the QSSs
are given for the cubic CGL equation). The third term contains the higher
order terms in the asymptotic expansion of the repelling QSS, and continued
integration by parts will yield them.

The fourth term is exponentially small for
$\mu\in[\delta,\omega_{0}-K\operatorname{\varepsilon}^{r})$, for some $K>0$
and any $0<r<1$. It is a classic Stokes type term. This term is not in the
expansion (2.15) of the attracting QSS (on $\mu<-\delta$) to all orders or in
the expansion of the repelling QSS (on $\mu>\delta$) to all orders. Rather, it
is beyond all orders,
$\mathcal{O}\left(e^{-\frac{\omega_{0}^{2}}{2\operatorname{\varepsilon}}}\right)$,
arising naturally from tracking solutions on (and near) the attracting QSS
along a contour over the saddle point in the complex $\mu$ plane and into the
regime of ${\rm Re}(\mu)>0$. It is a measure of the exponentially small
distance between the attracting and repelling QSS at $\mu=0$. In Section 3, we
will determine when it becomes $\mathcal{O}(1)$ (and then exponentially
large).

Overall, therefore, formulas (2.15) and (2.22) give the asymptotics of
solutions $A_{p}(x,\mu)$ for all $\mu\in(-\omega_{0},-\delta]$ and all
$\mu\in[\delta,\omega_{0}]$, respectively. They show that, for all $x$, the
solutions of the linear CGL equation with Gevrey regular data $A_{0}(x)$ given
at any $\mu_{0}\leq-\omega_{0}$ are near the attracting QSS until the Hopf
bifurcation; and then, once ${\tilde{\mu}}$ has become positive, they remain
near the repelling QSS at least until ${\tilde{\mu}}=\omega_{0}$ to leading
order. This completes the analysis of this subsection. Note that the solutions
are also close to the QSS on $(-\delta,\delta)$, as shown in Appendix B.

##  3 The space-time buffer curve

In this section, we track the particular solution $B_{p}(x,\mu)$ (and hence
also $A_{p}(x,\mu)$ via (2.2)) from the initial time $\mu_{0}$, satisfying
$\mu_{0}\leq-\omega_{0}$, to an $x$-dependent, maximal value of $\mu$, beyond
$\mu=\omega_{0}$. For each $x$, this maximal value, labeled $\mu_{\rm
stbc}(x)$, denotes the space-dependent value of $\mu$ at which
$|A_{p}(x,\mu)|=1$, i.e., at which the real part of the space-time dependent
phase of $A_{p}$ first vanishes. To leading order, the space-dependent time
$A_{p}(x,\mu)$ is exponentially small for $\mu\in(\mu_{0},\mu_{\rm stbc}(x))$
and then transitions to being exponentially large for $\mu>\mu_{\rm stbc}(x)$.
Hence, at each $x$, $\mu_{\rm stbc}(x)$ is the maximum of $\mu$ for which
solutions with initial data at $\mu_{0}\leq-\omega_{0}$ can remain near the
repelling QSS. We label the union of $\mu_{\rm stbc}(x)$ over all $x$ as the
space-time buffer curve.

###  3.1 Derivation of the space-time buffer curve, $\mu_{\rm stbc}(x)$

From the result of Section 2.3, we know that the curve lies to the right of
$\mu=\omega_{0}$, since solutions with $\mu_{0}\leq-\omega_{0}$ remain close
to the repelling QSS at least until $\mu=\omega_{0}$ to leading order at all
points $x$. To track the solutions past this value, we again deform the
contour in the complex $\mu$ plane, taking advantage of the saddle point at
${\tilde{\mu}}=-i\omega_{0}$ in (2.7). Several of the calculations needed here
follow directly from those performed above.

Figure 4:  The contour $C=C_{1}\cup C_{2}\cup C_{3}\cup C_{4}$ in the complex
${\tilde{\mu}}$ plane.

Let the contour

$C=C_{1}\bigcup C_{2}\bigcup C_{3}\bigcup C_{4}$

consist of the following four segments: $C_{1}=[\mu_{0},-\omega_{0}]$, along
the negative real axis; $C_{2}$ is the segment of the Stokes line
${\tilde{\mu}}_{I}=-{\tilde{\mu}}_{R}-\omega_{0}$ from $-\omega_{0}$ down to
the saddle at $-i\omega_{0}$; $C_{3}$ is the segment of the other Stokes line
${\tilde{\mu}}_{I}={\tilde{\mu}}_{R}-\omega_{0}$ from the saddle at
$-i\omega_{0}$ up to $\omega_{0}$; and, $C_{4}=[\omega_{0},\mu_{\rm
stbc}(x))$, along the positive real axis. See Figure 4. Note that
$C_{1}=C_{r1}$ and $C_{2}=C_{r2}$.

From (2.6)(a) and the composition of the contour $C$, one finds for any
arbitrary time $\mu$ on $C_{4}$,

$\begin{split}B_{p}(x,\mu)&=\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4},\qquad{\rm
where}\ \\\
\mathcal{I}_{j}&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{C_{j}}g(x,\mu-{\tilde{\mu}})e^{-\frac{1}{2\operatorname{\varepsilon}}({\tilde{\mu}}+i\omega_{0})^{2}}d{\tilde{\mu}},\qquad
j=1,2,3,\ \\\
\mathcal{I}_{4}&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{\omega_{0}}^{\mu}g(x,\mu-{\tilde{\mu}})e^{-\frac{1}{2\operatorname{\varepsilon}}({\tilde{\mu}}+i\omega_{0})^{2}}d{\tilde{\mu}},\quad\mu\in
C_{4}.\end{split}$ (3.1)

For each $j$, the value of $\mu$ in the integrand along $C_{j}$ is the same
fixed value on $C_{4}$.

Here, since $C_{1}=C_{r1}$, one finds by recalling (2.17),

$\mathcal{I}_{1}=\mathcal{I}_{r1}=\sqrt{\operatorname{\varepsilon}}e^{\frac{i\omega_{0}^{2}}{\operatorname{\varepsilon}}}\left(\frac{1+i}{2\omega_{0}}\right)g(x,\mu+\omega_{0})+\mathcal{O}(\operatorname{\varepsilon}^{\frac{3}{2}}).$
(3.2)

Next, since $C_{2}=C_{r,2}$, one finds by recalling (2.18),

$\mathcal{I}_{2}=\mathcal{I}_{r2}=\sqrt{\frac{\pi}{2}}g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}}).$
(3.3)

Then, from the definition of $C_{3}$, we have

$\mathcal{I}_{3}=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{-i\omega_{0}}^{\omega_{0}}g(x,\mu-{\tilde{\mu}})e^{-\frac{1}{2\operatorname{\varepsilon}}({\tilde{\mu}}+i\omega_{0})^{2}}d{\tilde{\mu}}.$

This integral may be evaluated in the same manner using stationary phase as
that in $\mathcal{I}_{r3}$, except here one integrates all the way up to
$\omega_{0}$ (instead of stopping at $q_{r}$),

$\begin{split}\mathcal{I}_{3}&=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{0}^{\omega_{0}}g(x,\mu-[{\tilde{\mu}}_{R}+i({\tilde{\mu}}_{R}-\omega_{0})])e^{-\frac{i}{\operatorname{\varepsilon}}{\tilde{\mu}}_{R}^{2}}(1+i)d{\tilde{\mu}}_{R}\\\
&=\sqrt{\frac{\pi}{2}}g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}}).\end{split}$
(3.4)

The contribution along the segment $C_{4}$ is
$\mathcal{I}_{4}=\frac{1}{\sqrt{\operatorname{\varepsilon}}}\int_{\omega_{0}}^{\mu}g(x,\mu-{\tilde{\mu}})e^{-\frac{1}{2\operatorname{\varepsilon}}({\tilde{\mu}}+i\omega_{0})^{2}}d{\tilde{\mu}}.$
The contour integral may be estimated in a manner similar to that used along
$C_{1}$,

$|\mathcal{I}_{4}|\leq\mathcal{O}(\sqrt{\operatorname{\varepsilon}}).$ (3.5)

Then, substituting the results for
$\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3},$ and $\mathcal{I}_{4}$ (see
(3.2) – (3.5)) into (3.1), we find

$B_{p}(x,\mu)=\sqrt{2\pi}g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}}),\quad{\rm
for}\ {\rm any}\ \mu\in C_{4}.$ (3.6)

This represents the particular solution valid for all $\mu$ on $C_{4}$.

Finally, using the change of variables (2.2), we see from (3.6) that the
particular solution of the linear CGL equation (2.1) with
$\mu_{0}\leq-\omega_{0}$ is

$A_{p}(x,\mu)=\left(\sqrt{2\pi}g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}})\right)e^{\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}},\qquad{\rm
for}\ \ \mu\in C_{4}.$ (3.7)

Therefore, one finds that $|A_{p}(x,\mu)|=1$ to leading order along the curve
$\mu_{\rm stbc}(x)$ defined by

$\left\\{(\mu_{\rm stbc}(x),x)|\quad{\rm Re}\left(\ln(\sqrt{2\pi}g(x,\mu_{\rm
stbc}(x)+i\omega_{0}))+\frac{1}{2\operatorname{\varepsilon}}(\mu_{\rm
stbc}(x)+i\omega_{0})^{2}\right)=0\right\\}.$ (3.8)

This is the space-time buffer curve, where the real part of the argument of
the space-time-dependent phase in the exponential function in (3.7) vanishes,
to leading order. It marks the transition between $A_{p}$ being exponentially
small for $\mu\in(\mu_{0},\mu_{\rm stbc}(x))$, to it being exponentially large
for $\mu>\mu_{\rm stbc}(x)$. Moreover, to leading order, the implicit form of
the analytical formula is

$(\mu_{\rm
stbc}(x))^{2}=\omega_{0}^{2}-\operatorname{\varepsilon}\ln(2\pi)-2\operatorname{\varepsilon}\ln\left|g(x,{\mu_{\rm
stbc}}(x)+i\omega_{0})\right|.$ (3.9)

In summary, formula (3.8) defines the space-time buffer curve, and (3.9) gives
the leading order asymptotics for solutions of (2.1) with any
$\mu_{0}\leq-\omega_{0}$ and the source terms $I_{a}(x)$ considered here.

Remark. In the limit $|d|\to 0$, the PDE (1.1) reduces to a one-parameter
family of ODEs in time, in which $x$ is the parameter through $I_{a}(x)$.
Here, we briefly show that, in this limit, the space-time buffer curve of the
PDE (3.8) reduces to the buffer point of the $x$-dependent ODE. In fact, in
this limit, the fundamental solution of the heat equation approaches a delta
function, and $A_{h}(x,\mu)\to
A_{0}(x)e^{\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}-(\mu_{0}+i\omega_{0})^{2}}.$
Then, (2.6) and (3.7) imply $g(x,\mu-{\tilde{\mu}})\to I_{a}(x)$, for all
$\mathcal{O}(1)$ values $\mu-{\tilde{\mu}}$, and
$A_{p}(x,\mu)=\left(\sqrt{2\pi}I_{a}(x)+\mathcal{O}(\sqrt{\operatorname{\varepsilon}})\right)e^{\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}},\
{\rm as}\ |d|\to 0.$ Hence, at each $x$, $A_{p}(x,\mu)$ is the same as the
solution of the corresponding $x$-dependent Shishkova ODE. Therefore, at each
point, the space-time buffer curve reduces to $\omega_{0}$, which is the
buffer point of the $x$-dependent ODE. See [31, 47, 58] for the general theory
of DHB and buffer points in analytic ODEs.

###  3.2 Examples of the space-time buffer curve, $\mu_{\rm stbc}(x)$

To study the space-time buffer curve (3.8), we give three examples, involving
different types of source terms: uni-modal (Gaussian), spatially-periodic, and
smoothed step function. The first example consists of the PDE (1.1) with a
Gaussian source term,

$I_{G}(x)=e^{\frac{-x^{2}}{4\sigma}},\quad\sigma>0.$ (3.10)

Gaussian source terms are simple models for spatially-localized sources in R-D
equations, such as the amplitude of a light-source in chemical pattern
formation, the refractive index of waveguides in nonlinear optics, and
spatially-localized electrical currents applied to arrays of nerve cells.

Figure 5:  ${\rm Re}(A(x,\mu))$ obtained from (1.1) with source term
$I_{G}(x)$. In the green region, the solution lies close to the repelling QSS.
Superimposed is the black space-time buffer curve obtained by solving (3.8)
with $g$ given by (3.11). The oscillations commence just before the space-time
buffer curve, where they have small amplitude. Beyond it, the oscillations
have large amplitude, since the amplitude of the stable limit cycles is large
by this time. The maxima and minima of the oscillations propagate to the
center of the interval, toward $x=0$, where they disappear. (The phase
velocities and group velocities of the periodic waves are calculated in
Section 11.2, where the nature of the defect is studied.) The instantaneous
Hopf bifurcation occurs at $\mu=0$ to leading order at all $x$. Here,
$\operatorname{\varepsilon}=0.01,\omega_{0}=\frac{1}{2},d_{R}=3,d_{I}=1$,
$\alpha=0$, and $\sigma=\frac{1}{4}$. The initial data at $\mu_{0}=-1.5$ is
$A_{0}(x)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{G}(x)}{\mu_{0}+i\omega_{0}}$.
Examples with other initial data are presented below.

