We now construct the jump map used to define the slow eigenvalue problem in
Def. 6.4. Our goal is to determine the fate of the slow components
$y_{s}(\xi,\lambda,\varepsilon):=(p(\xi,\lambda,\varepsilon),v(\xi,\lambda,\varepsilon))$
of $y(\xi,\lambda,\varepsilon)$ across the fast layer. The difficulty in
tracking the slow data in this inner layer is that
$y_{s}(\xi,\lambda,\varepsilon)$ remains $\mathcal{O}(\varepsilon)$ while
$u(\xi,\lambda,\varepsilon)$ grows to $\mathcal{O}(1)$ for a time interval
that is $\mathcal{O}(1/\varepsilon)$; in other words, the linearised solution
can be made to align arbitrarily closely to the fast fibres after crossing the
fold, and they remain close throughout most of the jump. The directional
information carried by the slow variables is not annihilated as
$\varepsilon\to 0$, however. The natural approach from the point of view of
GSPT is to perform an $\varepsilon$-dependent rescaling of the fast linearized
equations: for $\varepsilon>0$, let

$\displaystyle\varepsilon\beta_{1}$ $\displaystyle=p$ (83)
$\displaystyle\varepsilon\beta_{2}$ $\displaystyle=v.$

Then we have

$\displaystyle u^{\prime}$
$\displaystyle=\frac{1}{c}(\varepsilon\beta_{2}-(\varepsilon\lambda+D(\bar{u}))u)$
(84) $\displaystyle\beta_{1}^{\prime}$
$\displaystyle=(R^{\prime}(\bar{u})-\lambda)u$
$\displaystyle\beta_{2}^{\prime}$ $\displaystyle=cu-\varepsilon\beta_{1},$

where a factor of $\varepsilon$ has been cancelled from the latter two
equations. The resulting equations limit to the following linear system as
$\varepsilon\to 0$:

$\displaystyle u^{\prime}$ $\displaystyle=-\frac{1}{c}D(\bar{u})u$ (85)
$\displaystyle\beta_{1}^{\prime}$
$\displaystyle=(R^{\prime}(\bar{u})-\lambda)u$
$\displaystyle\beta_{2}^{\prime}$ $\displaystyle=cu.$

The first equation in (85) is the variational equation of the linearized layer
problem (43), and thus has a family of nontrivial bounded solutions
$u(\xi)=K(\bar{v}_{0}-F(\bar{u}(\xi)))=K(d\bar{u}/d\xi)$, where $\bar{v}_{0}$
is the $\bar{v}$-component of $F_{-}$. It is then possible to calculate
$\beta_{1}(\xi)$ and $\beta_{2}(\xi)$ explicitly. We find it more convenient
to perform the intermediate calculations in the projective space
$\mathbb{CP}^{2}$. This allows us to fix a free parameter by applying a
smoothness condition across the fold.

We consider the projection of the linear system (85) on a copy of
$\mathbb{CP}^{2}$, choosing the chart
$(s,g)=(\beta_{1}/\beta_{2},u/\beta_{2})$, $\beta_{2}\neq 0$:

$\displaystyle s^{\prime}$ $\displaystyle=(R^{\prime}(\bar{u})-\lambda)g-csg$
(86) $\displaystyle g^{\prime}$
$\displaystyle=-\frac{D(\bar{u})}{c}g-(R^{\prime}(\bar{u}-\lambda))g^{2}.$

The $g$-equation decouples, so we solve it directly to find

$\displaystyle g(\xi)$
$\displaystyle=\frac{\bar{v}_{F}-F(\bar{u}(\xi))}{c\bar{u}(\xi)+C},$

where $C\in\mathbb{C}$ is a constant. The $s$ equation is then given by

$\displaystyle s^{\prime}$
$\displaystyle=((R^{\prime}(\bar{u})-\lambda)-cs)\frac{\bar{v}_{F}-F(\bar{u}(\xi))}{c\bar{u}(\xi)+C}.$
(87)

Figure 7: Demonstration of the convergence of the projectivised dynamics
(blueish curves) onto the reduced dynamics (red curve) as $\varepsilon\to 0$
for $\lambda=15$. Solutions of reduced dynamical system are defined by
concatenating the solution of the projectivized flow of $dS/d\zeta$ (see
(89)), defined for $\bar{U}\in(0,\bar{U}_{F})\cup(\bar{U}_{J},1)$, with the
graph of the map (90) defined for $\bar{u}\in[\bar{u}_{F},\bar{u}_{J}]$. The
left dotted line denotes the fold $\bar{u}=\bar{u}_{F}$ and the right dotted
line denotes the jump curve $\bar{u}=\bar{u}_{J}$ (and hence the fast dynamics
takes place in between these dotted lines). The right dotted line is the jump
curve, where the jump condition is used to concatenate the appropriate
solutions. The wave speed was held at $c=c_{0}\approx 0.199362$ for each value
of $\varepsilon$.

Equation (87) is now a one-dimensional nonautonomous problem and can be solved
explicitly, but we must first fix the constant $C$ to define an unambiguous
jump condition. Such an ambiguity arises even at the projective level. For
each $\varepsilon>0$, the scale of the linearized solution along the wave is
set by one free parameter, whereas in the singular limit, we split the
linearized dynamics into subsystems defined along the singular heteroclinic
orbit. We are able to choose freely the scale of each corresponding linearized
solution segment. It follows that these free parameters of the reduced systems
may be constrained so that they are compatible with the scalings of the ‘full’
linearized solutions as $\varepsilon\to 0$.

A suitable compatibility condition is the $C^{1}$-differentiability of the
desingularized linearized flow across the fold. We compare (87) with the
projectivization of the desingularized slow eigenvalue problem (47), which we
write over the chart $S=P/V$, with $V\neq 0$, as

$\displaystyle\frac{dS}{d\zeta}$ $\displaystyle=(R^{\prime}(\bar{U})-\lambda-
cS+D(\bar{U})S^{2}).$ (88)

Using the chain rule, we write

$\displaystyle\frac{\partial
S}{\partial\bar{P}}\frac{d\bar{P}}{d\zeta}+\frac{\partial
S}{\partial\bar{U}}\frac{dS}{d\zeta}=\frac{dS}{d\zeta}$
$\displaystyle=(R^{\prime}(\bar{U})-\lambda-cS+D(\bar{U})S^{2})$ (89)
$\displaystyle\frac{ds}{d\bar{u}}$
$\displaystyle=\frac{1}{c\bar{u}-C}(R^{\prime}(\bar{U})-\lambda-cS).$

At the point $\Gamma_{0}\cap
F_{-}=\\{\bar{X}_{F}=(\bar{u}_{F},\bar{p}_{F},\bar{v}_{F})\\}$ where the
singular orbit intersects the fold, we have $D(\bar{U}_{F})=0$ and thus
$(dS/d\zeta)|_{\bar{X}_{F}}=(\partial
S/\partial\bar{U})(c\bar{u}_{F}-\bar{p}_{F})$. By matching $(\partial
S/\partial\bar{U})$ and $ds/d\bar{u}$ at $\bar{X}_{F}$ we find

$\displaystyle C$ $\displaystyle=\bar{p}_{F}.$

At the projective level, the jump map $u_{J}\mapsto s(\bar{u_{J}})$ is defined
from the solution $s(\bar{u})$ of the (complex) one-dimensional initial-value
problem

$\displaystyle\frac{ds}{d\bar{u}}$
$\displaystyle=\frac{1}{c\bar{u}-\bar{p}_{f}}(R^{\prime}(\bar{u})-\lambda-cs)$
$\displaystyle s(\bar{u}_{F})$ $\displaystyle=s_{0}.$

Note that $c\bar{u}-\bar{p}_{F}>0$ is bounded away from 0 for
$\bar{u}\in[\bar{u}_{F},\bar{u}_{J}]$ since $c\bar{u}_{F}-\bar{p}_{F}>0$ by
Hypothesis 2.8 and $\bar{u}\geq\bar{u}_{f}$ across the jump. Explicitly, we
have

$\displaystyle s(\bar{u},\lambda)$
$\displaystyle=\frac{R(\bar{u})-R(\bar{u}_{F})-\lambda(\bar{u}-\bar{u}_{F})+s_{0}(c\bar{u}_{F}-\bar{p}_{F})}{c\bar{u}-\bar{p}_{F}}$
(90)

for $\bar{u}_{F}\leq\bar{u}\leq\bar{u}_{J}$. See Fig. 7 for a demonstration of
the approximation of the ‘full’ linearized flow to that of the hybrid reduced
problem for a nonzero value of $\lambda$.

The linear jump map (55) can be extracted from (90) by using the chart map
$s=p/v$. For $a,b\in\mathbb{C}^{n}$, we define the equivalence relation $a\sim
b$ if $a=\gamma b$ for some complex number $\gamma\neq 0$. We then have

$\displaystyle(p_{0},v_{0})\mapsto(p,v)$ $\displaystyle=((p/v)v,v)=(sv,v)$
(91)
$\displaystyle\sim(s,1)=\left(\frac{R(\bar{u})-R(\bar{u}_{F})-\lambda(\bar{u}-\bar{u}_{F})+s_{0}(c\bar{u}_{F}-\bar{p}_{F})}{c\bar{u}-\bar{p}_{F}},1\right)$
$\displaystyle\sim\left(\frac{v_{0}(R(\bar{u})-R(\bar{u}_{F})-\lambda(\bar{u}-\bar{u}_{F}))+p_{0}(c\bar{u}_{F}-\bar{p}_{F})}{c\bar{u}-\bar{p}_{F}},v_{0}\right)$
$\displaystyle:=J(p_{0},v_{0},\lambda,\bar{u}).$

We can now define $J_{\lambda}(P,V):=J(P,V,\lambda,\bar{U}_{J})$. Our jump map
is defined up to linear equivalence and is therefore clearly nonunique;
however, any two linear choices induce the same projective jump map
$\hat{J}_{\lambda}:=s(\bar{u}_{J},\lambda)$ on $\mathbb{CP}$. This essentially
completes the proof of Theorem 7.1(b).

