# Unfolding a Hopf bifurcation in a linear reaction-diffusion equation with
strongly localized impurity – existence of breathing pulses

Ji Li School of Mathematics and Statistics, Huazhong University of Science and
Technology, Wuhan, Hubei, 430074, P.R. China<EMAIL_ADDRESS>Qing Yu
School of Mathematics and Statistics, Huazhong University of Science and
Technology, Wuhan, Hubei, 430074, P.R. China<EMAIL_ADDRESS>Qian Zhang
Corresponding author. School of Mathematics and Statistics, Huazhong
University of Science and Technology, Wuhan, Hubei, 430074, P.R. China,
<EMAIL_ADDRESS>
###### Abstract

This paper presents a general framework to derive the weakly nonlinear
stability near a Hopf bifurcation in a special class of multi-scale reaction-
diffusion equations. The main focus is on how the linearity and nonlinearity
of the fast variables in system influence the emergence of the breathing
pulses when the slow variables are linear and the bifurcation parameter is
around the Hopf bifurcation point. By applying the matching principle to the
fast and slow changing quantities and using the relevant theory of singular
perturbation, we obtain explicit expressions for the stationary pulses. Then,
the normal form theory and the center manifold theory are applied to give Hopf
normal form expressions. Finally, one of these expressions is verified by the
numerical simulation.

Key Words: Pinned solutions; Hopf bifurcation; Breathing pulse; Center
manifolds expansion; Normal forms.

Here, we consider systems of linear reaction-diffusion equations with the
impact of strong, spatially localized, nonlinear impurities. From an applied
perspective, it is feasible to see linear systems becoming locally nonlinear
with the introduction of strongly localized impurities. Besides, these
impurities exhibit the structure of Dirac delta-type function, whose role can
be viewed as adding fast components to the unperturbed systems. In other
words, we are studying the general singularly perturbed slow-fast reaction-
diffusion equations with linear slow flow. The aim of this paper is to study
localized patterns, such as breathing pulses, in the aforementioned systems
with linear/nonlinear impurities. It further confirms the fact that the most
general pulse destabilization scenario corresponds to a Hopf bifurcation. By
applying Normal Form Theory and Center Manifold Theory, we develop a mechanism
for weakly nonlinear analysis. It is worth noting that the calculations
provided in this paper are much simpler than the previous analysis.
Furthermore, our analysis sheds some light on the detailed unfolding of a
Bogdanov–Takens bifurcation or a Dumortier–Roussarie–Sotomayor bifurcation of
a localized pulse solution.

##  1 Introduction

In this paper, we consider the unfolding of a Hopf bifurcation in a linear
reaction-diffusion equation with strongly localized impurity, which was first
introduced by Arjen, Heijster and Shen in [12] as:

$\left\\{\begin{aligned} \frac{\partial U_{1}}{\partial
t}&=\frac{\partial^{2}U_{1}}{\partial x^{2}}-\mu
U_{1}+\frac{\alpha}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})G_{1}(U_{1},U_{2}),\\\
\frac{\partial U_{2}}{\partial t}&=D\frac{\partial^{2}U_{2}}{\partial
x^{2}}-bU_{1}-\mu
U_{2}+\frac{\beta}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})G_{2}(U_{1},U_{2}),\end{aligned}\right.$
(1.1)

where $(x,t)\in\mathbb{R}\times\mathbb{R^{+}}$,
$U_{1,2}(x,t):\mathbb{R}\times\mathbb{R^{+}}\rightarrow\mathbb{R^{\mathbb{missing}}}$,
$0<\varepsilon\ll 1$ is a sufficiently small parameter measuring the degree of
locality, $\alpha,\beta\in\mathbb{R}$ are parameters measuring the strength of
the impurities, $G_{1},G_{2}$ are sufficiently smooth functions satisfying
$G_{1,2}(\mathbf{0})\neq\mathbf{0}$, and $I$ is a Dirac delta-type impurity.
Here, $\mu$ is assumed to be positive to ensure the ground state 0 is stable,
which has been explained in [12] and [31]. Without missing any essential idea,
we assume exclusively $I(\xi)=\frac{1}{\sqrt{\pi}}e^{-\xi^{2}}$ all over this
paper. The reasons why we study this system can be summarized into three
parts. First, adding strongly localized impurities can turn the linear system
into a locally nonlinear one and in reality the linearity of the models may
break down under strong perturbations. Second, the system we present here
avoids the difficulty in analyzing the spectrum of differential operators with
nonconstant coefficients. Instead, the spectral stability problem is reduced
to linear algebra by matching the changing quantities in fast and slow scales
with the methods of GSPT. Third, the spatial heterogeneities eliminate the
translational eigenvalue $\lambda=0$, which makes the center manifold analysis
much easier.