For $\mu_{0}\leq{\tilde{\mu}}<\mu$, we find from (2.6)(b) that

$g(x,\mu-\tilde{\mu})=\sqrt{\frac{\sigma}{d(\mu-\tilde{\mu})+\sigma}}\ \
e^{\frac{-x^{2}}{4(d(\mu-\tilde{\mu})+\sigma)}}.$ (3.11)

The function $g$ is analytic along the contour $C$ and in a neighbourhood of
it, except at the branch point and along the cut. Application of the general
formula (3.6) for $\mu$ on $C_{4}$ yields

$B_{p}(x,\mu)=\sqrt{\frac{2\pi\sigma}{d(\mu+i\omega_{0})+\sigma}}\ {\rm
exp}\left[\frac{-x^{2}}{4(d(\mu+i\omega_{0})+\sigma)}\right]+\mathcal{O}(\sqrt{\operatorname{\varepsilon}}).$

Translating back using (2.2), we find for all $\mu$ on $C_{4}$,

$A_{p}(x,\mu)=\left(\sqrt{\frac{2\pi\sigma}{d(\mu+i\omega_{0})+\sigma}}\ {\rm
exp}\left[\frac{-x^{2}}{4(d(\mu+i\omega_{0})+\sigma)}\right]+\mathcal{O}(\sqrt{\operatorname{\varepsilon}})\right)e^{\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}.$
(3.12)

Therefore, by condition (3.8), $\mu_{\rm stbc}(x)$ is given implicitly to
leading order by

$\mu^{2}=\omega_{0}^{2}+\frac{\operatorname{\varepsilon}x^{2}(d_{R}\mu-
d_{I}\omega_{0}+\sigma)}{2\left((d_{R}\mu-
d_{I}\omega_{0}+\sigma)^{2}+(d_{R}\omega_{0}+d_{I}\mu)^{2}\right)}-\operatorname{\varepsilon}\ln(2\pi\sigma)+\frac{\operatorname{\varepsilon}}{2}\ln((d_{R}\mu-
d_{I}\omega_{0}+\sigma)^{2}+(d_{R}\omega_{0}+d_{I}\mu)^{2}),$ (3.13)

provided

${\rm Re}\left(\frac{d(\mu+i\omega_{0})+\sigma}{d(\mu+i\omega_{0})}\right)\geq
0.$

Condition (3.8) implicitly defines the space-time buffer curve $(\mu_{\rm
stbc}(x),x)$ along which $|A_{p}|=1$ for Gaussian sources in (1.1).

Figure 5 illustrates this result. For $0<\mu<\mu_{\rm stbc}(x)$, the solution
is exponentially close to the repelling QSS. Then, at each $x$, the solution
diverges from the repelling QSS there, and the post-DHB oscillations set in,
as soon as $\mu$ is $\mu_{\rm stbc}(x)$ to leading order for that $x$.
Overall, (3.13) and the results presented in Figure 5 show that $x=0$ is the
minimum of $\mu_{\rm stbc}(x)$, and the solution first diverges from the
repelling QSS there. Then, as $|x|$ increases, $\mu_{\rm stbc}(x)$ increases,
quadratically near the tip. Hence, the duration of the DHB (i.e., the time
when the solution leaves a neighborhood of the repelling QSS) grows
quadratically with $|x|$ near the tip.

Figure 6:  ${\rm Re}(A(x,\mu))$ obtained from (1.1) with the error function
source term $I_{\rm erf}(x)$, $I_{\rm ave}=0.5$, and $I_{\rm e}=0.125$. The
black space-time buffer curve is super-imposed, showing that it gives, to
leading order, the time of onset of the oscillations at all points $x$ in the
domain. Here, the initial data at $\mu_{0}=-1$ is
$A_{0}(x)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{\rm
erf}(x)}{\mu_{0}+i\omega_{0}}$, and the parameter values are
$\operatorname{\varepsilon}=0.01,$ $\omega_{0}=0.5,$ $\alpha=0$, $d_{R}=1$,
and $d_{I}=0$.

The second example of the space-time buffer curve is given by a smoothed step
function,

$I_{\rm erf}(x)=I_{\rm ave}+I_{\rm e}{\rm erf}(x),$ (3.14)

with $I_{\rm ave}>I_{\rm e}>0$. (The error function is ${\rm
erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$, and ${\rm
erf}(-x)=-{\rm erf}(x)$.) This is a simple form for R-D systems in which there
is (approximately) piecewise constant input, with some portion of the domain
receiving one level ($I_{\rm ave}+I_{\rm e}$) and a complementary part
receiving a different level ($I_{\rm ave}-I_{\rm e}$), with a smooth
transition in between. By (2.6)(b), one finds

$g(x,\mu-{\tilde{\mu}})=I_{\rm ave}+I_{\rm e}{\rm
erf}\left(\frac{x}{\sqrt{1+4d(\mu-{\tilde{\mu}})}}\right),$ (3.15)

see for example [49], provided that the argument of the error function lies in
$(-\frac{\pi}{4},\frac{\pi}{4})$. Hence, by (3.8), the space-time buffer curve
for (1.1) with $I_{\rm erf}(x)$ is to leading order

$(\mu_{\rm
stbc}(x))^{2}=\omega_{0}^{2}-2\operatorname{\varepsilon}\ln\left(I_{\rm
ave}+I_{\rm e}{\rm erf}\left(\frac{x}{\sqrt{1+4d(\mu_{\rm
stbc}(x)+i\omega_{0})}}\right)\right)-\operatorname{\varepsilon}\ln(2\pi).$
(3.16)

See Figure 6. Small-amplitude oscillations set in just before $\mu_{\rm
stbc}(x)$. Then, at each point $x$, the amplitude of the oscillations becomes
large as soon as $\mu$ reaches $\mu_{\rm stbc}(x)$, to leading order.

The third example of the space-time buffer curve consists of a spatially
periodic source term,

$I_{\rm per}(x)=p_{1}+p_{2}\cos(x),$ (3.17)

with $p_{1}>p_{2}>0$, $\mathcal{O}(1)$ independent of
$\operatorname{\varepsilon}$. Spatially-periodic forcing arises in various
pattern formation problems, see for instance [19, 28]. From the general
definition (2.6)(b) of $g$, one finds

$g(x,\mu-{\tilde{\mu}})=\frac{1}{\sqrt{4\pi
d(\mu-{\tilde{\mu}})}}\int_{R}e^{-\frac{(x-y)^{2}}{4d(\mu-{\tilde{\mu}})}}(p_{1}+p_{2}\cos(y))dy=p_{1}+p_{2}e^{-d(\mu-{\tilde{\mu}})}\cos(x).$
(3.18)

Now, with this elementary form of $g$, the integral (2.6)(a) for
$B_{p}(x,\mu)$ may be evaluated in closed form in this example. Specifically,
recalling (2.2), we find that the particular solution is

$\begin{split}A_{p}(x,\mu)&=\sqrt{\frac{\pi}{2}}p_{1}\left[{\rm
erf}\left(\frac{\mu+i\omega_{0}}{\sqrt{2\operatorname{\varepsilon}}}\right)-{\rm
erf}\left(\frac{\mu_{0}+i\omega_{0}}{\sqrt{2\operatorname{\varepsilon}}}\right)\right]e^{\frac{(\mu+i\omega_{0})^{2}}{2\operatorname{\varepsilon}}}\\\
&+\sqrt{\frac{\pi}{2}}p_{2}\cos(x)\left[{\rm
erf}\left(\frac{\mu+i\omega_{0}-\operatorname{\varepsilon}d}{\sqrt{2\operatorname{\varepsilon}}}\right)-{\rm
erf}\left(\frac{\mu_{0}+i\omega_{0}-\operatorname{\varepsilon}d}{\sqrt{2\operatorname{\varepsilon}}}\right)\right]e^{\frac{(\mu+i\omega_{0}-\operatorname{\varepsilon}d)^{2}}{2\operatorname{\varepsilon}}}.\end{split}$
(3.19)

The space-time buffer curve, derived from this exact solution, is shown in
Figure 7. There is good agreement with the onset of oscillations observed
numerically in the cubic CGL (1.1).

Figure 7:  ${\rm Re}(A)$ for the solution of (1.1) with the spatially-periodic
source term, $I_{\rm per}(x)$, $p_{1}=\frac{1}{3}$, and $p_{2}=\frac{1}{4}$.
The space-time buffer curve obtained by setting
$|A_{p}(x,\mu)|=\sqrt{\operatorname{\varepsilon}}$ is super-imposed (black
curve). Time series at $x=5\pi$ (a local minimum of $I_{\rm per}(x)$) and
$x=6\pi$ (a local maximum) are also shown. The initial data at $\mu_{0}=-1$ is
$A_{0}(x)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{\rm
per}(x)}{\mu_{0}+i\omega_{0}}$. Here, $\operatorname{\varepsilon}=0.01$,
$\omega_{0}=0.75$, $\alpha=0$, $d_{R}=1$, and $d_{I}=0$.

The space-time buffer curve in this case is determined by the condition
$|A_{p}(x,\mu)|=\sqrt{\operatorname{\varepsilon}}$ instead of the usual
$|A_{p}(x,\mu)|=1$ criterion. This choice was made because whilst the
nonlinear terms in the cubic CGL equation do not affect the onset of the
oscillations, they do influence the spatial phase of the oscillations. In this
case of a periodic source term, the cubic nonlinearities induce a phase shift
in space causing the buffer curve derived from $|A_{p}(x,\mu)|=1$ to be $\pi$
units out of phase in the $x$-direction with the numerically observed onset.
By setting the space-time buffer criterion to be
$|A_{p}(x,\mu)|=\sqrt{\operatorname{\varepsilon}}$, the contribution from the
nonlinearities to the phase shift is still small, and hence there is better
agreement between the space-time buffer curve prediction and the onset of
large-amplitude oscillations in the numerically-calculated solutions of (1.1).

Remark. For this example with a periodic source term, we have derived the
space-time buffer curve above from the exact, closed form expression for the
particular solution $A_{p}$, (3.19). One may also find the leading order
asymptotics using (3.8). In fact, recalling (3.7), we see that the leading
order term in the particular solution for each $\mu$ on $C_{4}$ is
$A_{p}(x,\mu)=(\sqrt{2\pi}g(x,\mu+i\omega_{0})+\mathcal{O}(\sqrt{\operatorname{\varepsilon}}))e^{\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}},$
with $g$ given by (3.18). Hence, to leading order, the space-time buffer curve
is given implicitly by $(\mu_{\rm
stbc}(x))^{2}=\omega_{0}^{2}-\operatorname{\varepsilon}\ln(2\pi)+2\operatorname{\varepsilon}\ln\left|p_{1}+p_{2}e^{-d(\mu_{\rm
stbc}(x)+i\omega_{0})}\cos(x)\right|.$

##  4 The homogeneous exit time curve, $\mu_{h}(x)$

In this section, we study the homogeneous component, $A_{h}(x,\mu)$, of the
solution of the linear CGL PDE (2.1). The main result is the homogeneous exit
time curve, along which $|A_{h}(x,\mu)|=1$, i.e., where $A_{h}$ transitions
from being exponentially small to large. We label it as $\mu_{h}(x)$.

Recalling the change of variables (2.2), we see that formula (2.5) gives

$A_{h}(x,\mu)=\frac{e^{\frac{1}{2\operatorname{\varepsilon}}\left((\mu+i\omega_{0})^{2}-(\mu_{0}+i\omega_{0})^{2}\right)}}{\sqrt{4\pi
d(\mu-\mu_{0})}}\int_{R}e^{\frac{-(x-y)^{2}}{4d(\mu-\mu_{0})}}A_{0}(y)dy,\qquad{\rm
for}\ \ \mu>\mu_{0}.$ (4.1)

The integral in (4.1) can be evaluated for many different types of bounded
initial data $A_{0}(x)$. Moreover, the function $A_{h}(x,\mu)$ is bounded for
$\mu\in(\mu_{0},-\mu_{0})$ and, for our examples, also analytic in a region of
the complex plane about this interval, excluding the branch point and cut.