### 7.4 Control from the reduced dynamics

Proof of Theorem 7.1(c). By Corollary B.4, the unique solution
$\hat{E}(\zeta,\lambda,\varepsilon)$ which tends to the unstable eigenvector
at $\hat{u}=0$ as $\zeta\to-\infty$ remains uniformly close to the
(projectivized) slow subbundle $\hat{\Sigma}_{s}$.

Fix an interval $a\leq|\zeta|\leq A$, where $0<a<A$ with $A\leq\infty$. For
fixed $\lambda,\varepsilon$, let $(\beta_{1}(\xi),\beta_{2}(\xi))$ denote the
local coordinates of the solution $\hat{E}(\zeta,\lambda,\varepsilon)$ which
coincide with the coordinates used to define the slow projectivized system
(32). Then by Lemma 4.2, for any $\delta>0$ we can find some sufficiently
small $\bar{\varepsilon}=\bar{\varepsilon}(\delta)$ so that
$|\beta_{1}-1/D(\bar{U})|\leq\delta$ for all $a\leq|\zeta|\leq A$ and for all
$0<\varepsilon\leq\bar{\varepsilon}$. This estimate can be read off from the
expressions for the reduced eigenvectors in (36).

Now consider the solution $\hat{Z}_{*}(\zeta,\lambda)$ written with the
coordinate $S_{*}(\zeta)=P(\zeta)/V(\zeta)$ with $V\neq 0$. The dynamics of
$S(\zeta)$ is determined by the system (46). At either limit $\zeta\to-\infty$
and $\zeta\to+\infty$, the corresponding asymptotic systems admit a pair of
hyperbolic (saddle) fixed points, interpreted in the projectivized system as
an attractor and a repeller. Let us consider the case $\zeta\to-\infty$ and
denote the projectivized attractor (corresponding to the unstable eigenvector
of the saddle) by $\hat{u}^{-}(\lambda)$ and the projectivized repeller
(stable eigenvector) by $\hat{s}^{-}(\lambda)$. We seek to verify that the
only possibility is that $\hat{Z}_{*}(\zeta,\lambda)\to\hat{u}^{-}(\lambda)$
as $\zeta\to-\infty$ (i.e. that
$\hat{E}_{*}(\zeta,\lambda)\to\hat{r}_{s,1}^{-}$, with $r_{s,1}^{-}$ as
defined in (36)).

So suppose that $\hat{E}_{*}(\zeta,\lambda)\to\hat{r}_{s,2}^{-}$ (the weak
stable eigenvector) instead. Since the corresponding critical point
$\hat{s}^{-}$ is a repeller, let $N$ denote a repelling neighborhood of radius
$\eta>0$. Then by hypothesis, there exists some $\zeta_{1}=\zeta_{1}(\eta)$ so
that $S_{*}(\zeta,\lambda,\varepsilon)\in N$ for all $\zeta\leq\zeta_{1}$. As
long as $V\neq 0$ remains bounded away from zero, for each sufficiently small
$\varepsilon>0$ we may choose a section $E\in\pi^{-1}\hat{E}$ so that

$\displaystyle E(\zeta,\lambda,\varepsilon)$
$\displaystyle=(1/D(\hat{u}),ac_{0}/D(\hat{u})+(b-a)\nu_{m,-},1)+\mathcal{O}(\delta)$

where $a+b=1$. Thus, $|\beta_{1}|$ and $|\beta_{2}|$ (the norms of the
coordinates of $\hat{E}$ with respect to the chart specified by (32)) remain
uniformly bounded while $S\in N$. Now, observe that the $\beta_{2}$ equation
in (32) may be written as follows:

$\displaystyle\dot{\beta_{2}}$
$\displaystyle=(R^{\prime}(\bar{U}^{-})-\lambda)-c\beta+D(\bar{U}^{-})S^{2}+d(\zeta,\lambda,\varepsilon),$

where $d(\zeta,\lambda,\varepsilon)$ consists of terms characterizing the
perturbation from the asymptotic system together with terms of the form
$(\beta_{1}-1/D(\bar{U}))$, which can be uniformly bounded as stated in the
beginning of the proof. Hence, $N$ remains a repelling neighborhood of
$\hat{s}^{-}$ for the system above. But this implies that
$\hat{E}(\zeta,\lambda,\varepsilon)$ can be chosen to remain in any small
neighborhood of $\hat{r}^{-}_{s,1}(\lambda)$ as $\zeta\to-\infty$, which
contradicts the fact that
$\hat{E}(\zeta,\lambda,\varepsilon)\to\hat{r}^{-}_{s,2}$ as $\zeta\to-\infty$.
Hence, $\hat{E}_{*}$ and $\hat{E}_{0}=\iota_{0}\hat{Z}_{0}$ coincide for
$\zeta<0$.

For the matching on the right hand-side, we have by Theorem 7.1(b) that the
jump condition is identical and uniquely determined, so in fact $\hat{E}_{*}$
and $\hat{E}_{0}$ coincide for $\zeta>0$ as well. It remains to check that
$\hat{E}$ tends uniformly to $\hat{r}_{s,1}^{+}$. This step proceeds
identically to the preceding argument, so we omit it. $\Box$

### 7.5 Continuing $\mathcal{E}_{\varepsilon}(K)$ to $\mathcal{E}_{0}(K)$

We now prove the following corollary of Theorem 7.1:

######  Corollary 7.10.

There exists $\varepsilon_{0}>0$ such that for each
$\varepsilon\in(0,\varepsilon_{0}]$,
$\mathcal{E}_{\varepsilon}(K)\cong\mathcal{E}_{0}(K)$. In particular,
$c_{1}(\mathcal{E}_{\varepsilon}(K))=c_{1}(\mathcal{E}_{0}(K))$.

We follow the approach of Sec. VI-D in [8]. The method is quite natural. With
respect to the compactified time $T$-scaling (see (57)), the fast dynamics
over the jump occurs over an $\mathcal{O}(\varepsilon)$-interval straddling
the midpoint of the interval $[-1,1]$. The approach is to first define two
hemispheric bundles over the complement of this small interval in $[-1,1]$,
using the flow itself to construct the gluing map. We then construct a
homotopy which closes this gap continuously as the homotopy parameter is
varied. Theorem 7.1 is then used to show that the gluing map and bundles have
nice singular limits: they are precisely the ones used to construct the
reduced augmented unstable bundle $\mathcal{E}_{0}(K)$.

Proof of Corollary 7.10: Fix the pair of real parameters
$S^{a,-}_{\varepsilon},\,S^{a,+}_{\varepsilon}$ with
$-1<S^{a,-}_{\varepsilon}<0<S^{a,+}_{\varepsilon}<1$. We work with the
parameter pair $S=(S^{a,-}_{\varepsilon},\sigma)$, where
$\sigma\in[S^{a,-}_{\varepsilon},S^{a,+}_{\varepsilon}]$ is the homotopy
parameter that we will continuously vary.

For $\varepsilon>0$ fixed, define the bundle $\mathcal{E}(S,\varepsilon)$ as
follows. The base space $B_{S}$ of $\mathcal{E}(S,\varepsilon)$ is a sphere
obtained by gluing the following two hemispheres along their boundaries:

$\displaystyle b_{-}(S^{a,-}_{\varepsilon})$ $\displaystyle=B\cap\\{T\leq
S^{a,-}_{\varepsilon}\\}$ $\displaystyle b_{+}(\sigma)$
$\displaystyle=B\cap\\{T\geq\sigma\\}.$

We define two hemispheric bundles over these base spaces as follows:

$\displaystyle\mathcal{E}^{-}(S^{a,-}_{\varepsilon},\varepsilon)$
$\displaystyle=\mathcal{E}_{\varepsilon}(K)|_{b_{-}(S^{a,-}_{\varepsilon})}$
$\displaystyle\mathcal{E}^{+}(\sigma,\varepsilon)$
$\displaystyle=\mathcal{E}_{\varepsilon}(K)|_{b_{+}(\sigma)}.$

We complete the construction by defining a gluing map
$\varphi_{S,\varepsilon}:\mathcal{E}^{-}(S^{a,-}_{\varepsilon},\varepsilon)|_{b_{-}\cap\,b_{+}}\to\mathcal{E}^{+}(\sigma,\varepsilon)$.
Here “$b_{-}\cap b_{+}$” refers to the intersection curve of the hemisphere
boundaries, corresponding to a copy of $K$ in $B_{S}$. Let $\zeta_{L}$ and
$\zeta_{\sigma}$ be the values of $\zeta$ corresponding to
$T=S^{a,-}_{\varepsilon}$ and $T=\sigma$, respectively, defined using the
initial value problem (57). Let $E(\zeta,\lambda,\varepsilon)$ be the
solution, which has data $E_{L}$ at $\zeta=\zeta_{L}$ (by a time translation,
if necessary), where $E_{L}$ lies in the fiber of
$\mathcal{E}^{-}(S^{a,-}_{\varepsilon},\varepsilon)$ over
$(S^{a,-}_{\varepsilon},\lambda)$. A suitable gluing map is then the obvious
one which is induced by the flow:

$\displaystyle\varphi_{S,\varepsilon}E_{L}$
$\displaystyle=E(\zeta_{\sigma},\lambda,\varepsilon).$ (92)

The bundle $\mathcal{E}(S,\varepsilon)$ is then defined as

$\displaystyle\mathcal{E}(S,\varepsilon)$
$\displaystyle:=\mathcal{E}^{-}(S^{a,-}_{\varepsilon},\varepsilon)\cup_{\varphi_{S,\varepsilon}}\mathcal{E}^{+}(\sigma,\varepsilon).$
(93)

Note that at $\sigma=S^{a,-}_{\varepsilon}$, the map $\varphi_{S,\varepsilon}$
is the identity (since $\zeta_{L}=\zeta_{\sigma}$). Furthermore, the flow map
is continuous for each $\varepsilon>0$, implying that
$\varphi_{S,\varepsilon}$ forms a homotopy of isomorphisms. These facts imply
(see eg. Atiyah [2]) that

$\displaystyle\mathcal{E}_{\varepsilon}(K)$
$\displaystyle\cong\mathcal{E}(S,\varepsilon)).$ (94)

Note that we have fixed $\varepsilon>0$ up to now. Let us now fix
$S^{a,-}_{\varepsilon}$ and $S^{a,+}_{\varepsilon}$ and take $\varepsilon>0$
sufficiently small that Theorem 7.1 holds. By Theorem 7.1 (a) and (c),

$\displaystyle\lim_{\varepsilon\to 0}\hat{E}(T,\lambda,\varepsilon)$
$\displaystyle=\hat{E}_{0}(T,\lambda)$

uniformly for $(T,\lambda)\in b_{-}(S^{a,-}_{\varepsilon})\cup
b_{+}(S^{a,+}_{\varepsilon})$. Observe here that the information about the
jump map is already encoded in the solution $\hat{E}_{0}(T,\lambda)$, as
described in Theorem 7.1(c). Let us now determine the singular limit of the
gluing map $\varphi_{S,\varepsilon}$.