As one of the most general pulse destabilisation scenarios [34], the mechanism
of the unfolding of a Hopf bifurcation can be developed by the local analysis
of the associated center manifold. With regard to the dynamic behavior near
this bifurcation point, numerical simulations can reveal stable, temporally
oscillating pulses. Such breathing localized pulses, also known as oscillons,
are fascinating topics of research in the fields of lasers [36, 24], optical
media [23, 25, 33, 16] and excitatory neural networks [19, 20, 3, 5, 4] over
the past few decades. These oscillons are localized waves that periodically
vary in shape and amplitude. They often occur in the singularly perturbed
reaction-diffusion systems. They are also excitatory localized structures that
can respond to external stimuli and change accordingly. Mathematical models of
breathing localized pulses are often complicated. The singular perturbation
theory and the center manifold reduction are required in helping analyzing
their stability and bifurcations. Their existence and formation mechanisms are
not fully understood and are interesting research topics in nonlinear
dynamics.

Breathing pulses were initially found in one-dimensional reaction-diffusion
systems by Koga and Kuramoto [29], in which a stationary localized pattern
destabilized around a Hopf bifurcation leading to a “breathing motion”.
Besides, it was found that multi-spike quasi-equilibrium solutions could also
undergo a Hopf bifurcation, resulting in oscillations in the spike amplitudes
on an $O(1)$ time scale in [7, 37, 38]. With respect to those oscillating
fronts in [19, 4], they are denoted by the terminology “breather”. In the
field of optics, “breathing pulses” can be used to describe the propagation
behavior of light pulses in optical fibers. In biology, “oscillons” can be
employed to model the activity patterns of cells or neural networks.

Recent studies [12, 31, 9] have explored the pattern formation processes in
models with linear or nonlinear structures outside the impurities based on
[28, 27]. It is worth noting that these systems can be viewed as the
simplifications of some canonical singularly perturbed reaction-diffusion
systems like: Gray-Scott [22, 8, 14], Gierer-Meinhardt [21, 13] and Klausmeier
[6, 2] models, due to the existence of fast and slow scales. For these
classical systems, the singularly perturbed structure induces the spatial
scale separation, which gives explicit leading order expressions for the
patterns under consideration. Moreover, with the help of the Hopf
eigenfunctions, explicit leading order expressions for Hopf normal form
coefficients can be obtained. As we have seen, [34] studies the emergence of
the breathing pulses in a slowly nonlinear Gierer-Meinhardt system [35], which
initially exhibits typical “subcritical” growth behavior and then a bounded
temporally oscillating pulse. Different from the pattern formation in Gierer-
Meinhardt system [21, 13], this additional nonlinearity has a profound impact
on both the stability analysis and dynamic behavior near the Hopf bifurcation.

The numerical simulation in [12] also reveals such oscillating pulses. The
addition of small nonlinearity $-\varepsilon^{3}U_{1}^{3}$ to linear $G_{1}$
prevents the profile from blowing up and forms a breathing pinned 1-pulse
solution. This sheds some light on performing a center manifold analysis near
spectral configurations of co-dimension one and higher. Hence, the unfolding
of a Bogdanov-Takens bifurcation [30] or controllable chaotic pulse dynamics
[16, 32] can be a natural next step. Besides, the oscillating pulses around
the Hopf bifurcation still need an analytical explanation. Hence, in this
paper, we focus on the conditions under which such breathing pulses arise for
general linear reaction-diffusion equations with the impact of spatial
defects. To accomplish this, we need to calculate the leading order
expressions for the target pulses, then apply the Hopf normal form theory to
derive the normal form coefficients and finally determine the pitchfork
bifurcation type. We emphasize that the main differences with [34] is that we
avoid the analysis of the direction spanned by the translational mode in the
Hopf center manifold and the computation of the inverse problems associated
with the scale separated structure.