Figure 8:  ${\rm Re}(A)$ of the solution obtained from the direct numerical
simulation of (1.1) with $\omega_{0}=1$ and source term
$I_{G}(x)=e^{-\frac{x^{2}}{4\sigma}}$, $\sigma=\frac{1}{4}$. The initial data
$A_{0}(x)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{G}(x)}{\mu_{0}+i\omega_{0}}$
is given at $\mu_{0}=-0.2$. For each $x$, the solution stays near the
repelling QSS (green region) for a long time past the instantaneous Hopf
bifurcation at $\mu=0$. Then, the exit time from a neighbourhood of the
repelling QSS is $x$-dependent. In the center, it is given by (4.3) (red
curve), where $\mu_{h}(x)<\mu_{\rm stbc}(x)$. In contrast, for $x$ outside
this interval, $\mu_{\rm stbc}(x)$ comes first, and the space-time buffer
curve (3.8) (black curve) determines the exit time. The inset shows a
magnification for $\mu>1.8$. The convex hull given by $(x,\mu={\rm
min}(\mu_{h}(x),\mu_{\rm stbc}(x)))$ agrees to leading order with the onset of
oscillations for all $x$. Here,
$\operatorname{\varepsilon}=0.01,\omega_{0}=1,\alpha=0.6,d_{R}=1,d_{I}=0.$

To give a first illustration, we choose
$A_{0}(x)=-\frac{\sqrt{\operatorname{\varepsilon}}I_{a}(x)}{\mu_{0}+i\omega_{0}}$,
which is the leading order term in the attracting QSS, and we use the Gaussian
source $I_{G}(x)=e^{-\frac{x^{2}}{4\sigma}}$. (Examples with more general
initial data and with other source terms will be given in Section 5.) The
integral in (4.1) yields

$A_{h}(x,\mu)=-\left(\frac{\sqrt{\operatorname{\varepsilon}}}{\mu_{0}+i\omega_{0}}\right)e^{\frac{1}{2\operatorname{\varepsilon}}\left[(\mu+i\omega_{0})^{2}-(\mu_{0}+i\omega_{0})^{2}\right]}\sqrt{\frac{\sigma}{d(\mu-\mu_{0})+\sigma}}e^{-\frac{x^{2}}{4(d(\mu-\mu_{0})+\sigma)}}.$
(4.2)

Hence, the argument of the total exponential in $A_{h}$ depends on both $x$
and $\mu$. Setting $|A_{h}(x,\mu)|=1$, we find that for $\mu>0$ the exit time
$\mu_{h}(x)$ is given implicitly by

$(\mu_{h}(x))^{2}=\mu_{0}^{2}+\frac{\operatorname{\varepsilon}x^{2}\left[d_{R}(\mu_{h}(x)-\mu_{0})+\sigma\right]}{2\left[(d_{R}(\mu_{h}(x)-\mu_{0})+\sigma)^{2}+d_{I}^{2}(\mu_{h}(x)-\mu_{0})^{2}\right]},$
(4.3)

where to leading order $\mu_{h}(x)=-\mu_{0}$. This curve is the homogeneous
exit time curve, $\mu_{h}(x)$. The logarithmic terms at
$\mathcal{O}(\operatorname{\varepsilon})$ are not reported here, but may be
calculated as in Section 3.2.

Figure 8 reveals the role played by the (red) curve (4.3) in determining the
exit time (and the time of onset of oscillations) for solutions of the cubic
CGL with Gaussian source term. There is a central interval about $x=0$ on
which $\mu_{h}(x)<\mu_{\rm stbc}(x)$. On this interval, the homogeneous
solution $A_{h}$ with the given initial data switches from being exponentially
small to being exponentially large before the particular solution $A_{p}$ does
so for this source. Hence, it determines the exit time (and the time of onset
of the oscillations) there. See the red curve in Figure 8. In contrast, for
$x$ outside this interval, the situation is reversed, with $\mu_{\rm stbc}(x)$
occurring first. Hence, at all points $x$ outside this interval, the exit time
(and onset time for the oscillations) is determined by the space-time buffer
curve $\mu_{\rm stbc}(x)$ given by (3.8). See the black curve in Figure 8 and
the inset.

##  5 The main cases of DHB: one case for each different type of outcome in
the competition between which of $A_{p}$ and $A_{h}$ ceases to be
exponentially small first

From the analysis in Sections 3 and 4, we see that at each point $x$ there is
a competition between which of $\mu_{\rm stbc}(x)$ and $\mu_{h}(x)$ comes
first, i.e., between which component, $A_{p}(x,\mu)$ or $A_{h}(x,\mu)$,
transitions first from being exponentially small to exponentially large.
Moreover, the formulas (3.9) and (4.3) for $\mu_{\rm stbc}(x)$ and
$\mu_{h}(x)$ show that these times depend on key parameters, $\omega_{0}$ and
$\operatorname{\varepsilon}$, the initial data $A_{0}(x)$ and time $\mu_{0}$,
as well as on the form of $I_{a}(x)$.

In this section, we analyze both of these functions and determine various
possible outcomes of the competition. Each different type of outcome leads to
a distinct type of delayed Hopf bifurcation (DHB). We begin in Subsection 5.1
with cases of DHB that arise for solutions of (1.1) with initial data given at
any $\mu_{0}\leq-\omega_{0}$. Then, in Subsection 5.2, we present a main case
of DHB that arises for solutions of (1.1) with initial data given at any
$-\omega_{0}<\mu_{0}\leq-\delta$, where $\delta>0$ is a small,
$\mathcal{O}(1)$ constant. Also, we illustrate all of these cases of DHB using
the different types of source terms introduced in Section 3: Gaussian,
spatially-periodic, and smoothed step function.

###  5.1 Cases 1-3 of DHB for solutions with initial data given at
$\mu_{0}\leq-\omega_{0}$

For solutions of (1.1) with initial data given at $\mu_{0}\leq-\omega_{0}$,
the competition between $A_{p}$ and $A_{h}$ can have three possible outcomes
depending on which ceases to be exponentially small first. These correspond to
the following three cases of DHB:

Case 1 of DHB. $\mu_{\rm stbc}(x)<\mu_{h}(x)$ for all $x\in[-\ell,\ell]$. In
this case, the parameters $\omega_{0}$ and $\operatorname{\varepsilon}$, the
initial data $A_{0}(x)$ and time $\mu_{0}$, and the source term $I_{a}(x)$ are
such that $A_{p}(x,\mu)$ ceases to be exponentially small first, before
$A_{h}(x,\mu)$ does, for all $x$, i.e., $A_{h}(x,\mu_{\rm stbc}(x))$ is
exponentially small at all points $x$. Hence, the full solution $A$ is
exponentially close to the repelling QSS until $\mu=\mu_{\rm stbc}(x)$ to
leading order, and the duration of the DHB and the time of onset of the
oscillations, $\mu_{\rm stbc}(x)$, is determined completely by $A_{p}$ on the
entire domain.

Case 1 of DHB is illustrated in Figures 5, 6, and 7, for the Gaussian,
spatially-periodic, and smoothed step function source terms, respectively. For
the Gaussian source term (with the Gaussian initial data), one finds that
$\mu_{\rm stbc}(x)<\mu_{h}(x)$ for all $x\in[-\ell,\ell]$. This is consistent
with formulas (3.12) and (3.13) derived above for $\mu_{\rm stbc}(x)$ and with
formula (4.3) for $\mu_{h}(x)$. See Figure 5.

Next, for the error function source term, $I_{\rm erf}(x)$ and the initial
data used above, one finds

$A_{h}(x,\mu)=\frac{-\sqrt{\operatorname{\varepsilon}}}{(\mu_{0}+i\omega_{0})}e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu_{0}+i\omega_{0})^{2}}\left[I_{\rm
ave}+I_{\rm e}{\rm
erf}\left(\frac{x}{\sqrt{1+4d(\mu-\mu_{0})}}\right)\right].$ (5.1)

Now, the homogeneous exit time curve $\mu_{h}(x)$ is obtained directly by
setting $|A_{h}(x,\mu)|=1$ with the exact solution. Hence, recalling that
$\mu_{\rm stbc}(x)$ is given by setting $|A_{p}(x,\mu)|=1$ with $g$ given
exactly by (3.15), we see that $\mu_{\rm stbc}(x)<\mu_{h}(x)$ for all
$x\in[-\ell,\ell]$. There is again good quantitative agreement between the
leading order space-time buffer curve and the numerically observed onset of
the large-amplitude oscillations, as shown in Figure 6.

The third example of Case 1 of DHB is given by the simulation with the
spatially-periodic source term, $I_{\rm per}(x)$, and initial data
$A_{0}(x)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{\rm
per}(x)}{\mu_{0}+i\omega_{0}}$. The homogeneous solution is
$A_{h}(x,\mu)=\sqrt{2}e^{-d(\mu-\mu_{0})}{\rm
exp}\left[\frac{1}{2\operatorname{\varepsilon}}\left((\mu+i\omega_{0})^{2}-(\mu_{0}+i\omega_{0})^{2}\right)\right]\cos(x).$
Hence, the homogeneous exit time curve is $\mu_{h}(x)=-\mu_{0}$ to leading
order, which is derived from this exact solution. There are also
$\mathcal{O}(\operatorname{\varepsilon})$ corrections, which are periodic in
space. Then, from (3.19), we see that $\mu_{\rm stbc}(x)<\mu_{h}(x)$ for all
points on the domain. See Figure 7.

Case 2 of DHB. $\mu_{\rm stbc}(x)<\mu_{h}(x)$ for some intervals of points $x$
on $[-\ell,\ell]$, and $\mu_{h}(x)<\mu_{\rm stbc}(x)$ on the complementary
intervals. This case arises when the parameters $\omega_{0}$ and
$\operatorname{\varepsilon}$, the initial data $A_{0}(x)$ and time $\mu_{0}$,
and the source term $I_{a}(x)$ are such that $\mu_{\rm stbc}(x)<\mu_{h}(x)$ at
some, but not all, points $x$, and $\mu_{h}(x)<\mu_{\rm stbc}(x)$ on the
complementary intervals, even though $\omega_{0}<-\mu_{0}$. Here, $A_{p}$
first causes the solution to diverge from the repelling QSS at points $x$
where $\mu_{\rm stbc}(x)<\mu_{h}(x)$, before the homogeneous component $A_{h}$
can do so. On the complementary intervals, where $\mu_{h}(x)<\mu_{\rm
stbc}(x)$, the homogeneous solution stops being exponentially small first, and
hence $A_{h}(x,\mu)$ determines the DHB.

Figure 9:  Illustration of DHB in Case 2 showing ${\rm Re}(A(x,\mu))$ obtained
from (1.1) with $I_{G}(x)$. The initial data is given at time $\mu_{0}=-1.2$.
For $|x|\lesssim 37.1$, $\mu_{\rm stbc}(x)<-\mu_{0}=+1.2$. Hence, the space-
time buffer curve given by the particular solution determines the exit time to
leading order for these points $x$. In contrast, for $|x|\gtrsim 37.1$,
$\mu_{\rm stbc}(x)>-\mu_{0}=+1.2$, so that for these $x$ the exit time is
determined to leading order by $A_{h}(x,\mu)$. Here,
$\operatorname{\varepsilon}=0.01,\omega_{0}=\frac{1}{2},d_{R}=3,d_{I}=1$,
$\alpha=0.6$, and $A_{0}(x)=\cos\left(\frac{10\pi x}{\ell}\right)$ with
$\ell=50$. Similar results are obtained for other initial data.

An example of DHB in Case 2 is presented in Figure 9. Here, the source term is
$I_{G}(x)=e^{-\frac{x^{2}}{4\sigma}}$ with $\sigma=\frac{1}{4}$, and the
initial data at $\mu_{0}=-1.2$ is $A_{0}(x)=\cos\left(\frac{n\pi
x}{\ell}\right)$, with $n=10$ and $\ell=50$. For this initial condition, we
find
$A_{h}(x,\mu)=\sqrt{2}e^{-\frac{dn^{2}\pi^{2}}{\ell^{2}}(\mu-\mu_{0})}{\rm
exp}\left[\frac{1}{2\operatorname{\varepsilon}}\left((\mu+i\omega_{0})^{2}-(\mu_{0}+i\omega_{0})^{2}\right)\right]\cos\left(\frac{n\pi
x}{\ell}\right)$. Hence, we find $\mu_{h}(x)$ directly from this exact
solution.

We observe that $\mu_{\rm stbc}(x)=-\mu_{0}$ at $|x|\approx 37.1.$ Here, to
leading order, the competition is a tie. For $|x|\lesssim 37.1$, $\mu_{\rm
stbc}(x)<\mu_{h}(x)$, so that the space-time buffer curve (3.13) predicts when
the solutions diverge from the repelling QSS and begin to oscillate. In
contrast, for all $|x|\gtrsim 37.1$, $\mu_{h}(x)<\mu_{\rm stbc}(x)$, i.e.,
$A_{h}$ first ceases to be exponentially small, while $A_{p}$ remains
exponentially small. Hence, for $|x|\gtrsim 37.1$, the solution diverges from
the repelling QSS as $\mu$ reaches $\mu_{h}(x)\sim-\mu_{0}$, to leading order.
See Figure 9.