Let $\zeta_{L}$ and $\zeta_{R}$ correspond (as described earlier) to
$T=S^{a,-}_{\varepsilon}$ and $T=S^{a,+}_{\varepsilon}$, respectively, and
choose a solution $Z_{0}(\zeta,\lambda)$ of the reduced problem so that

$\displaystyle\hat{E}_{0}$
$\displaystyle=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[width("\iota_{0}(\zeta,\lambda)Z_{0}(\zeta,\lambda)")]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\iota_{0}(\zeta,\lambda)Z_{0}(\zeta,\lambda)}{\tmpbox}$

It follows from Theorem 7.1(a) that

$\displaystyle\varphi_{S,0}[\iota_{0}(S^{a,-}_{\varepsilon},\lambda)Z_{0}(S^{a,-}_{\varepsilon},\lambda)]$
$\displaystyle=\iota_{0}(\zeta_{R},\lambda)Z_{0}(\zeta_{R},\lambda).$ (95)

We now define the ‘singular’ limit of the glued hemispheric bundles
$\mathcal{E}(S,\varepsilon)$ in the obvious way:

$\displaystyle\mathcal{E}(S,0)$
$\displaystyle:=\mathcal{E}_{0}|b_{-}(S^{a,-}_{\varepsilon})\cup_{\varphi_{S,0}}\mathcal{E}_{0}|b_{+}(S^{a,+}_{\varepsilon}).$
(96)

We emphasize here that this is a topological limit, and so
$\mathcal{E}(S,\varepsilon)\cong\mathcal{E}(S,0)$ directly—this is the
advantage of using the augmented unstable bundle. It thus follows from (94)
that $\mathcal{E}_{\varepsilon}(K)\cong\mathcal{E}(S,0)$ for each
$\varepsilon>0$ sufficiently small.

We have so far held $S^{a,-}_{\varepsilon}$ and $S^{a,+}_{\varepsilon}$ fixed.
Noting that the required smallness of $\varepsilon$ from the above argument is
independent of these parameters, we are free to send $S^{a,-}_{\varepsilon}$
to 0 from below and $S^{a,+}_{\varepsilon}$ to 0 from above. By Theorem
7.1(b), $\varphi_{S,0}$ approaches $\varphi^{H}$ as defined in (60). $\Box$

## 8 Counting the slow eigenvalues

We now compute the slow eigenvalues $\lambda\in\Omega$; see Def. 6.4. We will
adapt the technique presented in Section E of [8]; we split the projectivized
equation (88) for (47) into its real and imaginary parts, with the spatial
eigenvalue parameter written as $\lambda=\mu+i\omega$, and we consider
separately the cases $\omega\neq 0$ and $\omega=0$. Afterwards we may restrict
our analysis to the real line; using comparison of solutions together with a
Gronwall estimate, we demonstrate that there are no positive real eigenvalues.
Here we need the monotonicity hypothesis 2.8 to ensure that we can work with
the chart defined by $V\neq 0$ for $\lambda\geq 0$. Generally speaking, the
jump condition does not introduce any new complications with respect to these
arguments.

######  Theorem 8.1.

There are exactly two slow eigenvalues in $\Omega$, given by $\lambda_{0}=0$
and $\lambda_{1}$, with $\text{Im}(\lambda_{1})=0$ and
$R^{\prime}(0)<\lambda_{1}<0$. Both $\lambda_{0}$ and $\lambda_{1}$ are
simple.

######  Remark 8.2.

As far as spectral stability of the wave for $\varepsilon>0$ is concerned, it
suffices to verify that the eigenvalue with the largest real part in the point
spectrum is the simple translational eigenvalue $\lambda_{0}$. We resort to a
numerical calculation using a Riccati-Evans function to determine the
existence of the secondary eigenvalue $\lambda_{1}$, but the simplicity of
both eigenvalues are verified rigorously. $\triangle$

Proof that $\lambda_{0}=0$ is the eigenvalue of largest real part:

We will prove that all slow eigenvalues are real, and furthermore there is no
slow eigenvalue $\lambda$ with $\text{Re}(\lambda)>0$. The asymptotic systems
associated with the projectivized linearized slow flow (56) admit the fixed
points $\hat{u}^{-},\,\hat{s}^{-}$ at $\bar{U}=0$ and
$\hat{u}^{+},\hat{s}^{+}$ at $\bar{U}=1$, defined as follows:

$\displaystyle\hat{u}^{-}(\lambda)$
$\displaystyle=\frac{c-\sqrt{c^{2}+D(0)(R^{\prime}(0)-\lambda)}}{2D(0)}$ (97)
$\displaystyle\hat{s}^{-}(\lambda)$
$\displaystyle=\frac{c+\sqrt{c^{2}+D(0)(R^{\prime}(0)-\lambda)}}{2D(0)}$
$\displaystyle\hat{u}^{+}(\lambda)$
$\displaystyle=\frac{c-\sqrt{c^{2}+D(1)(R^{\prime}(1)-\lambda)}}{2D(1)}$
$\displaystyle\hat{s}^{+}(\lambda)$
$\displaystyle=\frac{c+\sqrt{c^{2}+D(1)(R^{\prime}(1)-\lambda)}}{2D(1)}.$

Linear analysis verifies that $\hat{u}^{\pm}$ are attractors and
$\hat{s}^{\pm}$ are repellers with respect to the projectivized asymptotic
dynamics, corresponding to the asymptotic unstable resp. stable
eigendirections. We remind the reader of the geometric characterisation of
eigenvalues of (47) in terms of their asymptotic behavior; if $\lambda$ is not
a slow eigenvalue, it suffices to show that ‘the’ unique solution
$S_{0}(\zeta,\lambda)$ which tends to $\hat{u}^{-}(\lambda)$ as
$\zeta\to-\infty$ does not tend to $\hat{s}^{+}(\lambda)$ as
$\zeta\to+\infty$.777Strictly speaking, $S_{0}$ refers to a pair of solutions
$S^{a,-}_{\varepsilon}$ and $S^{a,+}_{\varepsilon}$, defined for $\zeta\leq 0$
and $\zeta\geq 0$, which are uniquely defined by two constraints: the
aforementioned asymptotic constraint at $\zeta=-\infty$, and the jump
condition defined using (90).

We now write (56) in terms of its real and imaginary parts. Specifically,
writing $S=X+iY$ and $\lambda=\mu+i\omega$, we have

$\displaystyle\dot{X}$ $\displaystyle=R^{\prime}(\bar{U})-\mu-
cX+(X^{2}-Y^{2})D(\bar{U})$ (98) $\displaystyle\dot{Y}$
$\displaystyle=-\omega-cY+2XYD(\bar{U}).$

We consider the two subcases $\omega\neq 0$ and $\omega=0$.

The subcase $\omega\neq 0$. Let us focus on the case $\omega>0$; the case
$\omega<0$ is similar. We have $\text{Im}\,(\hat{u}^{\pm}(\lambda))<0$ and
$\text{Im}\,(\hat{s}^{\pm}(\lambda))>0$ for each $\lambda\in\Omega$, by
applying the inequality $R^{\prime}(1)<R^{\prime}(0)<\text{Re}\,(\lambda)$ on
this set to the expressions (97) for the asymptotic fixed points. On the other
hand, the half-plane $\\{Y\leq 0\\}$ is forward invariant since
$\dot{Y}=-\omega<0$ along $\\{Y=0\\}$. Furthermore, the jump condition is
monotone decreasing in $Y$, i.e.
$S^{a,+}_{\varepsilon}(0,\lambda)_{Y}=(J_{\lambda}(S^{a,-}_{\varepsilon}(0,\lambda)))_{Y}\leq
S^{a,-}_{\varepsilon}(0,\lambda)_{Y}$. Thus, if
$S(\zeta,\lambda)=X(\zeta,\lambda)+iY(\zeta,\lambda)$ remains on the chart
specified by $V\neq 0$, then it is impossible that $S(\zeta,\lambda)\to
s^{+}(\lambda)$ as $\zeta\to+\infty$.