The article is structured as follows. Section 2 introduces the relevant
notations, hypotheses and lemmas, mainly based on [26]. Section 3 is divided
into five parts. In subsection 3.1, we briefly review the basic mechanism of
the existence and stability for pinned pulse solution in system (1.1) and give
explicit expressions of pulse solutions and corresponding eigenfunctions. In
subsection 3.2, we study the local expansion of the center manifold when
$G_{1}$ and $G_{2}$ are both linear. The linearities of $G_{1}$ and $G_{2}$ do
not generate nonlinear remainder terms involving $U_{1}$ or $U_{2}$, which
prevents breathing pulses. In subsections 3.3 and 3.4, due to the difficulty
in analyzing the Hopf bifurcation conditions, we examine several simple
situations and discuss the cases where $G_{1}$ and $G_{2}$ are separately
nonlinear. Then in subsection 3.5, we calculate a specific example which ever
appeared in [12] and explain the pattern formations that emerged. Finally, in
section 4, we conclude with some remarks and suggestions for future research.

##  2 Notations, Hypotheses and Lemmas

We briefly introduce some required notations, hypotheses and lemmas in
connection with the normal form theory and the center manifold theory. As a
general reference, we would like to mention [26].

Let $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$ be (real or complex) Banach
spaces such that
$\mathcal{Z}\hookrightarrow\mathcal{Y}\hookrightarrow\mathcal{X}$, with
continuous embeddings. For a parameter-dependent differential equation in
$\mathcal{X}$ of the form

$\frac{du}{dt}=Lu+R(u,\mu),$ (2.1)

in which $L$ is a linear operator and $R$ is defined for $(u,\mu)$ in a
neighborhood of (0,0) in $\mathcal{Z}\times\mathbb{R}^{m}$. Here $\mu$ is a
parameter to be small.

######  Definition 2.1.

A linear operator $L:\mathcal{Z}\rightarrow\mathcal{X}$ is called a bounded
linear operator, if $L$ is continuous. The set of bounded linear operator is
denoted by $L\in\mathcal{L}(\mathcal{Z},\mathcal{X})$.

######  Hypothesis 2.1.

We assume that $L$ and $R$ in (2.1) have the following properties:

  1. (i)

$L\in\mathcal{L}(\mathcal{Z},\mathcal{X})$;

  2. (ii)

for some $k\geq 2$, there exist neighborhoods
$\mathcal{V}_{u}\subset\mathcal{Z}$ and
$\mathcal{V}_{\mu}\subset\mathbb{R}^{m}$ of 0 such that
$R\in\mathcal{C}^{k}(\mathcal{V}_{u}\times\mathcal{V}_{\mu},\mathcal{Y})$ and
$R(0,0)=0,\ D_{u}R(0,0)=0$.

######  Hypothesis 2.2.

(Spectral decomposition) Consider the spectrum $\sigma$ of the linear operator
$L$, and write $\sigma=\sigma_{+}\cup\sigma_{0}\cup\sigma_{-}$, in which
$\sigma_{+}=\\{\lambda\in\sigma;\rm{Re}\lambda>0\\},\
\sigma_{0}=\\{\lambda\in\sigma;\rm{Re}\lambda=0\\},\
\sigma_{-}=\\{\lambda\in\sigma;\rm{Re}\lambda<0\\}$. We assume that

  1. (i)

there exists a positive constant $\gamma>0$ such that
$\inf\limits_{\lambda\in\sigma_{+}}\rm{Re}$$\lambda>\gamma,\
\sup\limits_{\lambda\in\sigma_{-}}\rm{Re}$$\lambda<-\gamma$;

  2. (ii)

the set $\sigma_{0}$ consists of a finite number of eigenvalues with finite
algebraic multiplicities.