On the outer intervals, the source term $I_{G}(x)$ is essentially zero (below
round-off in the simulations). Hence, the cubic CGL PDE is effectively
symmetric under $A\to Ae^{i\theta}$ ($\theta$ real), including under $A\to-A$
($\theta=\pi$) here. Hence, to leading order, it has a symmetric way-in way-
out function, so the homogeneous exit time is $-\mu_{0}$, to leading order at
these $x$. Here, the amount of credit built up as $\mu$ increases from
$-\mu_{0}$ to zero and the solution spirals toward the attracting QSS
exponentially is exactly spent as $\mu$ increases from zero to $-\mu_{0}$, and
the solution spirals away exponentially from the repelling QSS. Simulations
with other values of $\mu_{0}$ in $(-1.5,-0.5)$ and with other initial data
show similar results for $\mu_{\rm stbc}(x)$ and $\mu_{h}(x)$, with the onset
being determined by $\mu_{\rm stbc}(x)$ in the central portion of the domain
and by $\mu_{h}(x)$ in the outer portions.

The parameter $\operatorname{\varepsilon}$ plays an important role in
determining the width of $\mu_{\rm stbc}(x)$, and hence whether a solution on
a finite domain exhibits Case 2 or Case 1 of DHB. This is illustrated in
Figure 10.

Figure 10:  On a finite domain, the parameter $\operatorname{\varepsilon}$
changes the width of space-time buffer curve. Here, we fix the values of all
of the parameters (except $\operatorname{\varepsilon}$) to be the same as in
Figure 9. Then, in comparison with Figure 9 where
$\operatorname{\varepsilon}=0.01$, we observe that (a) for
$\operatorname{\varepsilon}=0.025$ the space-time buffer curve is narrower,
since at any point $x$ the duration of the DHB $\mu_{\rm stbc}(x)$ increases
as $\operatorname{\varepsilon}$ gets larger, by (3.9). Conversely, (b) for
$\operatorname{\varepsilon}=0.005$, the space-time buffer curve is wider,
since at any point $x$ the duration of the DHB $\mu_{\rm stbc}(x)$ decreases
as $\operatorname{\varepsilon}$ gets smaller. In fact, for this smaller value
of $\operatorname{\varepsilon}$, $\mu_{\rm stbc}(x)<\mu_{h}(x)$ for all $x$ on
the domain, and the solution has shifted over to being in Case 1 of DHB with
this domain.

Finally, for DHB in Case 2 with a Gaussian source term, we observe that there
is a difference between the spatio-temporal dynamics of the large-amplitude
oscillations in $A(x,\mu)$ which are observed in the central portion of the
domain after the space-time buffer curve $\mu_{\rm stbc}(x)$ is crossed and
those which arise in the outer portions of the interval $[-\ell,\ell]$, after
the homogeneous exit time curve $\mu_{h}(x)$ is crossed. In the central
portion (where $A_{p}$ first becomes exponentially large, which is on
$|x|\lesssim 37.1$ in Figure 9), the large-amplitude oscillations propagate
spatially, initially to $x=0$ and then outward, away from $x=0$ for most $\mu$
($\mu\gtrsim 0.8$ in Figure 9). In contrast, outside the central portion
(where $A_{h}$ first becomes exponentially large i.e., where
$\mu_{h}(x)<\mu_{\rm stbc}(x)$), the oscillations do not propagate spatially.
Moreover, at the interfaces (e.g., at $|x|\sim 37.1$ in the figure), the
outward propagating pulses get absorbed by the regime in which the
oscillations do not propagate. The spatio-temporal dynamics of the post-DHB
oscillations is discussed briefly in 11.2.

Case 3 of DHB. $\mu_{h}(x)<\mu_{\rm stbc}(x)$ for all $x\in[-\ell,\ell]$. In
this case, the parameters $\omega_{0}$ and $\operatorname{\varepsilon}$, the
initial data $A_{0}(x)$ and time $\mu_{0}$, and the source term $I_{a}(x)$ are
all such that the homogeneous component $A_{h}$ stops being exponentially
small first at all points $x$. It causes the solution $A(x,\mu)$ to diverge
from the repelling QSS at the time $\mu_{h}(x)$, since at each point
$A_{p}(x,\mu_{h}(x))$ is exponentially small. Hence, the DHB is determined
completely by $A_{h}$. An example is given in Figure 11.

Figure 11:  ${\rm Re}(A(x,\mu))$ illustrating DHB in Case 3 for (1.1) with
$I_{G}(x)$ ($\sigma=\frac{1}{4}$). The curves $\mu_{h}(x)$ (red) and $\mu_{\rm
stbc}(x)$ (black) are super-imposed. For all $x$ in $[-\ell,\ell]$
($\ell=50$), $\mu_{h}(x)<\mu_{\rm stbc}(x)$. Hence, $A_{h}$ is the first to
stop being exponentially small and grow, before $A_{p}$ does, for each $x$.
The oscillations just to the left of the red curve are small-amplitude, and
they are large-amplitude as soon as $\mu$ reaches $\mu_{h}(x)$. The parameters
are
$\operatorname{\varepsilon}=0.02,\omega_{0}=\frac{1}{2},\alpha=0.6,d_{R}=3,d_{I}=1$.
The initial data, given at $\mu_{0}=-0.55$, is
$A_{0}(x)=-20e^{-\frac{x^{2}}{40}}$.

###  5.2 Case 4 of DHB for initial data given at any
$-\omega_{0}<\mu_{0}\leq-\delta$

Case 4 of DHB arises for solutions $A(x,\mu)$ of (1.1) with initial data given
at $\mu_{0}\in(-\omega_{0},-\delta]$, where $\delta>0$ is again any small,
$\mathcal{O}(1)$ constant. With this initial time, one has
$-\mu_{0}<\omega_{0}$. Hence, the left tip of $\mu_{h}(x)$ (where $|A_{h}|=1$,
as calculated from (4.1)) comes before the left tip of the space-time buffer
curve $\mu_{\rm stbc}(x)$ (where $|A_{p}|=1$, as calculated from (3.8)). In
this case, the source terms $I_{a}(x)$, parameters $\omega_{0}$ and
$\operatorname{\varepsilon}$, and initial data $A_{0}(x)$ are such that, for
some intervals of $x$, the homogeneous component $A_{h}$ stops being
exponentially small before $\mu$ reaches $\omega_{0}$, i.e., before $A_{p}$
can. Furthermore, it does so in a manner that $\mu_{h}(x)$ has non-trivial
spatial dependence. We illustrate this with two examples.

Figure 12:  Illustration of DHB in Case 4 with $I_{G}(x)$ and initial data
$A_{0}(x)=\cos(n\pi x/\ell)$ with $n=4$ and $\ell=50$, given at
$\mu_{0}=-0.2$. For all $x$, the exit time for solutions of the CGL PDE (1.1),
is $\mu_{h}(x)$ (red curve), as determined by $A_{h}$. We observe that that
$\mu_{h}(x)\sim-\mu_{0}=0.2$, and the
$\mathcal{O}(\operatorname{\varepsilon})$ corrections are spatially periodic.
The temporal oscillations which set in after $\mu_{h}(x)$ are stationary in
space; there are no propagating pulses. The parameter values are
$\operatorname{\varepsilon}=0.01,\omega_{0}=\frac{1}{2},d_{R}=1,d_{I}=0$, and
$\alpha=0.6$. The space-time buffer curve (black curve) is super-imposed to
illustrate that $\mu_{\rm stbc}(x)$ plays no role in the onset of the
oscillations.

The first example of Case 4 of DHB is given in Figure 8. Here, the solution
$A$ of the cubic PDE (1.1) stops being exponentially small, and the large-
amplitude oscillations set in, at $\mu={\rm min}(\mu_{h}(x),\mu_{\rm
stbc}(x))$ at each point. On the central portion of the interval,
$\mu_{h}(x)<\mu_{\rm stbc}(x)$, i.e., the homogeneous exit time curve (red)
lies to the left of the space-time buffer curve (black). Hence, this solution
fits in Case 4 of DHB, since the oscillations begin to set in at
$\mu=-\mu_{0}$ at $x=0$, where $|A_{h}(x,\mu)|$ first stops being
exponentially small and grows to one, well before $A_{p}$ can transition.
Then, outside this central portion, $\mu_{\rm stbc}(x)<\mu_{h}(x)$, i.e., the
space-time buffer curve (black) occurs before the homogeneous exit time curve
(red). See also the inset in Figure 8.

The second example of Case 4 of DHB is illustrated in Figure 12, for (1.1)
with $I_{G}(x)$. For the given choice of initial data at $\mu_{0}=-0.2$, which
lies inside the interval $(-\omega_{0},0)$ with $\omega_{0}=0.5$, we observe
that $\mu_{h}(x)<\mu_{\rm stbc}(x)$ and $\mu_{h}(x)\sim-\mu_{0}$ for all
$x\in[-\ell,\ell]$. That is, $A_{h}(x,\mu)$ transitions from being
exponentially small to large at $\mu_{h}(x)\sim-\mu_{0}$, at which time
$A_{p}$ is still exponentially small. Therefore, in this case, the time at
which the solution exits from a neighbourhood of the repelling QSS is
$-\mu_{0}$ to leading order on $[-\ell,\ell]$. The
$\mathcal{O}(\operatorname{\varepsilon})$ corrections to
$\mu_{h}(x)\sim-\mu_{0}$ are spatially periodic, and by zooming in on the
homogeneous exit time curve one can see these wiggles, as well. Also, at each
point $x$, the large-amplitude oscillations in $A(x,t)$ which set in after
$\mu$ reaches $\mu_{h}(x)$ do not propagate spatially in this case, with a
Gaussian source term.

Remark. The spatial dependence of the DHB in Case 4 for solutions with initial
data at $\mu_{0}\in(-\omega_{0},\delta]$ is a non-trivial extension to PDEs of
what is known for analytic ODEs with initial conditions given at the same
time. Consider for example, the Shishkova ODE (aka Stuart-Landau ODE with
slowly-varying bifurcation parameter). As mentioned above, it corresponds to
setting $d=0$ in (1.1) and replacing $I_{a}(x)$ with an analytic function
$h(\mu)$ satisfying $h(-i\omega_{0})\neq 0$, see [31, 47, 58]). For solutions
with initial conditions given at $\mu_{0}$, where $\mu_{0}$ is any value in
$(-\omega_{0},-\delta]$, the exit time from a neighbourhood of a repelling
slow manifold is $-\mu_{0}>0$ to leading order. This is because, for each
$\mu_{0}\in(-\omega_{0},0)$, there is a Stokes line in the complex plane that
connects it to the point $-\mu_{0}$ on the positive real axis, without any
saddle point or turning point in between. Recall Figure 4. Hence, for these
solutions, the exit time of $-\mu_{0}$ occurs before the buffer point created
by the particular solution, and one says that the entry-exit function (aka
way-in way-out function) of this analytic ODE is symmetric to leading order
for any solution with $\mu_{0}\in(-\omega_{0},-\delta]$. For the PDE, the same
elliptic contour is used, however the exit time is generally spatially
dependent.

To conclude this section with the examples of DHB, we observe that there is
good agreement between the theory derived for the linear PDE (2.1) and the
results in all of the numerical simulations of the nonlinear PDE (1.1) which
we carried out. This indicates that, for the nonlinear PDE, the cubic terms in
$A$ are higher order, and this will be confirmed by the analysis of the
nonlinear terms in Section 6. Moreover, we note that in this respect the
phenomenon of DHB in the CGL PDE is similar to that for DHB in the Shishkova
ODE and in other analytic ODEs, where the cubic and other nonlinear terms are
also higher order. See for example Section 3 of [31].

##  6 DHB and the space-time buffer curve for the cubic CGL

In this section, we build on the results for the linear CGL equation (2.1)
established in Section 3 to study the full nonlinear CGL equation (1.1) in the
base case in which $\beta=\frac{1}{2}$ and $\gamma=1$,

$\operatorname{\varepsilon}A_{\mu}=(\mu+i\omega_{0})A-(1+i\alpha)|A|^{2}A+\operatorname{\varepsilon}^{\frac{1}{2}}I_{a}(x)+\operatorname{\varepsilon}dA_{xx},$
(6.1)

with complex-valued $d$ and $d_{R}>0$, and with $I_{a}(x)\geq 0$ for all $x$.
We demonstrate that solutions with initial data $A(x,\mu_{0})=A_{0}(x)$ given
at time $\mu_{0}<-\omega_{0}$ in DHB Case 1 stay near the attracting and
repelling quasi-stationary states for
$\mu\in[\mu_{0},\omega_{0}-\tilde{\delta})$, for some $\tilde{\delta}>0$ small
but $\mathcal{O}(1)$ with respect to $\operatorname{\varepsilon}$. Hence, the
nonlinear solution exhibits DHB and the space-time buffer curve for this
nonlinear equation is the same to leading order as the curve (3.8) for the
linear CGL equation (2.1).