It could happen that the solution $S_{0}(\zeta,\lambda)$ leaves the chart by
blowing up in the $Y\to-\infty$ direction, emerging ‘on the other side’ of the
half-plane $\\{Y>0\\}$ from $Y=+\infty$ to make a connection to
$s^{+}(\lambda)$. Let us show that any such blow-up leads to a contradiction.
Following the style of Lemma 6.6 in [8], we observe that if the blow-up
happens at $\zeta_{0}=+\infty$, then $S_{0}(\zeta,\lambda)$ remains inside the
closure of the image of $\\{Y\leq 0\\}$ on the Riemann sphere, and
$s^{+}(\lambda)$ is bounded away from this closed set; hence, we suppose that
$\zeta_{0}\in\mathbb{R}$, and that $\zeta_{0}$ is the smallest real number so
that $S_{0}(\zeta,\lambda)$ becomes unbounded as $\zeta\to\zeta_{0}$ but for
which $S_{0}(\zeta,\lambda)$ remains finite within the lower half-plane for
all $\zeta<\zeta_{0}$. Then $T_{0}(\zeta,\lambda):=S_{0}(\zeta,\lambda)^{-1}$
remains well-defined for all $\zeta$ sufficiently close to $\zeta_{0}$. This
motivates the corresponding change in chart

$\displaystyle s=\frac{X}{X^{2}+Y^{2}},$ $\displaystyle\hskip
28.45274ptt=\frac{-Y}{X^{2}+Y^{2}}.$ (99)

The corresponding dynamical system (98) expressed in the new chart is

$\displaystyle\dot{s}$ $\displaystyle=-D(\bar{U})+cs-2\omega
st-(R^{\prime}(\bar{U})-\mu)(s^{2}-t^{2})$ (100) $\displaystyle\dot{t}$
$\displaystyle=ct-2st(R^{\prime}(\bar{U})-\mu)+\omega(s^{2}-t^{2}),$

with the solution written as $T_{0}(\zeta)=s(\zeta)+it(\zeta)$. If
$Y\to-\infty$, then both $s$ and $t$ tend to 0 as $\zeta\to\zeta_{0}$;
furthermore, $t(\zeta)>0$ for values of $\zeta<\zeta_{0}$ sufficiently close
to the blow-up value according to (99). The contradiction will arise by
considering the behavior of $t(\zeta)$ for $\zeta$ near $\zeta_{0}$ using
Taylor expansions. In the following argument, we suppose that $\zeta_{0}\neq
0$, i.e. we are not exactly at the jump and all two-sided limits of the
relevant functions exist; we discuss the case $\zeta_{0}=0$ later. By Taylor
expanding $s(\zeta)$ around $\zeta=\zeta_{0}$ and using the first equation in
(100), we find that

$\displaystyle s(\zeta)$
$\displaystyle=-D(\bar{U}(\zeta_{0}))(\zeta-\zeta_{0})+\mathcal{O}(\zeta-\zeta_{0})^{2}.$
(101)

Using the second equation in (100), we note that $\dot{t}=\ddot{t}=0$ at
$\zeta=\zeta_{0}$; however, we have $\dddot{t}(\zeta_{0})=2\omega
D(\bar{U}(\zeta_{0}))^{2}$, and so the Taylor expansion of $t(\zeta)$ gives

$\displaystyle t(\zeta)$ $\displaystyle=\frac{\omega
D(\bar{U}(\zeta_{0}))^{2}}{3}(\zeta-\zeta_{0})^{3}+\mathcal{O}(\zeta-\zeta_{0})^{4}.$
(102)

But this implies that $t(\zeta)<0$ for $\zeta<\zeta_{0}$ and $\zeta$
sufficiently close to $\zeta_{0}$, which produces a contradiction.

If the blow-up occurs exactly at the point of discontinuity $\zeta_{0}=0$, the
argument above survives by constructing continuous extensions of
$D(\bar{U}(\zeta))$ and $R^{\prime}(\bar{U}(\zeta))$, since their right limits
still exist. Altogether, we have shown that there are no slow eigenvalues with
nonzero imaginary part.

The subcase $\omega=0$. The (un)stable eigenvectors associated with the
asymptotic subspaces are real, and by (98) the subset $\\{Y=0\\}$ is invariant
when $\omega=0$; hence, we are able to focus on the real one-dimensional
problem given by the first equation in (98)

$\displaystyle\dot{X}$
$\displaystyle=f(X,\bar{U};\mu):=R^{\prime}(\bar{U})-\mu-cX+X^{2}D(\bar{U}).$
(103)

By Remark 6.5, $\mu=0$ is a slow eigenvalue. Now suppose $\mu>0$. Noting that
$\partial f/\partial\mu<0$ and that $\partial\hat{u}^{-}/\partial\mu<0$ from
(97), any solution $X(\zeta,\lambda)$ which tends to $\hat{u}^{-}(\lambda)$ as
$\zeta\to-\infty$ satisfies the comparison

$\displaystyle X(\zeta,\mu^{\prime})\leq X(\zeta,\mu)$ (104)

for all $\zeta\in\mathbb{R}$ whenever $0<\mu\leq\mu^{\prime}$ (note that this
comparison is preserved across the jump). Each such solution for any given
$\mu>0$ has a common upper bound, given by the solution for $\lambda=\mu=0$,
which does not blow up by Hypothesis 2.8. We also have
$\partial\hat{s}^{+}/\partial\mu>0$ for $\mu>0$ from (97); thus,
$\hat{s}^{+}(\mu)$ lies on the other side of the connection for $\mu=0$
(relative to the solution $X(\zeta,\mu)$) . As long as we can show that the
solution for $\mu>0$ does not blow up (potentially allowing $X(\zeta,\mu)$ to
connect to $s^{+}$ by ‘looping around’ the Riemann sphere), it is impossible
for the solution $X(\zeta,\lambda)$ to approach $\hat{s}^{+}(\mu)$ as
$\zeta\to\infty$, and we will be done.

According to the comparison to the common upper bound stated above, the
solution $X(\zeta,\lambda)$ can only possibly blow up in the direction
$X\to-\infty$. This does not happen since for each $\mu>0$, a specific lower
bound is given by Gronwall’s inequality and a comparison to the system
$\dot{X}=\min_{0\leq\bar{U}\leq 1}R^{\prime}(\bar{U})-\mu-cX$, whose solutions
do not blow up. $\Box$

The existence of $\lambda_{1}$. We now verify the remaining statements in
Theorem 8.1. It is a straightforward task to numerically integrate the one-
dimensional projection of the system (98) along the invariant subspace of real
solutions $\\{Y=0\\}$. The solution can blow up by leaving the chart given by
$S=P/V$, $V\neq 0$. If this happens, the system (100) can be numerically
integrated without issue on the chart $T=V/P$, $P\neq 0$, around a small
interval surrounding the pole, and then we can return to the original chart.

(a) (b)

Figure 8: Numerical integration of (98) along $\\{Y=0\\}$ for (a)
$\lambda=0.1$, (b) $\lambda=\lambda_{0}=0$, (c) $\lambda=-0.1$, (d)
$\lambda=-0.3$, (e) $\lambda=-0.8$, (f) $\lambda=\lambda_{1}\approx-0.80925$,
(g) $\lambda=-0.85$. Red resp. black dots: (projectivisation of) the unstable
resp. stable eigenvectors of the saddle-point at $\bar{U}=1$. Magenta
segments: concatenated fast jumps defined by the map (90).

(c) (d) (e) (f) (g)

These numerics are depicted in Fig. 8 for seven values of $\lambda$ within the
range $[-0.85,0.1]$. As shown in Fig. 8(a)–(c) and (e)–(g), a connection is
formed between the unstable eigendirection of the saddle-point at $\bar{U}=0$
and the stable eigendirection of the saddle-point at $\bar{U}=1$ for the
distinguished values $\lambda=\lambda_{0}=0$ and $\lambda_{1}\approx-0.80925$.
Simultaneously, we observe the winds that are necessarily generated upon
crossing these eigenvalues. These winds can be continued in the parameter
$\lambda$—such coordinate singularities can of course be characterised as
zeroes of the flow with respect to a suitably chosen chart. They propagate to
the left and are preserved across the jump, as is shown in Fig. 8(c)–(d).

The simplicity of the eigenvalues in the reduced problem. We have now
determined the existence of eigenfunctions as solutions to the Riccati
formulation (103) of the linearised problem (47). Recall that the jump
conditions for $X$ are given in Def. 6.4, and boundary conditions for $X$
given in (97), i.e.

$\lim_{\zeta\to\pm\infty}X=\hat{u}^{-}(\mu),\hat{s}^{+}(\mu).$

In terms of the original reduced problem (47),

this means that we have a value $\lambda=\mu$, and a solution to (47)
satisfying the boundary conditions

$\lim_{\zeta\to\pm\infty}P,V=0,$

and appropriate jump conditions.

Differentiating the equation for $V$ with respect to $\zeta$, and substituting
in for $\dot{P}$, and multiplying through by

$g(\zeta):=\exp\left(-\int^{\zeta}(\log(D(\bar{U})))_{s}+cds\right)=\frac{e^{-c\zeta}}{D(\bar{U})}$

we are led to the ‘Sturm-Liouville’ form of the reduced problem on the slow
manifold:

$\displaystyle
e^{c\zeta}\frac{d}{d\zeta}\left(\frac{e^{-c\zeta}}{D(\bar{U})}\frac{d}{d\zeta}V\right)+\tilde{Q}(\zeta)V=\mu
V.$ (105)

where $\tilde{Q}(\zeta)$ is defined as:

$\tilde{Q}(\zeta):=R^{\prime}(\bar{U})+\frac{cD^{\prime}(\bar{U})\bar{U}_{\zeta}}{D(\bar{U})^{2}}$

Letting $L$ be the symmetric linear operator :

$L[y]:=e^{c\zeta}\frac{d}{d\zeta}\left(\frac{e^{-c\zeta}}{D(\bar{U})}\frac{dy}{d\zeta}\right)+\tilde{Q}(\zeta)y$

we are interested in establishing the simplicity of the eigenvalues of such an
operator where $U(\zeta)$ is given as the solution on the slow manifold with
the appropriate boundary conditions, and jump at $\zeta=0$.

######  Remark 8.3.

As $D(\bar{U})$ is discontinuous at $\zeta=0$, the domain of the linear
operator $L$ is a function space that will necessarily incorporate a jump
condition which is compatible with the one from Def 6.4. Further, the
weighting function $e^{c\zeta}$ will define the inner product space on which
$L$ is symmetric and well-defined. Here we only aim to establish simplicity of
any eigenvalues/eigenfunctions, which a priori exist. $\triangle$

First we note that because $L$ is symmetric on eigenfunctions, the Fredholm
alternative means that the geometric multiplicity of any eigenvalues must be
the same as the algebraic multiplicity, and in particular if
$(L-\mu)[y_{1}]=0$, there can be no non-zero solutions to $(L-\mu)[y]=y_{1}$.