As a consequence of Hypothesis (ii)(ii), we can define the spectral projection
$P_{0}\in\mathcal{L}(\mathcal{X})$, corresponding to $\sigma_{0}$, by the
Dunford integral formula $P_{0}=\frac{1}{2\pi
i}\int_{\Gamma}(\lambda\mathbb{I}-L)^{-1}d\lambda$, where $\Gamma$ is a
simple, oriented counterclockwise, Jordan curve surrounding $\sigma_{0}$ and
lying entirely in $\\{\lambda\in\mathbb{C}:|\rm{Re}\lambda|<\gamma\\}$. Then
we define projection $P_{h}:\mathcal{X}\rightarrow\mathcal{X}$ by
$P_{h}=\mathbb{I}-P_{0}$. The spectral subspaces associated with these two
projections are denoted by
$\mathcal{E}_{0}=\rm{Ran}$$P_{0}$$=\rm{Ker}$$P_{h}$$\subset\mathcal{X},\
\mathcal{X}_{h}=\rm{Ran}$$P_{h}$$=\rm{Ker}$$P_{0}$$\subset\mathcal{X}$. Also,
we set $\mathcal{Z}_{h}=P_{h}\mathcal{Z}\subset\mathcal{Z},\
\mathcal{Y}_{h}=P_{h}\mathcal{Y}\subset\mathcal{Y}$, and denote by $L_{0}$ and
$L_{h}$ the restrictions of $L$ to $\mathcal{E}_{0}$ and $\mathcal{Z}_{h}$. As
an immediate consequence, the spectrum of $L_{0}$ is $\sigma_{0}$ and the
spectrum of $L_{h}$ is $\sigma_{h}=\sigma_{+}\cup\sigma_{-}$.

######  Hypothesis 2.3.

(Resolvent estimates) Assume that
$\tilde{\mathcal{X}}=\mathcal{X}\times\mathbb{R}^{m}$,
$\tilde{\mathcal{Y}}=\mathcal{Y}\times\mathbb{R}^{m}$,
$\tilde{\mathcal{Z}}=\mathcal{Z}\times\mathbb{R}^{m}$ are Hilbert spaces, and
there exist positive constants $\omega_{0}>0$, $c>0$ and $\alpha\in[0,1)$ such
that for all $\omega\in\mathbb{R}$, with $|\omega|\geq\omega_{0}$, we have
that $i\omega$ belongs to the resolvent set of $L$ and
$||(i\omega\mathbb{I}-\tilde{L}_{h})^{-1}||_{\mathcal{L}(\mathcal{X})}\leq\frac{c}{|\omega|}$,
where $\tilde{L}_{h}:=L_{h}+D_{\mu}P_{h}R(0,0)$.

######  Lemma 2.1.

(Parameter-dependent center manifolds) (P.46 of [26]) Assume that Hypotheses
2.1, 2.2 and 2.3 hold. Then there exists a map
$\Psi\in\mathcal{C}^{k}(\mathcal{E}_{0}\times\mathbb{R}^{m},\mathcal{Z}_{h})$,
with $\Psi(0,0)=0,\ D_{u}\Psi(0,0)=0$, and a neighborhood
$\mathcal{O}_{u}\times\mathcal{O}_{\mu}$ of (0,0) in
$\mathcal{Z}\times\mathbb{R}^{m}$ such that for $\mu\in\mathcal{O}_{\mu}$, the
manifold
$\mathcal{M}_{0}(\mu)=\\{u_{0}+\Psi(u_{0},\mu):u_{0}\in\mathcal{E}_{0}\\}$ has
the following properties:

  1. (i)

$\mathcal{M}_{0}(\mu)$ is locally invariant.

  2. (ii)

$\mathcal{M}_{0}(\mu)$ contains the set of bounded solutions of (2.1) staying
in $\mathcal{O}_{u}$ for all $t\in\mathbb{R}$.