We use the same dependent variable given by (2.2),
$B(x,\mu)=A(x,\mu)e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}},$
to transform the cubic CGL to

$\operatorname{\varepsilon}B_{\mu}=-(1+i\alpha)E(\mu)|B|^{2}B+\sqrt{\operatorname{\varepsilon}}I_{a}(x)e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}+\operatorname{\varepsilon}dB_{xx},$
(6.2)

where

$E(\mu)=e^{\frac{1}{\operatorname{\varepsilon}}(\mu_{R}^{2}-(\mu_{I}+\omega_{0})^{2})}.$
(6.3)

Next, we subtract off the linear particular solution $B_{p}$, recall (2.21),
substituting $B=B_{p}+b$ into (6.2) to obtain

$\displaystyle\operatorname{\varepsilon}b_{\mu}=\operatorname{\varepsilon}db_{xx}-(1+i\alpha)E(\mu)(B_{p}+b)|B_{p}+b|^{2},\qquad\mu\in[\mu_{0},-\mu_{0}].$
(6.4)

We suppress the $x$ dependence in the solutions to keep the formulas more
manageable. As it is needed throughout this section, we note the general
expansion of the nonlinearity $N[b]:=b|b|^{2}$

$N[f+g]=(f+g)^{2}(\overline{f+g})=f^{2}\bar{f}+2f\bar{f}g+f^{2}\bar{g}+g^{2}\bar{f}+2fg\bar{g}+g^{2}\bar{g}.$

We consider mild solutions of (6.4) using the variation of constants formula

$\displaystyle b(\mu)$
$\displaystyle=G_{d}(\mu-\mu_{0})*\left(e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu_{0}+i\omega_{0})^{2}}A_{0}\right)-\frac{(1+i\alpha)}{\operatorname{\varepsilon}}\int_{\mu_{0}}^{\mu}E(\tilde{\mu})G_{d}(\mu-\tilde{\mu})*N\left[B_{p}(\tilde{\mu})+b(\tilde{\mu})\right]d\tilde{\mu},$
(6.5)

where $G_{d}(\mu-\mu_{0})$ denotes the Green’s function and $*$ denotes the
convolution. We let
$H\left[v\right](\mu):=B_{h}(\mu)+\tilde{H}\left[v\right](\mu)$ denote the
right member of this equation with

$\displaystyle\tilde{H}\left[v\right](\mu)$
$\displaystyle:=-\frac{(1+i\alpha)}{\operatorname{\varepsilon}}\int_{\mu_{0}}^{\mu}E(\tilde{\mu})G_{d}(\mu-\tilde{\mu})*N\left[v(\tilde{\mu})\right]d\tilde{\mu},$
$\displaystyle B_{h}(\mu)$
$\displaystyle:=G_{d}(\mu-\mu_{0})*\left(e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu_{0}+i\omega_{0})^{2}}A_{0}\right).$
(6.6)

We shall assume that the initial data $A_{0}$ is bounded and the inhomogeneity
$I_{a}$ is smooth with uniformly bounded derivatives. That is, we assume there
exists a constant $C>0$ with

$|\partial_{x}^{j}I_{a}(x)|\leq\begin{cases}C,&\qquad|\omega_{0}|\geq 1\\\
C|\omega_{0}|^{2},&\qquad|\omega_{0}|<1\end{cases}\qquad
j\in\mathbb{N}_{0},x\in\mathbb{R}.$ (6.7)

This is a rather strong assumption which allows us to readily bound remainder
terms occurring below uniformly in $x$. We strongly suspect that similar
results can be obtained for less restrictive assumptions on $I_{a}$.

###  6.1 Iterative framework and base iterate

To construct an approximate solution to the mild formulation (6.5), we use an
iterative approach. We set $b_{0}(\mu):=0$ and then iteratively define

$b_{j+1}(\mu):=H\left[B_{p}+b_{j}\right](\mu),\qquad j\geq 0.$ (6.8)

In this section, we estimate $b_{1}$. We claim

$b_{1}(\mu)=e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\left(-\operatorname{\varepsilon}^{3/2}\frac{(1+i\alpha)I_{a}(\cdot)^{3}}{(\mu+i\omega_{0})^{2}(\mu^{2}+\omega_{0}^{2})}+\mathcal{O}(\operatorname{\varepsilon}^{5/2})\right)+\mathcal{O}(e^{\frac{1}{2\operatorname{\varepsilon}}(\omega_{0}^{2}-\mu_{0}^{2})}),\quad\mu\in[-\mu_{0},\omega_{0}-\tilde{\delta})$
(6.9)

for some $\tilde{\delta}>0$ fixed and small, with error terms uniform in $x$.
This gives the leading order terms in (6.9). To obtain this estimate, we note
the linear term $B_{h}$ defined in (2.5) is exponentially small for all
$\mu\in[\mu_{0},-\mu_{0}]$ provided $A_{0}$ is bounded. Hence, it suffices to
estimate the nonlinear term.

The formula for $B_{p}$ is given by (2.14) for $\mu\in[\mu_{0},-\delta]$, by
the formulas in Appendix B for $\mu\in(-\delta,\delta)$, and by (2.21) for
$\mu\in[\delta,\omega_{0}]$. Overall, for all $\mu\in[\mu_{0},\omega_{0}]$, we
may write the asymptotic expansion for $B_{p}$ as

$B_{p}=e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\left(-\operatorname{\varepsilon}^{1/2}\frac{I_{a}(\cdot)}{\mu+i\omega_{0}}+\mathcal{O}(\operatorname{\varepsilon}^{3/2})\right)+\tilde{c}(\mu)\left(g(\cdot,\mu+i\omega_{0})+\mathcal{O}(\operatorname{\varepsilon}^{1/2})\right)=:B_{p,1}+B_{p,2}$

where $\tilde{c}(\mu)$ is a bounded, monotonic function with
$\tilde{c}(\mu)\equiv 0$ for $\mu\leq-\delta$,
$\tilde{c}(\mu)\rightarrow\sqrt{\pi/2}$ as $\mu\rightarrow 0^{+}$, and
$\tilde{c}(\mu)\equiv\sqrt{2\pi}$ for all $\mu\geq\delta$. Here, $B_{p,1}$ is
the linear contribution to the QSS and $B_{p,2}$ is defined to be the term
that arises in the same solution for $\mu\geq 0$ due to passing through the
saddle at $-i\omega_{0}$. We remark that, by the assumptions on $I_{a}$, the
error terms are uniform in $x$.

Next, re-write the expansion as

$B_{p}=e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\left(-\operatorname{\varepsilon}^{1/2}\frac{I_{a}(\cdot)}{\mu+i\omega_{0}}+\mathcal{O}(\operatorname{\varepsilon}^{3/2})+e^{\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}(\tilde{c}(\mu)g(\cdot,\mu+i\omega_{0})+\mathcal{O}(\operatorname{\varepsilon}^{1/2}))\right).$

Also note that this factorization, which moves
$e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}$ outside of
all terms in $B_{p}$, illuminates what remains when transitioning back to the
$A$ coordinates. Since $\tilde{c}(\mu)g(\cdot,\mu+i\omega_{0})$ is
$\mathcal{O}(1)$, the corresponding term in $A$-coordinates,
$e^{\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\tilde{c}(\mu)g(\cdot,\mu+i\omega_{0})$
is exponentially small for $\mu\in[\mu_{0},\omega_{0}-\tilde{\delta}]$ for
some fixed $\tilde{\delta}>0$ which is small but $\mathcal{O}(1)$ with respect
to $\operatorname{\varepsilon}$.

To estimate the nonlinear term, we use this expansion and work separately on
$\mu<0$ and on $\mu>0$, beginning with the former,

$\displaystyle\tilde{H}[B_{p}]$
$\displaystyle:=-\frac{1+i\alpha}{\operatorname{\varepsilon}}\int_{\mu_{0}}^{\mu}E(\tilde{\mu})G_{d}(\mu-\tilde{\mu})*\left[B_{p}(\tilde{\mu})|B_{p}(\tilde{\mu})|^{2}\right]d\tilde{\mu}$
$\displaystyle=-\frac{1+i\alpha}{\operatorname{\varepsilon}}\int_{\mu_{0}}^{\mu}\left(-\frac{\tilde{\mu}+i\omega_{0}}{\operatorname{\varepsilon}}e^{-\frac{1}{2\operatorname{\varepsilon}}(\tilde{\mu}+i\omega_{0})^{2}}\right)\left(\frac{-\operatorname{\varepsilon}}{\tilde{\mu}+i\omega_{0}}G_{d}(\mu-\tilde{\mu})*[-\operatorname{\varepsilon}^{3/2}\frac{I_{a}(\cdot)^{3}}{(\tilde{\mu}+i\omega_{0})(\tilde{\mu}^{2}+\omega_{0}^{2})}+\mathcal{O}(\operatorname{\varepsilon}^{5/2})]\right)d\tilde{\mu}$
$\displaystyle=-[e^{-\frac{1}{2\operatorname{\varepsilon}}(\tilde{\mu}+i\omega_{0})^{2}}\operatorname{\varepsilon}^{3/2}G_{d}(\mu-\tilde{\mu})*\frac{(1+i\alpha)I_{a}(\cdot)^{3}}{(\tilde{\mu}+i\omega_{0})^{2}(\tilde{\mu}^{2}+\omega_{0}^{2})}+\mathcal{O}(\operatorname{\varepsilon}^{5/2})\Big{]}_{\tilde{\mu}=\mu_{0}}^{\mu}$
$\displaystyle\qquad\qquad+\int_{\mu_{0}}^{\mu}e^{-\frac{1}{2\operatorname{\varepsilon}}(\tilde{\mu}+i\omega_{0})^{2}}\partial_{\tilde{\mu}}\left(G_{d}(\mu-\tilde{\mu})*[\operatorname{\varepsilon}^{3/2}\frac{(1+i\alpha)I_{a}(\cdot)^{3}}{(\tilde{\mu}+i\omega_{0})^{2}(\tilde{\mu}^{2}+\omega_{0}^{2})}+\mathcal{O}(\operatorname{\varepsilon}^{5/2})]\right)d\tilde{\mu}$
$\displaystyle=e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\left(-\operatorname{\varepsilon}^{3/2}\frac{(1+i\alpha)I_{a}(\cdot)^{3}}{(\mu+i\omega_{0})^{2}(\mu^{2}+\omega_{0}^{2})}+\mathcal{O}(\operatorname{\varepsilon}^{5/2})\right)+\mathcal{O}(e^{\frac{1}{2\operatorname{\varepsilon}}(\omega_{0}^{2}-\mu_{0}^{2})}).$
(6.10)

Note that in the second line, we multiplied the integrand by one in a useful
manner and used the approximation (2.14). In the third line, we have
integrated by parts; and, in the fourth line, we have used the fact that the
$\tilde{\mu}=\mu_{0}$ boundary term is exponentially small (in particular
$\mathcal{O}(e^{\frac{1}{2\operatorname{\varepsilon}}(\omega_{0}^{2}-\mu_{0}^{2})})$)
while the remaining integral is
$\mathcal{O}(\operatorname{\varepsilon}^{5/2})$, uniformly in $x$. This last
claim can be obtained by integrating by parts once more and using the fact
that the imaginary part of the phase
$-(2\operatorname{\varepsilon})^{-1}(\tilde{\mu}+i\omega_{0})^{2}$ is non-
stationary for $\tilde{\mu}\in[\mu_{0},\mu]$; see for example [8, §6].

A similar estimate holds for $\mu\in[0,\omega_{0}-\delta]$, as the term
$B_{p,2}$ only contributes exponentially small effects here relative to
$B_{p,1}$, for $\mu\leq\omega_{0}-\tilde{\delta}$. To see this in more detail,
we estimate a few of the terms in the expansion of $N[B_{p,1}+B_{p,2}]$. For
instance, consider the term $B_{p,2}|B_{p,2}|^{2}$:

$\displaystyle\Bigg{\|}-\frac{1+i\alpha}{\operatorname{\varepsilon}}$
$\displaystyle\int_{\mu_{0}}^{\mu}E(\tilde{\mu})G_{d}(\mu-\tilde{\mu})*\left[\tilde{c}(\tilde{\mu})^{3}g(\cdot,\tilde{\mu}+i\omega_{0})|g(\cdot,\tilde{\mu}+i\omega_{0})|^{2}+\mathcal{O}(\operatorname{\varepsilon}^{1/2})\right]d\tilde{\mu}\Big{\|}_{L^{\infty}}$
$\displaystyle=\Bigg{\|}-\frac{1+i\alpha}{\operatorname{\varepsilon}}\int_{-\delta}^{\mu}E(\tilde{\mu})G_{d}(\mu-\tilde{\mu})*\left[\tilde{c}(\tilde{\mu})^{3}g(\cdot,\tilde{\mu}+i\omega_{0})|g(\cdot,\tilde{\mu}+i\omega_{0})|^{2}+\mathcal{O}(\operatorname{\varepsilon}^{1/2})\right]d\tilde{\mu}\Bigg{\|}$
$\displaystyle\leq|1+i\alpha|\operatorname{\varepsilon}^{-1}E(\mu)\tilde{c}(\mu)^{3}\int_{-\delta}^{\mu}\|G_{d}(\mu-\tilde{\mu})*\left[g(\cdot,\tilde{\mu}+i\omega_{0})|g(\cdot,\tilde{\mu}+i\omega_{0})|^{2}+\mathcal{O}(\operatorname{\varepsilon}^{1/2})\right]\|_{L^{\infty}}d\tilde{\mu}$
$\displaystyle\leq
CE(\mu)(\operatorname{\varepsilon}^{-1}+\mathcal{O}(1))\|I_{a}\|_{L^{\infty}}^{3}$
(6.11)

for some constant $C>0$, possibly dependent on $\omega_{0}$. We recall that
$\tilde{c}(\mu)$ increases monotonically. Note the last line remains
exponentially small for $\operatorname{\varepsilon}\ll 1$, uniformly for
$\mu\in[0,\omega_{0}-\tilde{\delta}]$. In these inequalities we have
repeatedly used the estimate on the heat evolution
$\|G_{d}(\mu-\tilde{\mu})*f\|_{L^{\infty}}\leq C\|f\|_{L^{\infty}}$ for
$\mu-\tilde{\mu}\geq 0$. Terms which are quadratic in $B_{p,2}$ can also be
bound in a similar way by a term of the form
$C\operatorname{\varepsilon}^{-1}E(\mu)^{1/2}\|I_{a}\|_{L^{\infty}}^{2}$.