Now, suppose that $v_{1}$ and $v_{2}$ are eigenfunctions of $L$ with the same
eigenvalue $\mu$, so $L[v_{j}]=\mu v_{j}$. On the one hand, we have

$\displaystyle v_{1}L[v_{2}]-v_{2}L[v_{1}]=\mu(v_{1}v_{2}-v_{2}v_{1})=0,$
(106)

while on the other we have

$\displaystyle v_{1}L[v_{2}]-v_{2}L[v_{1}]$
$\displaystyle=e^{c\zeta}\frac{d}{d\zeta}\left(g(\zeta)(v_{1}\frac{dv_{2}}{d\zeta}-\frac{dv_{1}}{d\zeta}v_{2})\right).$
(107)

Thus, the quantity

$\displaystyle
g(\zeta)(v_{1}\frac{dv_{2}}{d\zeta}-\frac{dv_{1}}{d\zeta}v_{2})=:\frac{e^{-c\zeta}}{D(\bar{U})}W(v_{1},v_{2})$
(108)

is a constant. Evaluating at $\zeta=+\infty$ we have

$\lim_{\zeta\to+\infty}\frac{e^{-c\zeta}}{D(\bar{U})}W(v_{1},v_{2})=0.$

We observe that

$v_{1,2}(\zeta)\sim
e^{\frac{1}{2}\left(c+\sqrt{c^{2}+4D(0)(\mu-R^{\prime}(0))}\right)\zeta}$

as $\zeta<0$ grows very large in magnitude, and so in particular

$\lim_{\zeta\to-\infty}\frac{e^{-c\zeta}}{D(\bar{U})}W(v_{1},v_{2})=0$

when $\mu>R^{\prime}(0)$. Thus
$\frac{e^{-c\zeta}}{D(\bar{U})}W(v_{1},v_{2})\equiv 0$ on both sides of the
jump and hence we have that $W(v_{1},v_{2})\equiv 0$. Since the Wronskian of
the two eigenfunctions is identically zero, they must be linearly dependent.
We conclude that the slow eigenvalues $\lambda_{0}$ and $\lambda_{1}$ are both
simple. $\Box$

Figure 9: Evaluation of the Riccati-Evans function (109) along circles $C_{z}$
of radius 0.03 centered at the point $z\in\mathbb{C}$. The winding numbers
along each contour are given in the legend and are also depicted by the
arrowheads. Note that $\lambda_{0}$ and $\lambda_{1}$ denote the two
eigenvalues and $\lambda_{p}=-0.08$ is the approximate location of the pole of
$E^{\Sigma}(\lambda)$.

We supplement our proofs with a numerical demonstration of the existence of
the simple eigenvalues $\lambda_{0},\,\lambda_{1}$, by means of a Riccati-
Evans function. Define the cross section $\Sigma=\\{\bar{U}=0.95\\}.$ The
Riccati-Evans function associated with the Riccati equation (98) on the
coordinate chart $\\{V\neq 0\\}$ is

$\displaystyle E^{\Sigma}(\lambda)$
$\displaystyle:=s^{\Sigma}_{1}(\lambda)-u^{\Sigma}_{0}(\lambda),$ (109)

where $s^{\Sigma}_{1}(\lambda)$ denotes the (first) intersection of the unique
nontrivial solution $s_{1}(\zeta,\lambda)$ of (98) which converges
asymptotically to the stable eigenvector of the saddle point at $\bar{U}=1$ as
$\zeta\to+\infty$, and similarly for $u^{\Sigma}_{0}(\lambda)$, which connects
to the unstable eigendirection of the saddle point as $\zeta\to-\infty$.
Compare our definition to the general construction in [10].

We highlight a few key points about our function (109). The function is
meromorphic (hence satisfying the argument principle), and it vanishes on
eigenvalues $\lambda$. Its zeroes are intrinsic, depending neither on the
choice of section nor on the choice of chart. On the other hand, as we can
anticipate from the dynamics depicted in Fig. 8, the poles arise for values of
$\lambda$ where the solution leaves the chart by winding. In other words, the
poles are an artifact of the choice of coordinate chart as well as the choice
of section, and can usually be moved (or even removed entirely) by a judicious
selection of the chart and the section (with our choice of $\Sigma$, we locate
a pole near $\lambda_{p}=-0.08$). See [10] for a general exposition.

In this analysis we find it instructive to work with the ‘naive’ chart
$S=P/V$, $V\neq 0$, for all values of $\lambda$—the dimensionality of the slow
eigenvalue problem is low enough that we can easily demonstrate the utility of
the argument principle. As shown in Fig. 9, the multiplicity of each
eigenvalue can be readily computed by evaluating the corresponding winding
number $W(E^{\Sigma},C)$ along simple closed contours $C$. Contours which
surround poles, corresponding to blow-up of solutions off the chosen
coordinate chart, will map under $E$ to contours with a negative winding
number (corresponding to a clockwise orientation). Here we demonstrate that
the poles are also simple. This completes our numerical verification of
Theorem 8.1.

## 9 Concluding remarks

We have given a complete characterisation of the spectral stability problem
for shock-fronted travelling waves of the regularized system (1). But other
types of high-order regularization terms can be applied to the underlying
system that exhibits shock solutions; it can further be shown that these
nonequivalent regularizations pick out distinct one-parameter families of
shock-fronted travelling waves, which limit to singular solutions satisfying
different rules.

Consider the following system which applies to both a viscous relaxation and a
fourth-order nonlocal regularization, where the new parameter $a\geq 0$
characterises the relative weighting of the two regularizations.

$\displaystyle\frac{\partial\bar{U}}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
x}\left(D(\bar{U})\frac{\partial\bar{U}}{\partial
x}\right)+R(\bar{U})+\varepsilon a\frac{\partial^{3}\bar{U}}{\partial
x^{2}\partial t}-\varepsilon^{2}\frac{\partial^{4}\bar{U}}{\partial x^{4}}.$
(110)

Setting $a=0$ recovers a regularized system with ‘purely’ nonlocal
regularization. In this case, it can be shown that the corresponding one-
parameter family of travelling waves solutions has a singular limit as a
shock-fronted travelling wave which satisfies a so-called ‘equal area rule’–
the shock connection across disjoint branches of the potential function
$F(\bar{U})$ is selected so that the area bounded above and below the shock
height are exactly balanced. Recent work [4] has shown that families of shock-
fronted waves persist robustly for $a>0$, with each such family satisfying a
generalized equal area rule in the singular limit. Furthermore, there is a
finite value $a=a_{V}>0$ for which the shock connection is formed at the fold,
again resulting in viscous-type shocks. The problem in this paper can be
thought of as the regularization in the (scaled) limit as $a\to\infty$.

Figure 10: Plot of the essential spectrum (shaded regions) of the of the
linearised operator L in (112), which shows $\Omega$ and the $\mathcal{A}_{i}$
partitioning the complex plane via the Fredholm borders (continuous spectrum).
The shape of the essential spectrum means that the operator $L$, for non-zero
$\varepsilon$ is sectorial. The dashed line (red online) is one of the
Fredholm borders coming from the linearisation of the reduced problem about
the (constant) steady-state $\bar{U}^{-}$ c.f. the right figure in Fig 4. For
the figure, $\varepsilon=0.1$ and $c=a=1$, with $D$ and $R$ as in (3) and (4).

Thus, let $a>0$ be given and assume the existence of such a family of
travelling waves for (110). As before, we can find such waves as standing wave
solutions $\bar{U}(z,t)\equiv\bar{U}(z)$ to:

$\displaystyle(\bar{U}-\varepsilon
a\bar{U}_{zz})_{t}=-\varepsilon^{2}\bar{U}_{zzzz}-\varepsilon
ac\bar{U}_{zzz}+(F(\bar{U}))_{zz}+c\bar{U}_{z}+R(\bar{U}).$ (111)

We now discuss how the stability problem changes in this more general case.
Linearising about a standing wave solution $\bar{U}(z)$ to (111) leads to the
eigenvalue problem

$\displaystyle\lambda p-\varepsilon a\lambda
p_{zz}=-\varepsilon^{2}p_{zzzz}-\varepsilon
acp_{zzz}+(D(\bar{U})p)_{zz}+cp_{z}+R^{\prime}(\bar{U})p.$ (112)

Defining the variables

$\displaystyle s$ $\displaystyle:=\varepsilon^{2}p_{zzz}+\varepsilon
acp_{zz}-((D(U)+\varepsilon a\lambda)p)_{z}-cp$ (113) $\displaystyle r$
$\displaystyle:=\varepsilon^{2}p_{zz}+\varepsilon acp_{z}-(D(U)+\varepsilon
a\lambda)p$ (114) $\displaystyle q$ $\displaystyle=\varepsilon p_{z}+acp,$
(115)

we can write the closed system

$\displaystyle\begin{pmatrix}\varepsilon p\\\ \varepsilon q\\\ r\\\
s\end{pmatrix}_{z}$ $\displaystyle=\begin{pmatrix}-ac&1&0&0\\\
D(\bar{U})+\varepsilon a\lambda&0&1&0\\\ c&0&0&1\\\
R^{\prime}(\bar{U})-\lambda&0&0&0\end{pmatrix}\begin{pmatrix}p\\\ q\\\ r\\\
s\end{pmatrix},$ (116)

with two fast and two slow variables. The calculations for the essential (and
absolute) spectrum follow the same ideas as in Section 5. Presuming the
existence of a wave $\bar{U}$ which exponentially approaches its end states
$\bar{U}^{\pm}$, we have the dispersion relations are the pair of parametrised
equations given by $k\in\mathbb{R}$:

$\lambda_{\pm}=\frac{-\varepsilon^{2}k^{4}-D(\bar{U}^{\pm})k^{2}+R^{\prime}(\bar{U}^{\pm})}{1+a\varepsilon
k^{2}}+ick.$

These are a pair of curves in the complex plane which are always opening
leftward and intersecting the real axis at the points
$(R^{\prime}(\bar{U}^{\pm}),0)$ in the left half plane. The dispersion
relations again form the Fredholm borders and make up the continuous spectrum.
As before, $\mathbb{C}$ is partitioned into five regions: the region $\Omega$
which contains the right half plane, together with $\mathcal{A}_{j}$ for
$j=1,2,3,4$. The essential spectrum in this case can again be determined by
considering the signatures of the asymptotic matrices, and can be seen to
consist of the regions $\mathcal{A}_{1,2,3}$, which is the region “between”
the Fredholm borders. We have

Region | $\textrm{sgn}(A^{-})$ | $\textrm{sgn}(A^{+})$
---|---|---
$\Omega$ | $(-,-,++)$ | $(-,-,+,+)$
$\mathcal{A}_{1}$ | $(-,-,-,+)$ | $(-,-,+,+)$
$\mathcal{A}_{2}$ | $(-,-,++)$ | $(-,-,-,+)$
$\mathcal{A}_{3}$ | $(-,-,+,+)$ | $(-,-,-,+)$
$\mathcal{A}_{4}$ | $(-,-,-,+)$ | $(-,-,-,+)$

Again, the Fredholm borders will be close to the Fredholm borders of the
linearised reduced problem, linearised about the steady-states corresponding
to the end states of the unperturbed problem. Eventually, as with the third
order perturbation problem, the higher order modes in the dispersion relations
will dominate, and the essential spectrum will diverge from the appropriate
continuous spectrum of the unperturbed problem (see Fig: 10). In contrast to
the case of ‘pure’ viscous relaxation, however, the Fredholm borders do not
asymptote to vertical lines in the complex plane and so the operator is
sectorial, and spectral stability in this case would indeed imply linearised
stability of the perturbation problem.

Having shown that the essential spectrum remains remarkably well-behaved in
this more general context, let us now highlight key differences (and
difficulties) in calculating the point spectrum. Here, the problem is now
four-dimensional, and it can be verified with direct calculation that the two-
dimensional slow manifolds on which lie saddle-type equilibria at $\bar{U}=0$
and $\bar{U}=1$ are now saddle-type (i.e. there is now both a fast stable and
fast unstable direction). The augmented unstable bundle construction for
$\varepsilon>0$ now defines a complex 2-plane bundle over the sphere.

It is natural to ask whether the present problem is amenable to the splitting
techniques introduced in [8]: namely, is it possible to decompose this 2-plane
bundle into ‘fast unstable’ and ‘slow unstable’ line bundles which are
controlled by separated reduced eigenvalue problems?

We conjecture that such a separation is not possible: as in the ‘pure’ viscous
case, it can be directly calculated that the eigenvalue problem again
introduces the eigenvalue parameter $\lambda$ only ‘weakly,’ i.e. through
$\mathcal{O}(\varepsilon\lambda)$ terms. Thus, the reduced fast eigenvalue
problem again degenerates. As mentioned in the introduction, this poses issues
for the construction of a fast elephant trunk over the entire wave;
consequently, it does not appear to be feasible with the present techniques to
uniformly separate a fast unstable line bundle from a slow unstable one.
Furthermore, any such reduced line bundle over the fast layer is governed by
the layer flow, which connects an unstable eigendirection to a stable one in
the singular limit (i.e. we are not able to construct an augmented unstable
line bundle for the reduced problem).

Nonetheless, we assert that estimates similar to Lemma 7.7 can be brought to
bear to regain control of the unstable 2-plane bundle as we track it across
the fast shock layer. It would be of interest to write down sharper (i.e.
exponential closeness) estimates to the invariant manifold near the slow
subbundle, as discussed in Remark 7.8. The calculation in Appendix D gives a
concrete example of such an exponential closeness result in an elementary
problem. Such calculations appear to be very cumbersome in the general case.
It would also be interesting to situate the construction of such invariant
manifolds within the context of general invariant manifold results in
nonautonomous dynamical systems (see e.g. [19]). These are topics of ongoing
work.

## Acknowledgement

Both authors acknowledge support from the Australian Research Council
Discovery Project grant DP200102130. The authors thank Martin Wechselberger at
the University of Sydney for several insightful discussions. The first author
thanks Martin Rasmussen at Imperial College London for a clarifying discussion
about invariant manifolds of nonautonomous dynamical systems.

##  Appendix A Construction of the fast and slow elephant trunks

Proof of Lemma 7.4: We follow the strategy to the proof of Lemma 4.2 in [8].
The goal is to verify the following four conditions for the projectivization
of the linearized system $\eqref{eq:visclinfast}$:

  * •

($(4.6)_{GJ}$ in Sec. IV of [8]): There exists $\alpha>0$ independent of
$\varepsilon$ and $\xi\in I$ so that the curve of critical points
$\beta_{0}(\gamma,\varepsilon)$ associated with the frozen family (71)
satisfies

$\text{Re}\,\sigma[G_{\beta}(\beta_{0}(\gamma,\varepsilon),\gamma,\varepsilon)]<-\alpha;$

  * •

($(4.7)_{GJ}$ in Sec. IV of [8]): for any $d>0$ there exists
$\varepsilon_{0}=\varepsilon_{0}(d)$ and a nested family of subintervals
$I(d)\subset I$ (i.e. $I(d_{1})\subset I(d_{2})$ for $d_{1}<d_{2}$) with

$\text{sup}_{(\beta,\xi,\varepsilon)\in
C}\\{|G|_{\xi},||G_{\beta,\varepsilon}||,|\beta_{0\xi}|\\}<d,$

where the set $C$ can be taken as

$C=\\{(\beta,\xi,\varepsilon):|\beta-\beta_{0}(\xi,\varepsilon)|<c_{0},\xi\in
I(d),0<\varepsilon\leq\varepsilon_{0}\\}$

for some $c_{0}>0$;

  * •

($(4.8)_{GJ}$ in Sec. IV of [8]): we have

$K=\text{sup}_{C}||D^{2}G(\beta,\xi,\varepsilon)||<\infty;\text{~{}and}$

  * •

($(4.9)_{GJ}$ in Sec. IV of [8]): let
$A(\gamma):=G_{\beta}(\beta_{0}(\gamma,\varepsilon),\gamma,\varepsilon)$. Then
there exists some invertible matrix $Y(\gamma)$ depending smoothly on
$\gamma$, such that the following holds for some $a>0$ depending only on
$\alpha$ (from condition $(4.6)_{GJ}$):

$\displaystyle\text{Re}(A(\gamma)\beta,\beta)_{\gamma}$
$\displaystyle<-a|\beta|_{\gamma}^{2},$ $\displaystyle||Y(\gamma)||_{\gamma}$
$\displaystyle=1,$ $\displaystyle||Y_{\gamma}(\gamma)||_{\gamma}$
$\displaystyle\leq d,$

where we define the inner product
$(\beta_{1},\beta_{2})_{\gamma}:=Y(\gamma)\beta_{1}\cdot\overline{Y(\gamma)\beta_{2}}$
and the norm $|\beta|_{\gamma}=(\beta,\beta)^{1/2}_{\gamma}$.

Lemma 4.1 in [8] then gives the result. By Lemma 4.2, there exists
$\bar{\xi}>0$ such that the second bullet point in the Lemma above holds for
all $|\xi|\geq\bar{\xi}$ and all sufficiently small $\varepsilon>0$. This
verifies the condition $(4.6)_{GJ}$. Furthermore, the branch of equilibria
$\beta_{0}(\xi,\lambda,\varepsilon)$ for the corresponding frozen family is
uniformly bounded for all $|\xi|>\bar{\xi}$ and all $\lambda\in K$, with
$K\subset\Omega$ a fixed contour. Hence condition $(4.8)_{GJ}$ holds for some
constant $K$, with $c_{0}$ in the definition equal to 1.

Let us check the remaining conditions $(4.7)_{GJ}$ and $(4.9)_{GJ}$. Take the
interval $I$ to be either
$I_{-}=\\{\xi\in\mathbb{R}:-\infty<\xi<-\bar{\xi}\\}$ or
$I_{+}=\\{\xi\in\mathbb{R}:\bar{\xi}<\xi<+\infty\\}$. On either interval, the
projectivized vector field $G(\beta,\xi,\lambda,\varepsilon)$ depends on $\xi$
through components of the (nonautonomous) linearisation matrix, in particular
through the diffusion and reaction terms (i.e. through $\bar{u}$). Hence for
each constant $L>0$ there exists $\varepsilon_{0}:=\varepsilon_{0}(L)$, so
that for $0<\varepsilon\leq\varepsilon_{0}$ we have

$\max\\{|G_{\xi}|,||G_{\xi,\beta}||,|\beta_{0,\xi}|\\}<L\max\\{|u^{\prime}(\xi,\varepsilon)|\\}.$

Now let $n(\delta)$ denote a $\delta$-tube around the singular limit of the
travelling wave and assume that $I=I_{-}$. Resetting $\varepsilon_{0}$ again
if necessary, we can assume that the travelling wave $x(\xi,\varepsilon)$ lies
within $n(\delta)$ for each $\xi\leq\xi_{L}(\delta)$ . Then there exists a
constant $M>0$ so that $|\bar{u}^{\prime}(\xi,\varepsilon)|\leq M\delta$ for
all $\xi\leq\xi_{L}(\delta)$, since $\bar{u}^{\prime}=\bar{v}-\Phi(\bar{u})$
is uniformly bounded along the entire wave. Hence we may take $\delta(d)=d/M$
and define

$I_{-}(d)=\\{\xi:\xi_{L}(\delta(d))\\}.$

Note that $\delta(d)$ depends smoothly on $d$ and that $\delta(0)=0$, and
furthermore that the definition for $I_{-}(d)$ provides nested intervals. This
verifies condition $(4.7)_{GJ}$.

To verify the remaining condition $(4.9)_{GJ}$, it is enough to note that the
linearization
$A(\xi,\lambda,\varepsilon)=A_{0}(\xi,\lambda,\varepsilon)+\mathcal{O}(\delta)$
of $G$ (the projectivization of the linear system), evaluated along the strong
unstable eigendirection, is a $2\times 2$ matrix which has two
$\mathcal{O}(1)$ negative eigenvalues. This matrix
$A(\xi,\lambda,\varepsilon)$, is necessarily negative definite for $I_{L}(d)$,
and hence we can take $Y=A_{0}^{-1}$ such that $Y^{-1}AY$ produces a matrix
with diagonal entries of uniformly negative real part and remaining entries of
$\mathcal{O}(\delta)$. This immediately implies the needed inequalities.
Applying Lemma 4.1 in [8], we have constructed the necessary elephant trunk
$\Omega_{-}^{f}$ over $S^{a,-}_{\varepsilon}$.