It remains to consider terms which are quadratic in $B_{p,1}$. For example,
consider the term $2B_{p,2}|B_{p,1}|^{2}$, where we can estimate

$\displaystyle\Bigg{\|}-\frac{1+i\alpha}{\operatorname{\varepsilon}}$
$\displaystyle\int_{\mu_{0}}^{\mu}E(\tilde{\mu})G_{d}(\mu-\tilde{\mu})*\left[2B_{p,2}|B_{p,1}|^{2}\right]d\tilde{\mu}\Big{\|}_{L^{\infty}}$
$\displaystyle=2\Bigg{\|}-\frac{1+i\alpha}{\operatorname{\varepsilon}}\int_{-\delta}^{\mu}G_{d}(\mu-\tilde{\mu})*\left[(\tilde{c}(\tilde{\mu})g(\cdot,\tilde{\mu}+i\omega_{0})+\mathcal{O}(\operatorname{\varepsilon}^{1/2}))\cdot\left|\frac{\operatorname{\varepsilon}^{1/2}I_{a}}{\tilde{\mu}+i\omega_{0}}+\mathcal{O}(\operatorname{\varepsilon}^{3/2})\right|^{2}\right]d\tilde{\mu}\Bigg{\|}_{L^{\infty}}$
$\displaystyle\leq
2|1+i\alpha|\tilde{c}(\mu)\int_{-\delta}^{\mu}\Big{\|}G_{d}(\mu-\tilde{\mu})*\left[g(\cdot,\tilde{\mu}+i\omega_{0})\cdot\left|\frac{I_{a}}{\tilde{\mu}+i\omega_{0}}\right|^{2}+\mathcal{O}(\operatorname{\varepsilon})\right]\Big{\|}_{L^{\infty}}d\tilde{\mu}$
$\displaystyle\leq
C\|I_{a}\|_{L^{\infty}}^{3}+\mathcal{O}(\operatorname{\varepsilon}).$ (6.12)

Hence, we may conclude the desired estimate (6.9) by using (6.10) for $\mu<0$,
the similar estimate (6.11) for $\mu>0$, and the estimates of the terms
quadratic in $B_{p,1}$, such as (6.12), as well as by noting that the terms
coming from $B_{p,2}$ are exponentially small relative to
$e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}$ on
$\mu\in[-\delta,\omega_{0}-\tilde{\delta})$, and thus contained in the
$\mathcal{O}(\operatorname{\varepsilon}^{5/2})$ term in (6.9).

###  6.2 Inductive step

We claim inductively that

$\displaystyle
b_{j+1}(\mu)=b_{j}(\mu)+e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\left(C_{j+1}(\mu,\omega_{0})(\operatorname{\varepsilon}^{1/2}I_{a})^{2(j+1)+1}+\mathcal{O}(\operatorname{\varepsilon}^{\frac{2j+5}{2}})\right),$
(6.13)

where $C_{j}$ is function of $\mu\in\mathbb{R}$ and $\omega_{0}$ for which
$I_{a}^{2j+1}C_{j}$ is uniformly bounded in $x$ and for real $\mu$. By (6.9),
the claim holds for $j=0$ with
$C_{1}(\mu,\omega_{0})=-\frac{1+i\alpha}{(\mu+i\omega_{0})^{2}(\mu^{2}+\omega_{0}^{2})}$.
Note, that by (6.7) we have that $|C_{1}(\mu,\omega_{0})I_{a}^{3}|\leq C$ for
some fixed constant $C>0$.

We assume formula (6.13) holds for all $0\leq k\leq j$, and prove it holds for
$k=j+1$. Expand

$\displaystyle b_{j+1}$
$\displaystyle=B_{h}(\mu)+\left(\tilde{H}[B_{p}+b_{j}]-\tilde{H}[B_{p}+b_{j-1}]\right)+\tilde{H}[B_{p}+b_{j-1}]$
$\displaystyle=b_{j}+\left(\tilde{H}[B_{p}+b_{j}]-\tilde{H}[B_{p}+b_{j-1}]\right)$
$\displaystyle=b_{j}-\frac{(1+i\alpha)}{\operatorname{\varepsilon}}\int_{\mu_{0}}^{\mu}E(\tilde{\mu})G_{d}(\mu-\tilde{\mu})*(N[B_{p}+b_{j}]-N[B_{p}+b_{j-1}])d\tilde{\mu}$
(6.14)

where the difference of nonlinearities above can be expanded as

$N[B_{p}+b_{j}]-N[B_{p}+b_{j-1}]=2|B_{p}|^{2}(b_{j}-b_{j-1})+B_{p}^{2}(\bar{b}_{j}-\bar{b}_{j-1})+\bar{B}_{p}(b_{j}^{2}-b_{j-1}^{2})+2B_{p}(|b_{j}|^{2}-|b_{j-1}|^{2})+(b_{j}^{2}\bar{b}_{j}-b_{j-1}^{2}\bar{b}_{j-1}).$

Also, note that by our inductive hypothesis, and the fact that the homogeneous
term $B_{h}$ is exponentially small, we find

$b_{j}-b_{j-1}=e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\left(C_{j}(\mu,\omega_{0})(\operatorname{\varepsilon}^{1/2}I_{a})^{2j+1}+\mathcal{O}(\operatorname{\varepsilon}^{\frac{2j+3}{2}})\right).$

Next, we notice that the leading order terms in $\operatorname{\varepsilon}$
of $N[B_{p}+b_{j}]-N[B_{p}+b_{j-1}]$ are

$\displaystyle 2|B_{p}|^{2}(b_{j}-b_{j-1})$
$\displaystyle=2E(\mu)^{-1}e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\left(\frac{C_{j}(\mu,\omega_{0})}{\mu^{2}+\omega_{0}^{2}}(\operatorname{\varepsilon}^{1/2}I_{a})^{2j+3}+\mathcal{O}(\operatorname{\varepsilon}^{\frac{2j+5}{2}})\right),$
(6.15) $\displaystyle B_{p}^{2}(\bar{b}_{j}-\bar{b}_{j-1})$
$\displaystyle=E(\mu)^{-1}e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\left(\frac{\overline{C_{j}(\mu,\omega_{0})}e^{4i\omega_{0}\mu}}{(\mu+i\omega_{0})^{2}}(\operatorname{\varepsilon}^{1/2}I_{a})^{2j+3}+\mathcal{O}(\operatorname{\varepsilon}^{\frac{2j+5}{2}})\right).$
(6.16)

Defining
$\tilde{C}_{j}(\mu,\omega_{0})=\frac{2C_{j}(\mu,\omega_{0})}{\mu^{2}+\omega_{0}^{2}}+\frac{\overline{C_{j}(\mu,\omega_{0})}e^{4i\omega_{0}\mu}}{(\mu+i\omega_{0})^{2}}$,
inserting the expansions for the leading order terms, and using integration by
parts we find

$\displaystyle b_{j+1}(\mu)-b_{j}(\mu)$
$\displaystyle=-\frac{1+i\alpha}{\operatorname{\varepsilon}}\int_{\mu_{0}}^{\mu}E(\tilde{\mu})G_{d}(\mu-\tilde{\mu})*\left[N(B_{p}+b_{j})-N(B_{p}+b_{j-1})\right]d\tilde{\mu}$
$\displaystyle=-\frac{1+i\alpha}{\operatorname{\varepsilon}}\int_{\mu_{0}}^{\mu}e^{-\frac{1}{2\operatorname{\varepsilon}}(\tilde{\mu}+i\omega_{0})^{2}}G_{d}(\mu-\tilde{\mu})*\left[\operatorname{\varepsilon}^{\frac{2j+3}{2}}\tilde{C}_{j}(\tilde{\mu},\omega_{0})I_{a}^{2j+3}+\mathcal{O}(\operatorname{\varepsilon}^{\frac{2j+5}{2}})\right]d\tilde{\mu}$
$\displaystyle=e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\frac{(1+i\alpha)\tilde{C}_{j}(\mu,\omega_{0})}{(\mu+i\omega_{0})}\operatorname{\varepsilon}^{\frac{2j+3}{2}}I_{a}^{2j+3}+\mathcal{O}(\operatorname{\varepsilon}^{\frac{2j+5}{2}})$
$\displaystyle\qquad-\frac{(1+i\alpha)}{\operatorname{\varepsilon}}\int_{\mu_{0}}^{\mu}e^{-\frac{1}{2\operatorname{\varepsilon}}(\tilde{\mu}+i\omega_{0})^{2}}\partial_{\tilde{\mu}}\left(\frac{\operatorname{\varepsilon}}{(\tilde{\mu}+i\omega_{0})}G_{d}(\mu-\tilde{\mu})*\left[\operatorname{\varepsilon}^{\frac{2j+3}{2}}\tilde{C}_{j}(\tilde{\mu},\omega_{0})I_{a}^{2j+3}+\mathcal{O}(\operatorname{\varepsilon}^{\frac{2j+5}{2}})\right]\right)d\tilde{\mu}$
$\displaystyle=e^{-\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}C_{j+1}(\mu,\omega_{0})\operatorname{\varepsilon}^{\frac{2j+3}{2}}I_{a}^{2j+3}+\mathcal{O}(\operatorname{\varepsilon}^{\frac{2j+5}{2}})$
(6.17)

where we have defined
$C_{j+1}(\mu,\omega_{0})=\frac{(1+i\alpha)\tilde{C}_{j}(\mu,\omega_{0})}{(\mu+i\omega_{0})}$.
Observe that due to the definition of $\tilde{C}_{j}$, we have that
$|C_{j+1}|\leq C|\omega_{0}|^{-3(j+1)-1}$. Then, by the assumption (6.7), and
by the boundedness of $C_{j}I_{a}^{2j+1}$, the term $C_{j+1}I_{a}^{2j+3}$ is
bounded in $\mu$ and $x$, uniformly in $\omega_{0}$. Hence, the difference
$b_{j+1}-b_{j}$ is $\mathcal{O}(\operatorname{\varepsilon}^{\frac{2j+3}{2}})$,
which becomes small as $j\rightarrow+\infty$.

From this iterative process, we observe that each approximation $b_{j}$
successively reveals the $\operatorname{\varepsilon}^{\frac{2j+1}{2}}$-order
terms in the expansion of the nonlinear attracting and repelling QSSs for
$\mu<0$ and $\mu>0$, respectively. Furthermore, this formal iterative method
makes it clear that solutions with bounded data $B_{0}(x,\mu_{0})$ remain
exponentially close to the QSSs for all
$\mu\in[\mu_{0},\omega_{0}-\tilde{\delta})$. As $\mu$ approaches $\omega_{0}$
from below, while terms coming from $B_{h}$ remain exponentially small, the
$B_{p,2}$ terms coming from tracking the solution over the saddle point are no
longer exponentially small in the original $A$-coordinates. For the
$x$-dependent value of $\mu\geq\omega_{0}$ given to leading order by $\mu_{\rm
stbc}(x)$, they induce the delayed Hopf bifurcation. Furthermore, we conclude
in Case 1 of DHB that, since the contribution from the nonlinear terms is
higher order, $A_{p}(x,\mu)$ mediates the spatially dependent bifurcation. A
similar analysis may be done in the other cases of DHB.

##  7 The $\mathcal{O}(\operatorname{\varepsilon})$ value of $\mu_{\rm
Hopf}(x)$ in the base case

In the analysis of the base case of the PDE (1.1) ($\beta=1/2$ and $\gamma=1$)
in Sections 2-5, we used that $\mu_{\rm Hopf}(x)=0$ for all $x$ to leading
order. In this section, we calculate the
$\mathcal{O}(\operatorname{\varepsilon})$ term in the value of $\mu_{\rm
Hopf}(x)$, and we identify the role this asymptotically small correction plays
in determining the spatial dependence of the observed onset of oscillations.
The calculations here are performed for general sources $I_{a}(x)$, and
examples are given with Gaussian and spatially-periodic terms.

Recall from formula (1.2) that the attracting and repelling QSS on
$\mu<-\delta$ and $\mu>\delta$, respectively, are
$\mathcal{O}(\sqrt{\operatorname{\varepsilon}})$ to leading order. Thus, the
linearisation about the small-amplitude QSS is consistent. We set $A=A_{\rm
QSS}+\operatorname{\varepsilon}^{a}\mathcal{A}$, with $a>1/2$, and the
linearised equation for $\mathcal{A}$ is

$\operatorname{\varepsilon}\mathcal{A}_{\mu}=(\mu+i\omega_{0})\mathcal{A}-(1+i\alpha)\left(2|A_{\rm
QSS}|^{2}\mathcal{A}+A_{\rm
QSS}^{2}\bar{\mathcal{A}}\right)+\operatorname{\varepsilon}d\mathcal{A}_{xx}.$
(7.1)

In terms of the real and imaginary parts,
$\mathcal{A}=\mathcal{U}+i\mathcal{V}$ and $A_{\rm QSS}(x)=u_{\rm Q}+iv_{\rm
Q}$, the linearised equation for $\mathcal{A}$ may be expressed as

$\operatorname{\varepsilon}\begin{bmatrix}{\mathcal{U}}_{\mu}\\\
{\mathcal{V}}_{\mu}\end{bmatrix}=M\begin{bmatrix}\mathcal{U}\\\
\mathcal{V}\end{bmatrix}+\operatorname{\varepsilon}\begin{bmatrix}d_{R}{\mathcal{U}}_{xx}-d_{I}{\mathcal{V}}_{xx}\\\
d_{R}{\mathcal{V}}_{xx}+d_{I}{\mathcal{U}}_{xx}\end{bmatrix}$

where $\mu=\mu_{R}+i\mu_{I}$ and

$M=\begin{bmatrix}\mu_{R}-3u_{\rm Q}^{2}-v_{\rm Q}^{2}+2\alpha u_{\rm Q}v_{\rm
Q}&(\mu_{I}+\omega_{0})+\alpha u_{\rm Q}^{2}+3\alpha v_{\rm Q}^{2}-2u_{\rm
Q}v_{\rm Q}\\\ (\mu_{I}+\omega_{0})-3\alpha u_{\rm Q}^{2}-\alpha v_{\rm
Q}^{2}-2u_{\rm Q}v_{\rm Q}&\mu_{R}-u_{\rm Q}^{2}-3v_{\rm Q}^{2}-2\alpha u_{\rm
Q}v_{\rm Q}\end{bmatrix}.$

The trace of $M$ is

${\rm tr}(M)=2\mu_{R}-4(u_{\rm Q}^{2}+v_{\rm
Q}^{2})=2\mu_{R}-\frac{4\operatorname{\varepsilon}I_{G}^{2}(x)}{\mu_{R}^{2}+(\mu_{I}+\omega_{0})^{2}}+\mathcal{O}(\operatorname{\varepsilon}^{2}),$
(7.2)

and

${\rm det}(M)=(\mu_{R}^{2}+(\mu_{I}+\omega_{0})^{2})-4\mu_{R}(u_{\rm
Q}^{2}+v_{\rm Q}^{2})(1+\mu_{I}+\omega_{0})+3(1+\alpha^{2})(u_{\rm
Q}^{2}+v_{\rm Q}^{2})^{2},$

so that ${\rm det}(M)>0$ for all $\mu_{R}<0$, as well as for a range of values
of $\mu_{R}>0$.

Figure 13:  Onset of oscillations near the tip of the space-time buffer curve,
of (2.1) in a neighbourhood of $x=0$ in the base case $\beta=1/2$ and
$\gamma=1$ of (1.1) with Gaussian source ($\sigma=\frac{1}{4}$). By (7.3),
$\mu_{\rm
Hopf}(x)=\frac{2\operatorname{\varepsilon}(I_{G}(x))^{2}}{\omega_{0}^{2}}+\mathcal{O}(\operatorname{\varepsilon}^{2})$.
The maximum is at $x=0$, and $\mu_{\rm Hopf}(x)$ decays rapidly for $|x|>0$.
Hence, near $x=0$, the delay in the onset of oscillations at each $x$ is
slightly longer, creating the “fork in the tongue”. In contrast, away from the
center of the domain, the magnitude of $\mu_{\rm Hopf}(x)$ is negligible, and
hence the space-time buffer curve (3.13) determines the DHB and the delayed
onset of the oscillations there. Also, this figure illustrates the effect of
the logarithmic terms at $\mathcal{O}(\operatorname{\varepsilon})$ in (3.9)
and (3.13). Namely, (a) with $d_{R}=1$ and $d_{I}=0$, the entire buffer curve
is shifted leftward from $\mu=\omega_{0}=\frac{1}{2}$ by an
$\mathcal{O}(\operatorname{\varepsilon})$ amount, as is most visible near the
tip, and (b) with $d_{R}=3$ and $d_{I}=1$, the logarithmic terms shift the
buffer curve rightward. The parameters are
$\operatorname{\varepsilon}=0.01,\omega_{0}=\frac{1}{2}$, and $\alpha=0$. The
initial data at $\mu_{0}=-1$ is
$A_{0}(x)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{G}(x)}{\mu_{0}+i\omega_{0}}$.

Therefore, the Hopf bifurcation curve for the solutions of (1.1), which is
obtained by setting ${\rm tr}(M)=0$, is given to leading order by

$\mu_{\rm
Hopf}(x)=\frac{2\operatorname{\varepsilon}(I_{a}(x))^{2}}{\omega_{0}^{2}}+\mathcal{O}(\operatorname{\varepsilon}^{2}).$
(7.3)

This asymptotic formula holds for general $I_{a}(x)$ in the base case of the
PDE (1.1).

Figure 14:  ${\rm Re}(A(x,\mu))$ for an example of Case 1 of DHB in the PDE
(1.1) with spatially-periodic source term, $I_{\rm per}(x)=2+\cos(x)$. To the
left of the exact space-time buffer curve (black curve), the solution is near
the small-amplitude, repelling, spatially-periodic QSS (yellow and green
spatially-periodic pattern). To the right, the oscillations have set in to
leading order, with the amplitude of the oscillations increasing rapidly from
small (yellow and medium blue) to large (red and dark blue). The Hopf
bifurcation curve $\mu_{\rm Hopf}(x)$ (red curve) is super-imposed. At all
points $x$, this term creates an $\mathcal{O}(\operatorname{\varepsilon})$
increase in the duration of the DHB and the onset of the oscillations past the
space-time buffer curve. At the minima, $x=(2k+1)\pi$ for each $k$, the onset
occurs first, and the maxima, which occur at $x=2k\pi$ for each $k$, show a
pronounced delay beyond the space-time buffer curve, since the numerical
values of $\mu_{\rm Hopf}(x)$ are approximately 0.4 at the maxima, which is of
the same size as $4\sqrt{\operatorname{\varepsilon}}$ here. The lower frames
show time traces at $x=5\pi$ (a local maximum of the space-time buffer curve)
and $x=6\pi$ (a local minimum of the space-time buffer curve), illustrating
the hard onset of the oscillations (the black markers). The effect of
$\mu_{\rm Hopf}(x)$ is particularly visible at even integer multiples of
$\pi$. Here,
$\operatorname{\varepsilon}=0.01,\omega_{0}=\frac{2}{3},d_{R}=1,d_{I}=0$, and
$\alpha=0$. The initial data at $\mu_{0}=-1$ is
$A_{0}(x)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{\rm
per}(x)}{\mu_{0}+i\omega_{0}}$. Similar results are observed for other
$A_{0}(x)$ and for other values of $\mu_{0}<-\omega_{0}$.

The spatial dependence of $\mu_{\rm Hopf}(x)$ is illustrated with two
different source terms. First, in Figure 13, we show the results obtained with
a Gaussian source term, $I_{G}(x)$. In a small interval about $x=0$, the
solution of the PDE remains near the repelling QSS (green region just inside
the tip of the space-time buffer curve) for an amount of time equal to
$\mu_{\rm Hopf}(x)$. Here, the onset of the oscillations is delayed slightly
longer than predicted by the space-time buffer curve by that same amount of
time, and this is manifested by the “fork in the tongue” centered at $x=0$.
Then, for $x$ further away from $x=0$, the amplitude of the Gaussian source
term, $I_{G}(x)$, is negligibly small. Hence, here the function $\mu_{\rm
Hopf}(x)$ is negligibly small, and the onset of the oscillations coincides
with the space-time buffer curve. In the numerical simulations, the small-
amplitude oscillations (light yellow and light blue) are observed right before
the space-time buffer curve, just as is the case for DHB in analytic ODEs.
Further, one sees that the amplitudes of the oscillations have become large
(orange, red, dark blue, and purple) immediately after the space-time buffer
curve.

Second, Figure 14 shows the results obtained with a spatially-periodic source
term, $I_{\rm per}(x)$. The influence of the
$\mathcal{O}(\operatorname{\varepsilon})$ value of $\mu_{\rm Hopf}(x)$, (7.3),
manifests more here, since the maximum numerical value of $(I_{\rm
per}(x))^{2}$ over all $x$ is nine, which is of the same size numerically as
$\frac{1}{\sqrt{\operatorname{\varepsilon}}}$.

Remark. In Figures 9, 11, and 12, the
$\mathcal{O}(\operatorname{\varepsilon})$ effect of $\mu_{\rm Hopf}(x)$ is
also visible in the center of the domains, near the tips of the space-time
buffer curve and the homogeneous exit time curve.

##  8 Spatially growing and sign-changing source terms

In this section, we push somewhat beyond the theory and examples for the
nonlinear PDE (1.1) in the base case ($\beta=\frac{1}{2}$ and $\gamma=1$), as
presented above in Sections 2–5. There, the source terms $I_{a}(x)$ are taken
to be positive with uniformly bounded derivatives at all points. Here, we
study the PDE (1.1) in the base case with an algebraically-growing source and
with a sign-changing source.

###  8.1 An algebraically-growing source term

In this section, we analyze (1.1) in the base case with an algebraically-
growing source term

$I_{\rm ag}(x)=x^{2}.$ (8.1)

Figure 15:  ${\rm Re}(A(x,\mu))$ of the solution of the cubic CGL (1.1) with
the algebraically-growing source term, $I_{\rm ag}(x)$, along with the exact
space-time buffer curve (black curve, given by the exact solution
$A_{p}(x,\mu)$ in (8.5)). DHB and the attendant oscillations are observed in
the central portion of the domain $(|x|\lesssim 1.55)$ to the right of the
exact space-time buffer curve. These oscillations propagate outward from
$x=0$, until they reach $|x|\approx 1.5$. For $|x|\approx 1.5$, $\mu_{\rm
stbc}(x)\approx 0$. Then, outside this central region, the solution stays near
the repelling QSS, at least until $\mu=10.5$. Note that the color scale
differs from that in the previous figures. The initial data at $\mu_{0}=-1$ is
$A_{0}(x)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{ag}(x)}{\mu_{0}+i\omega_{0}}$.
Here, $\omega_{0}=\frac{1}{2}$, $\alpha=0$, $d_{R}=1$, and $d_{I}=0$.

We start by observing that the QSS for the cubic PDE (1.1) with $I_{\rm
ag}(x)$ is, to leading order,

$A_{\rm
QSS}(x,\mu)=\left\\{{\frac{-\sqrt{\operatorname{\varepsilon}}x^{2}}{\mu+i\omega_{0}}\quad{\rm
for}\ \
|x|\ll\operatorname{\varepsilon}^{-\frac{1}{4}}}\hfill\atop{\frac{\operatorname{\varepsilon}^{\frac{1}{6}}x^{\frac{2}{3}}(1-i\alpha)}{(1+\alpha^{2})^{\frac{2}{3}}}+\frac{\operatorname{\varepsilon}^{-\frac{1}{6}}x^{-\frac{2}{3}}}{3(1+\alpha^{2})^{\frac{4}{3}}}\left((3-\alpha^{2}-4\alpha
i)(\mu+i\omega_{0})-2\mu_{\rm R}(1+\alpha^{2})\right)\quad{\rm for}\ \
|x|\gg\operatorname{\varepsilon}^{-\frac{1}{4}}}.\right.$ (8.2)

Here, we use the fact that for
$|x|\ll\operatorname{\varepsilon}^{-\frac{1}{4}}$ the QSS is determined to
leading order by balancing the linear term in $A$ and the source term in
(1.1), since the cubic term is higher order in this region. Hence, in this
region with $|x|\ll\operatorname{\varepsilon}^{-\frac{1}{4}}$, the
linearisation about $A=0$ is valid, as is the formula for the leading order
space-time buffer curve. Also, we note that the higher order terms in the QSS
are $\mathcal{O}(\operatorname{\varepsilon}^{\frac{3}{2}})$ and depend on $x$
and $\mu$, recall (1.2).