The preceding discussion applies identically for the case $I=I_{+}$ to produce
a fast elephant trunk along $S^{a,+}_{\varepsilon}$. $\Box$

##  Appendix B Lemmas for estimates near the slow subbundle

######  Lemma B.1.

Let $\delta,\,a>0$ be fixed. There exists $\bar{\varepsilon}>0$ sufficiently
small and $T>0$, both depending only on $\delta$ and $a$, so that for each
$0<\varepsilon\leq\bar{\varepsilon}$ and $|\gamma|\geq a/\varepsilon$, each
solution $\hat{y}(s)$ of (79) must satisfy at least one of the following:

* (ii)

$\hat{y}(-T)\in N_{\delta}(\hat{f}_{1}(\gamma,\lambda,\varepsilon))$.

######  Remark B.2.

Fix some $\gamma$ as above and let $\hat{y}(s)$ denote a solution to the
corresponding member of the family of projectivized frozen systems in (79),
and let $y(s)\in\pi^{-1}(\hat{y}(s))$ be chosen so that $|y(0)|_{\infty}=1$.
The corresponding frozen linearized system in (79) is autonomous and linear,
and thus its solution can be written explicitly as a linear combination of
eigenvectors

$\displaystyle y(s)$
$\displaystyle=g_{1}f_{1}e^{s\mu_{1,s}}+g_{2}f_{2}e^{s\mu_{2,s}}+g_{f}f_{f}e^{s\mu_{f}}.$
(123)

It thus follows from invariance of (generalized) eigenspaces that if (i) in
Lemma B.1 above does not hold for some fixed $\gamma$, then $g_{f}\neq 0$, and
so there is some $T>0$ so that (ii) holds. The point of this lemma is to show
that for all sufficiently small $\varepsilon$, such a $T>0$ exists which can
be chosen depending only on $\delta$ and $a$. $\triangle$

Proof of Lemma B.1: Assume that item (i) does not hold for some $\gamma$ as
given in the Lemma, so that there exists $T$ so that item (ii) holds for that
value of $\gamma$. Using the normalization of $y(s)$ in Remark B.2, we note
that $|g_{i}|\leq 1$. By Lemma 3.1, there exists some $K>0$, depending only on
the metric $\rho$, so that

$\displaystyle|g_{f}|\geq K\delta.$ (124)

From (123) we may write

$\displaystyle y(-T)$ $\displaystyle=e^{-\mu_{f}T}(g_{f}e_{f}+R(T)),$ (125)

where

$\displaystyle R(T)$
$\displaystyle=g_{1}f_{1}e^{-T(\mu_{1,s}-\mu_{f})}+g_{2}f_{2}e^{-T(\mu_{2,s}-\mu_{f})}.$

At this stage we remind the reader that the $\mu_{i}$ are still $\gamma$\- and
$\varepsilon$-dependent. We now apply the asymptotic estimate in Lemma 4.2 to
find $\bar{\varepsilon}\geq 0$, and $\alpha>0$ depending only on
$\bar{\varepsilon}$ and $a$, so that for each
$\varepsilon\in(0,\bar{\varepsilon}]$ and $|\gamma|\geq a/\varepsilon$, we
extract the uniform bound
$\min\\{\text{Re}(\mu_{1,s}-\mu_{f}),\,\text{Re}(\mu_{2,s}-\mu_{f})\\}>\alpha$.
It thus follows from the triangle inequality and the estimates $|g_{i}|\leq
1,\,|f_{i}|=1$ that

$\displaystyle|R(T)|$ $\displaystyle<2e^{-\alpha T}.$

For each $T^{\prime}>T$, we have $|R(T^{\prime})|<|R(T)|$. This estimate and
(124) applied to the factored form (125) then implies that

$\displaystyle\hat{y}(-T)\in
N_{\delta}(\hat{f}_{f}(\gamma,\lambda,\varepsilon))$

for $T$ depending only on $\alpha,\,\delta,\,a,\,\text{and }\rho$. $\Box$

We now turn to the dynamics on the slow timescale. Our objective here is to
strengthen the estimate in Lemma B.1 slightly, so that $a(\varepsilon)\equiv
a$ can be chosen independently of $\varepsilon>0$ such that the solution
$\hat{E}$ remains uniformly near the slow subbundle for $|\zeta|\geq a$. This
will be crucial in making the comparison to the linearized reduced flow
defined on $S^{a,-}_{\varepsilon},S^{a,+}_{\varepsilon}$.

From the construction in Lemmas 7.4 and 7.5, there exists $\delta_{1}>0$ such
that for each sufficiently small $\varepsilon$,

$\displaystyle
N_{\delta_{1}}(\hat{f}_{f}(\xi,\lambda,\varepsilon))\subset\hat{\Omega}_{f}(\xi)$
$\displaystyle\text{~{}for~{}}\xi\leq-a/\varepsilon\text{~{}and}$
$\displaystyle
N_{\delta_{1}}(\hat{\sigma}_{s}(\xi,\lambda,\varepsilon))\subset\hat{\Omega}_{s}(\xi)$
$\displaystyle\text{~{}for~{}}\xi\geq a/\varepsilon,$

where $\hat{\Omega}_{f}(\xi)$ and $\hat{\Omega}_{s}(\xi)$ denote the
projectivizations of the slices of the corresponding elephant trunks within
$\mathbb{CP}^{2}\times\\{\xi\\}$.

######  Lemma B.3.

Set $\delta>0$ with $\delta<\delta_{1}$, with $\delta_{1}$ as above, and fix
$a>0$. Suppose that $\hat{Y}$ is a solution to the projectivization of the
slow linearized equations

$\displaystyle\dot{\hat{Y}}$
$\displaystyle=\hat{A}(\hat{Y},\zeta,\lambda,\varepsilon)$ (126)

so that for each $\varepsilon$ sufficiently small, there exists
$A(\varepsilon)>a$ with the property that

$\displaystyle\hat{Y}(\zeta,\lambda,\varepsilon)\in
N_{\delta}(\Sigma_{s}(\zeta,\lambda,\varepsilon)).$ (127)

Then there exists $\bar{\varepsilon}>0$ such that for each
$0<\varepsilon<\bar{\varepsilon}$ and $|\zeta|\geq a$ we have

$\displaystyle\hat{Y}(\zeta,\lambda,\varepsilon)\in
N_{\delta}(\Sigma_{s}(\zeta,\lambda,\varepsilon)).$ (128)

Proof of Lemma B.3:

Our strategy will be to verify the uniform closeness estimate separately over
the slow branches $S^{a,-}_{\varepsilon}$ and $S^{a,+}_{\varepsilon}$. The
argument for $S^{a,-}_{\varepsilon}$ follows the indirect proof in [8]
closely. With our modified exchange lemma in hand, the remaining closeness
estimate over $S^{a,+}_{\varepsilon}$ is direct.

Suppose the lemma were false over $S^{a,-}_{\varepsilon}$; then there would
exist sequences $\zeta_{n}<-a$ and $\varepsilon_{n}\to 0$ so that (128) fails
to hold for all $n$. By passing to subsequences, we can assume that the
following sequences converge simultaneously:

$\displaystyle\zeta_{n}$ $\displaystyle\to\bar{\zeta}\text{
~{}~{}~{}where~{}~{} }a\leq-\bar{\zeta}<\infty,$ $\displaystyle
X(\zeta_{n},\varepsilon_{n})$ $\displaystyle\to\bar{x}~{}~{}~{}\in
S^{a,-}_{\varepsilon},\text{ and}$
$\displaystyle\hat{Y}(\zeta_{n},\lambda,\varepsilon_{n})=\hat{Y}_{n}$
$\displaystyle\to\hat{Y}_{*}~{}~{}\in\mathbb{CP}^{2}.$

Let us first suppose that $\bar{x}\in S^{a,-}_{\varepsilon}$. After centering
the linearized equations near $\bar{\zeta}$ by making the change of variables
$s=(\bar{\zeta}-\zeta_{n})/\varepsilon$ and
$\gamma_{n}=\zeta_{n}/\varepsilon_{n}$, we arrive at the following recentered
equations on the fast timescale:

$\displaystyle\frac{d\hat{z}}{ds}$
$\displaystyle=\hat{a}(\hat{z},\gamma_{n}+s,\lambda,\varepsilon_{n})$
$\displaystyle\hat{z}(0)$ $\displaystyle=\hat{Y}_{n}.$

Proceeding as before, we define an associated family of frozen systems. Let
$\hat{z}_{*}(s,\lambda,n)$ denote the solution of the corresponding frozen
system

$\displaystyle\frac{d\hat{z}}{ds}$
$\displaystyle=\hat{a}(\hat{z}_{*},\gamma_{n},\lambda,\varepsilon_{n})$
$\displaystyle\hat{z}_{*}(0,\lambda,n)$ $\displaystyle=\hat{Y}_{n}.$

Now fix $\delta_{2}>0$ such that $\delta<\delta_{2}<\delta_{1}$. By
assumption, for each $n$ we have $\hat{Y}_{n}\notin
N_{\delta}(\hat{\sigma}_{s}(\gamma_{n},\lambda,\varepsilon_{n}))$, and so it
must be true by Lemma B.1 that

$\displaystyle\hat{z}(\gamma_{n}-T,\lambda,n)$ $\displaystyle\in
N_{\delta_{2}}(\hat{f}_{f}(\gamma_{n}-T,\lambda,\varepsilon_{n})).$

Since $\delta_{2}<\delta_{1}$, $\hat{y}$ must enter the fast elephant trunk
about $\hat{f}_{f}$ over $S^{a,-}_{\varepsilon}$, which contradicts the
assumed behavior of $\hat{y}$ as $\zeta\to-\infty$.