In contrast, for $|x|\gg\operatorname{\varepsilon}^{-\frac{1}{4}}$, the QSS in
(8.2) is determined by balancing the cubic term and the source term in (1.1).
Hence, here the QSS has significantly larger amplitude, and linearisation
should be about the large-amplitude QSS, and no longer about $A=0$. The higher
order terms are
$\mathcal{O}(\operatorname{\varepsilon}^{-\frac{1}{2}}x^{-2})$. Also, to
leading order here,

$|A_{\rm QSS}(x,\mu_{\rm
R})|=\frac{\operatorname{\varepsilon}^{\frac{1}{6}}x^{\frac{2}{3}}}{(1+\alpha^{2})^{\frac{1}{6}}}+\frac{\operatorname{\varepsilon}^{-\frac{1}{6}}x^{-\frac{2}{3}}(\mu_{\rm
R}+\omega_{0}\alpha)}{3(1+\alpha^{2})^{\frac{5}{6}}}.$ (8.3)

Figure 16:  Spatial profiles of $\left|A\right|$ (blue curves) for the same
solution as in Figure 15, at $\mu=1.5$ and $\mu=10.5$. The oscillations are
seen in the center of the domain, and the width of this central region
increases slowly: $-2\lesssim x\lesssim 2$ at $\mu=1.5$ and $-3\lesssim
x\lesssim 3$ at $\mu=10.5$. Outside these central intervals, the solution is
close to the repelling QSS, which is given to leading order by (8.2)(b)
(dashed red curve). Super-imposed is the numerically obtained solution (green
curve) of the leading order QSS equation,
$r^{2}\left((\mu-r^{2})^{2}+(\omega_{0}-\alpha
r^{2})^{2}\right)=\operatorname{\varepsilon}x^{4},$ valid for all $x$, as
obtained from (1.1) by neglecting the time and space derivative terms (which
are higher order) and then squaring the modulus. Here, $r=|A|$, $\mu$ is real,
and $\alpha=0$. On $|x|\gg\operatorname{\varepsilon}^{-\frac{1}{4}}$, the
green curve lies essentially on top of the blue and red curves. For
$\mu<\omega_{0}$, the green curve does not have any fold points, and there are
no oscillations in the PDE solution, consistent with the DHB analysis. Then,
the fold points first appear when $\mu$ reaches $\omega_{0}$ to leading order,
and they are present (data not shown) for all $\mu$ up until at least 10.5.
These fold points delimit the interval on which the oscillations are observed.

We now determine the space-time buffer curve for the region in which
$|x|\ll\operatorname{\varepsilon}^{-\frac{1}{4}}$, where the QSS has small-
amplitude, so that the analysis of Section 3 applies. We require ${\rm
Re}(d(\mu-{\tilde{\mu}}))>0$ and $|{\rm
arg}(\frac{x}{2d(\mu-{\tilde{\mu}})})|<\pi$. Evaluating the integral in
definition (2.6)(b), we find

$g(x,\mu-{\tilde{\mu}})=x^{2}+2d(\mu-{\tilde{\mu}}).$ (8.4)

With this elementary form of $g$, the integral (2.6)(a) for $B_{p}(x,\mu)$ may
be evaluated in closed form in this example, and hence also $A_{p}(x,\mu)$ may
be found in closed form. Specifically, carrying out the integration in (2.6)
and recalling (2.2), we find

$\begin{split}A_{p}(x,\mu)&=\sqrt{\frac{\pi}{2}}\left(x^{2}+2d(\mu+i\omega_{0})\right)\left[{\rm
erf}\left(\frac{\mu+i\omega_{0}}{\sqrt{2\operatorname{\varepsilon}}}\right)-{\rm
erf}\left(\frac{\mu_{0}+i\omega_{0}}{\sqrt{2\operatorname{\varepsilon}}}\right)\right]e^{\frac{1}{2\operatorname{\varepsilon}}(\mu+i\omega_{0})^{2}}\\\
+&{}2d\sqrt{\operatorname{\varepsilon}}\left(1-e^{\frac{1}{2\operatorname{\varepsilon}}(\mu^{2}-\mu_{0}^{2}+2i\omega_{0}(\mu-\mu_{0}))}\right).\end{split}$
(8.5)

Hence, by taking the real part of the complex-valued, space-time-dependent
phase of the solution to be zero, we find the exact space-time buffer curve
for $|x|\ll\operatorname{\varepsilon}^{-\frac{1}{4}}$. This curve is plotted
in Figure 15, along with ${\rm Re}(A)$ from the numerical simulation of the
full nonlinear PDE (1.1) with this same source term. Here, we observe that
$\operatorname{\varepsilon}^{-\frac{1}{4}}\approx 3.16$ for
$\operatorname{\varepsilon}=0.01$. Excellent agreement is observed between the
onset of the oscillations and the exact space-time buffer curve in the region
$|x|\lesssim 1.55$.

In Figure 16, we see that for
$|x|\gg\operatorname{\varepsilon}^{-\frac{1}{4}}$, the solution of the PDE
(blue curve) is near the repelling QSS (red curve) at least until $\mu=10.5$,
where the QSS is given by (8.2)(b) for
$|x|\gg\operatorname{\varepsilon}^{-\frac{1}{4}}$. Moreover, for
$|x|\gg\operatorname{\varepsilon}^{-\frac{1}{4}}$, the Hopf bifurcation
(determined by the linearisation about the non-trivial QSS here, instead of
about $A=0$) is delayed. See also Section 9.

###  8.2 A sign-changing source term

In this section, we analyze (1.1) with a sign-changing source term

$I_{\rm sc}(x)=\cos(x).$ (8.6)

Figure 17:  $|A(x,\mu)|$ for the solution of (1.1) with the sign-changing
source term, $I_{\rm sc}(x)=\cos(x)$. This is an example of Case 2 of DHB. To
the left of the exact space-time buffer curve (black curve), the solution is
near the small-amplitude, repelling, spatially-periodic QSS. About each point
$x=k\pi$, there is a wide interval on which $\mu_{\rm stbc}(x)<\mu_{h}(x)$,
and the hard onset of the large-amplitude oscillations is determined to
leading order by the space-time buffer curve. On the narrow, complementary
intervals (about $x=\frac{(2n+1)\pi}{2}$), $\mu_{h}(x)<\mu_{\rm stbc}(x)$ so
that the oscillations set in at $\mu_{h}(x)\sim-\mu_{0}$, as determined to
leading order by $A_{h}$, and there are
$\mathcal{O}(\operatorname{\varepsilon})$ amplitude wiggles about this leading
order result, due to the periodicity of the initial data. Here,
$\operatorname{\varepsilon}=0.01,\omega_{0}=\frac{2}{3},d_{R}=1,d_{I}=0$, and
$\alpha=0$. The initial data at $\mu_{0}=-1$ is
$A_{0}(x)=-\sqrt{\operatorname{\varepsilon}}\frac{I_{\rm
sc}(x)}{\mu_{0}+i\omega_{0}}$.

One finds

$g(x,\mu-{\tilde{\mu}})=e^{-d(\mu-{\tilde{\mu}})}\cos(x).$ (8.7)

and

$A_{p}(x,\mu)=\sqrt{\frac{\pi}{2}}\cos(x)\left[{\rm
erf}\left(\frac{\mu+i\omega_{0}-\operatorname{\varepsilon}d}{\sqrt{2\operatorname{\varepsilon}}}\right)-{\rm
erf}\left(\frac{\mu_{0}+i\omega_{0}-\operatorname{\varepsilon}d}{\sqrt{2\operatorname{\varepsilon}}}\right)\right]e^{\frac{(\mu+i\omega_{0}-\operatorname{\varepsilon}d)^{2}}{2\operatorname{\varepsilon}}}.$
(8.8)

The resultant space-time buffer curve (which here is also determined exactly
by setting the real part of the complex-valued, space-time-dependent phase of
$A$ to zero) is shown in Figure 17.

This is an example of DHB in Case 2. About each point $x=k\pi$ (where
$|\cos(x)|=1$) there is a wide interval on which $\mu_{\rm
stbc}(x)<\mu_{h}(x)$, and the hard onset of oscillations on these intervals is
determined to leading order by the space-time buffer curve. See Figure 17. On
the complementary intervals, $\mu_{h}(x)<\mu_{\rm stbc}(x)$, so that $A_{h}$
stops being exponentially small first, at $\mu_{h}(x)\sim-\mu_{0}$, to leading
order. The existence of these narrow intervals may be understood from the
asymptotics of $\mu_{\rm stbc}(x)$. In particular, from (8.7), one sees that
for this example with a sign-changing source term there are infinitely many
points $x=\frac{(2n+1)\pi}{2}$ at which $g(x,\mu+i\omega_{0})$ vanishes, and
hence where $\ln|g|\to-\infty$. This causes $\mu_{\rm stbc}(x)$ to diverge at
these points, recall (3.8). Therefore, for any solution with
$\mu_{0}\leq-\omega_{0}$, there is a (narrow) interval about each point
$x=\frac{(2n+1)\pi}{2}$ on which $\mu_{h}(x)<\mu_{\rm stbc}(x)$, so that
$A_{h}$ determines the onset time at these points to be
$\mu=\mu_{h}(x)\sim-\mu_{0}$.

##  9 Asymptotically large source terms

In this section, we study the CGL equation (1.1) with an asymptotically large
$\mathcal{O}(\frac{1}{\sqrt{\operatorname{\varepsilon}}})$ source term, i.e.,
with $\beta=-\frac{1}{2}$ in (1.1),

$\operatorname{\varepsilon}A_{\mu}=(\mu+i\omega_{0})A-(1+i\alpha)|A|^{2}A+\frac{1}{\sqrt{\operatorname{\varepsilon}}}I_{\tilde{a}}(x)+\operatorname{\varepsilon}dA_{xx},$
(9.1)

while retaining the $\mathcal{O}(\operatorname{\varepsilon})$ diffusivity,
i.e., $\gamma=1$, as in the base case. The source term, which is denoted by
$I_{\tilde{a}}(x)$ in this section, is taken to be strictly positive. We find
that the Hopf bifurcation occurs along an $x$-dependent curve $\mu_{\rm
Hopf}(x)$, and we derive the asymptotics for it, showing that the large-
amplitude source term causes the bifurcation to occur well to the right of
$\mu=0$. We quantify how $\mu_{\rm Hopf}(x)$, together with the space-time
buffer curve, determines the DHB duration at each $x$, focusing on regions
where $I_{\tilde{a}}(x)=\mathcal{O}(1)$. We remark that in regions where
$I_{\tilde{a}}(x)$ is effectively small, i.e. of the size of
$\mathcal{O}(\operatorname{\varepsilon})$, then the Hopf term only affects the
higher order terms. Overall, the analysis here reveals that, by choosing
$I_{\tilde{a}}(x)$ appropriately, one has region-specific control over the
duration of the DHB, which can be useful in system design for postponing the
onset of undesirable oscillations.

The attracting and repelling QSS are given on $\mu<-\delta$ and $\mu>\delta$,
respectively, by

$A(x,\mu)=\frac{1}{\operatorname{\varepsilon}^{\frac{1}{6}}}A_{\rm
QSS}(x)+\operatorname{\varepsilon}^{\frac{1}{6}}\mathcal{A}(x,\mu,\operatorname{\varepsilon}),\quad{\rm
where}\quad A_{\rm
QSS}(x)=\frac{1-i\alpha}{(1+\alpha^{2})^{\frac{2}{3}}}(I_{\tilde{a}}(x))^{\frac{1}{3}},$
(9.2)

$\mathcal{A}=\mathcal{O}(1)$ for all $x$ and $\mu$ uniformly in
$\operatorname{\varepsilon}$ for sufficiently small
$\operatorname{\varepsilon}>0$. (See also the Remark below.) The linearised
equation for $\mathcal{A}$ is

$\operatorname{\varepsilon}^{\frac{4}{3}}\mathcal{A}_{\mu}=\operatorname{\varepsilon}^{\frac{1}{3}}(\mu+i\omega_{0})\mathcal{A}-(1+i\alpha)\left(2|A_{\rm
QSS}|^{2}\mathcal{A}+A_{\rm
QSS}^{2}\bar{\mathcal{A}}\right)+\operatorname{\varepsilon}^{\frac{4}{3}}d\mathcal{A}_{xx}+(\mu+i\omega_{0})A_{\rm
QSS}+\operatorname{\varepsilon}d(A_{\rm QSS})_{xx}.$ (9.3)

In terms of the real and imaginary parts, $A_{\rm QSS}(x)=u_{\rm Q}+iv_{\rm
Q}$ and $\mathcal{A}=\mathcal{U}+i\mathcal{V}$, the linearised equation for
$\mathcal{A}$ may be expressed as

$\operatorname{\varepsilon}^{\frac{4}{3}}\begin{bmatrix}{\mathcal{U}}_{\mu}\\\
{\mathcal{V}}_{\mu}\end{bmatrix}=M\begin{bmatrix}\mathcal{U}\\\