We now know that $\hat{Y}(\zeta,\lambda,\varepsilon)\in
N_{\delta}(\hat{\Sigma}(\zeta,\lambda,\varepsilon))$ for each $\zeta\leq-a$.
Then set $\delta>0$ small enough that for each sufficiently small
$\varepsilon>0$, Lemma 7.7 applies to $\hat{Y}(\zeta,\lambda,\varepsilon)$ as
the wave enters the vicinity of $\bar{x}:=X(a,\varepsilon)\in
S^{a,+}_{\varepsilon}$ from the fast layer. In particular, we have the
asymptotics that $\hat{Y}$ lies $\mathcal{O}(\varepsilon)$-close to
$\hat{\Sigma}$ at the time $\zeta=a$, with respect to the Fubini-Study metric,
and so for each sufficiently small $\varepsilon>0$, we have $\hat{Y}\in
N_{\delta}(\hat{\Sigma}_{s}(a,\lambda,\varepsilon))$. Simultaneously,
$\hat{Y}$ enters the slow elephant trunk over $S^{a,+}_{\varepsilon}$ since
$\delta<\delta_{1}$, and thus remains $\delta$-close to the slow subbundle for
each $\zeta>a$. This completes the proof. $\Box$

######  Corollary B.4.

Fix $\lambda\in\mathbb{C}$ and $a,\delta>0$. Then there exists
$\bar{\varepsilon}>0$ sufficiently small so that for each
$\varepsilon\in(0,\bar{\varepsilon}]$, the unique solution
$\hat{E}(\zeta,\lambda)$ of the projectivized linearized equations for which
$\hat{E}(\zeta,\lambda)\to\hat{e}_{s,2}^{-}$ (the unstable eigenvector) as
$\zeta\to-\infty$, also satisfies

$\displaystyle\hat{E}(\zeta,\lambda,\varepsilon)\in
N_{\delta}(\hat{\Sigma}_{s}(\zeta,\lambda,\varepsilon)).$ (129)

for $|\zeta|\geq a$.

##  Appendix C Proof of Lemma 7.7

In this proof we focus on clarifying the essential steps in the estimate,
omitting unwieldy calculations while noting that they can be traced from [17].
The key step is to write down a slightly modified version of the estimate
given in Proposition 8 in [17]; namely, there exist positive constants
$B_{1},\,B_{2},\,B_{3}$, and $T_{0}$ depending only on the width $\Delta$ of
the defining box (75), so that for all $T\geq T_{0}$ and for each $q\in
Q_{T}$, we have the upper bounds

$\displaystyle|dy(q(t))|$ $\displaystyle\leq B_{1}e^{\alpha t}$ (130)
$\displaystyle|db(q(t))|$ $\displaystyle\leq
B_{2}e^{(\alpha-\kappa)t}+B_{3}\varepsilon\lambda e^{\alpha t}$

for all $t\in[0,T]$, where $\alpha>0$ and $0<\kappa<|\gamma_{0}|$ are growth
rate constants characterizing respectively slow growth versus fast exponential
contraction (see Proposition 8 and Corollary 2 in [17]). We highlight the
essential modification: whereas in the standard exchange lemma, the component
$db(t)$ of the tangent vector that is aligned along the fast fibers can be
arranged to shrink exponentially quickly over long times, i.e. with growth
rate $(\alpha-\kappa)<0$ (in the case of a slow manifold with only attracting
directions), there is now an extra, slowly-growing error term, arising from
the new $\varepsilon\lambda$ term in the eigenvalue problem.

The strategy to prove (130) proceeds essentially as in the proof of
Proposition 8 in [17]. By smoothness of solutions to the ODEs, there exists
some $T^{\prime}>0$ such that the upper bound for $|db(t)|$ holds for
$t\in[0,T^{\prime}]$ if we choose $B_{2}$ and $B_{3}$ sufficiently large. We
seek to show that we can take $T^{\prime}=T$ for choices $B_{2},\,B_{3}$ which
are independent of the initial condition. Assume that there is a maximal
$T^{\prime}>0$ such that the error estimate is attained (otherwise there is
nothing to prove). We show that this leads to a contradiction if $T$ is large
enough.

By a similar calculation to that in the proof of Proposition 8 in [17], we
find that

$\displaystyle|dy(t)|$ $\displaystyle\leq\bar{B}_{1}e^{\alpha t},$

where the new constant $\bar{B}_{1}>0$ depends upon the width of the defining
box $\Delta>0$, as well as the growth rate constants. There is also an extra
term of the form $e^{(\alpha-\kappa)t}$ arising from applying the Duhamel
principle (see Lemma 5.1 in [17]) to the extra term of the form
$\varepsilon\lambda M_{2}db$, where $M_{2}$ is a smooth bounded function
defined by compositions of the Fenichel straightening diffeomorphism and its
inverse. We subsume this extra error term into the constant $\bar{B}_{1}$.

By the Duhamel principle and the bounding estimates on points in $Q_{T}$
together with the basic flow estimates for slowly-varying nonautonomous linear
systems (see Proposition 2, Corollary 1, Proposition 6, and Proposition 7 in
[17]), we then have

$\displaystyle|db(t)|$
$\displaystyle\leq\bar{K}_{X}e^{(\alpha-\kappa)t}\bar{M}_{2}+\varepsilon\lambda\int_{0}^{t}e^{-\mu(t-\zeta)}|M_{2}(b(\zeta),y(\zeta),\varepsilon)||dy(\zeta)|d\zeta$
$\displaystyle\leq\bar{K}_{X}e^{(\alpha-\kappa)t}\bar{M}_{2}+\varepsilon\lambda
M_{\Delta,\bar{\varepsilon}}e^{-\mu
t}\bar{B}_{1}\frac{e^{(\mu+\alpha)t}}{\mu+\alpha}$
$\displaystyle<B_{2}e^{(\alpha-\kappa)t}+B_{3}e^{\alpha t}$

if $T$ is chosen large enough and for $B_{2},\,B_{3}$ defined from the
coefficients as in the above calculation. This contradicts the assumed
maximality of $T^{\prime}$, and so we conclude that $db(t)$ satisfies the
required inequality for all sufficiently large $t$.

The remainder of the proof follows the outlines of the main Theorem 6.5 in
[17]. By assumption we have that $|dy(0)|\geq M\varepsilon$ for some $M>0$,
and let $\beta<0$ be fixed. It follows from general considerations about
slowly varying nonautonomous linear systems (see Proposition 2 in [17])
together with the Duhamel principle that

$\displaystyle|dy(\mathcal{T}_{\varepsilon})|\geq K_{y}\varepsilon
e^{\beta\mathcal{T}_{q}}.$

The result follows from applying the estimate for $|dy(t)|$ above and
calculating the distance between
$(db(\mathcal{T}_{\varepsilon}),dy(\mathcal{T}_{\varepsilon}))$ and
$(0,dy(\mathcal{T}_{\varepsilon}))$ in the Fubini-Study metric. $\Box$

######  Remark C.1.

An elephant trunk lemma is used in Gardner and Jones to prove an analogous
result. We point out that only partial elephant trunk-type estimates over
$S^{a,-}_{\varepsilon}$ and $S^{a,+}_{\varepsilon}$ are required to provide an
estimate of the type proven above. $\triangle$

##  Appendix D Example for modified exchange lemma estimates

We illustrate the assertions in Remark 7.8 with the following toy problem
defined in a suitable box $\Delta$ in $\mathbb{R}^{2}\times\mathbb{C}^{2}$,
where $\lambda\in\mathbb{C}$ is taken to be a fixed constant:

$\displaystyle b^{\prime}$ $\displaystyle=-b$ (131) $\displaystyle y^{\prime}$
$\displaystyle=\varepsilon y$ $\displaystyle db^{\prime}$
$\displaystyle=-db+\varepsilon\lambda y\,dy$ $\displaystyle dy^{\prime}$
$\displaystyle=\varepsilon\,dy.$

The critical manifold $\mathcal{M}_{0}$ of this system is given by
$\\{b=0,\,db=0\\}$. This manifold is normally hyperbolic and attracting since
the real parts of the nontrivial eigenvalues of the corresponding layer
problem of (131) along $\mathcal{M}_{0}$ are both negative in $\Delta$, and
hence there exists a one-parameter family $\mathcal{M}_{\varepsilon}$ of
attracting slow manifolds for sufficiently small values of $\varepsilon>0$.
Without loss of generality we think of this family as parametrized by the slow
variables $(y,\,dy)$, and evidently $b=0$ specifies one of the defining
equations for $\mathcal{M}_{\varepsilon}$. The slow subbundle along
$\mathcal{M}_{\varepsilon}$ in terms of this coordinate representation can be
computed by calculating the eigenvector corresponding to the
$\mathcal{O}(\varepsilon)$ eigenvalue of the $2\times 2$ Jacobian of the
latter two equations in (131). It is given by the span of the vector

$\displaystyle\begin{pmatrix}\frac{\varepsilon\lambda y}{1+\varepsilon}\\\
1\end{pmatrix}$

as $y$ varies. On the other hand, the system (131) can be explicitly solved
for $(db(t),dy(t))$ given initial values
$(b(0),y(0),db(0),dy(0))=(b_{0},y_{0},db_{0},dy_{0})$ to find

$\displaystyle db(t)$
$\displaystyle=\left(\frac{1}{1+2\varepsilon}\right)\left(e^{-t}[db_{0}(1+2\varepsilon)-\varepsilon\lambda
y_{0}dy_{0}]+e^{2\varepsilon t}\varepsilon\lambda y_{0}dy_{0}\right)$
$\displaystyle dy(t)$ $\displaystyle=dy_{0}e^{\varepsilon t}.$

From this calculation, it follows that $\mathcal{M}_{\varepsilon}$ is given by
$\\{b=0,\,db=\varepsilon\lambda y\,dy/(1+2\varepsilon)\\}$.

For times $t=\mathcal{T}=\mathcal{O}(1/\varepsilon)$, We find that the tangent
vector $(db(t),dy(t))$ lies exponentially close to

$\displaystyle\begin{pmatrix}\frac{\varepsilon\lambda
y_{0}dy_{0}}{1+2\varepsilon}e^{2\varepsilon\mathcal{T}}\\\
dy_{0}e^{\varepsilon\mathcal{T}}\end{pmatrix}.$

Evidently, this vector is $\mathcal{O}(\varepsilon)$ close to the tangent
bundle of the slow manifold, which has the local representation

$\begin{pmatrix}0\\\ 1\end{pmatrix}.$

Further comparing the angle of this vector to the slow subbundle at the point
$y=y_{0}e^{\varepsilon\mathcal{T}}$ by using the distance estimate Lemma 3.1
in [17], we find that the angle is no larger than

$\displaystyle\frac{2\varepsilon^{2}\lambda
y_{0}}{(1+2\varepsilon)(1+\varepsilon)},$

i.e. the angle scales as $\mathcal{O}(\varepsilon^{2})$. This error estimate
is sharper than that given by the tangent bundle of the slow manifold.

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