Specifically, when the unstable spectrum $\sigma_{+}$ of $L$ is empty, there
is

######  Lemma 2.2.

(Parameter-dependent center manifolds theorem for empty unstable spec-trum)
(P.59 of [26]) Assume that $\sigma_{+}=\emptyset$, Hypotheses 2.1, 2.2 and 2.3
hold. Then in addition to the properties in Lemma 2.1 the following holds.
The local center manifold $\mathcal{M}_{0}(\mu)$ is locally attracting. More
precisely, if $u(0)\in\mathcal{O}_{u}$ and the solution $u(t;u(0))$ of (2.1)
satisfies $u(t;u(0))\in\mathcal{O}_{u}$ for all $t>0$, then there exists
$\tilde{u}\in\mathcal{M}_{0}\cap\mathcal{O}_{u}$ and $\gamma^{\prime}>0$ such
that $u(t;u(0))=u(t;\tilde{u})+O(e^{-\gamma^{\prime}t})\ as\
t\rightarrow\infty$.

Next, we give the hypothesis and lemmas about the normal form theorem in
$\mathbb{R}^{n}$:

######  Hypothesis 2.4.

Assume that $L$ and $R$ have the following properties:

  1. (i)

$L$ is a linear map in $\mathbb{R}^{n}$;

  2. (ii)

for some $k\geq 2$, there exist neighborhoods
$\mathcal{V}_{u}\subset\mathbb{R}^{n}$ and
$\mathcal{V}_{\mu}\subset\mathbb{R}^{m}$ of 0 such that
$R\in\mathcal{C}^{k}(\mathcal{V}_{u}\times\mathcal{V}_{\mu},\mathbb{R}^{n})$
and $R(0,0)=0,\ D_{u}R(0,0)=0$.

######  Lemma 2.3.

(Normal form for perturbed vector fields) (P.110 of [26]) Assume that Hypothe-
sis 2.4 holds. Then for any positive integer $p$, $2\leq p\leq k$ , there
exist neighborhoods $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ of 0 in
$\mathbb{R}^{n}$ and $\mathbb{R}^{m}$, respectively, such that for any
$\mu\in\mathcal{O}_{2}$, there is a polynomial
$\Phi_{\mu}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ of degree $p$ with the
following properties:

  1. (i)

The coefficients of the monomials of degree $q$ in $\Phi_{\mu}$ are functions
of $\mu$ of class $\mathcal{C}^{k-p}$, and $\Phi_{0}(0)=0,\
D_{u}\Phi_{0}(0)=0$;

  2. (ii)

For $v\in\mathcal{O}_{1}$, the polynomial change of variable
$u=v+\Phi_{\mu}(v)$, transforms equation (2.1) into the “normal form”
$\frac{dv}{dt}=Lv+N_{\mu}(v)+\rho(v,\mu)$, and the following properties hold:

    1. a.

For any $\mu\in\mathcal{O}_{2}$, $N_{\mu}$ is a polynomial
$\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ of degree $p$ and $N_{0}(0)=0,\
D_{v}N_{0}(0)=0$;

    2. b.

The equality $N_{\mu}(e^{tL^{*}}v)=e^{tL^{*}}N_{\mu}(v)$ holds for all
$(t,v)\in\mathbb{R}\times\mathbb{R}^{n}$ and $\mu\in\mathcal{O}_{2}$;

    3. c.

The map $\rho$ belongs to
$\mathcal{C}^{k}(\mathcal{O}_{1}\times\mathcal{O}_{2},\mathbb{R}^{n})$, and
$\rho(v,\mu)=o(\|v\|^{p})$ for all $\mu\in\mathcal{O}_{2}$.

######  Lemma 2.4.

(P.11 of [26]) Let f be a complex-valued function of class $\mathcal{C}^{k}$,
$k\geq 1$, defined in a neighborhood $\mathcal{O}$ of the origin in
$\\{(z,\bar{z}):z\in\mathbb{C}\\}$, and which verifies

$f(e^{i\omega t}z,e^{-i\omega t}\bar{z})=e^{i\omega t}f(z,\bar{z})\ for\ any\
t\in\mathbb{R}\ and\ (z,\bar{z})\in\mathcal{O}.$

Then there exists an even, complex-valued function $g$ of class
$\mathcal{C}^{k-1}$ defined in a neighborhood of 0 in $\mathbb{R}$ such that
$f(z,\bar{z})=zg(|z|)$. Furthermore, if $f$ is a polynomial, then $g$ is an
even polynomial, $g(|z|)=\phi(|z|^{2})$ for a polynomial $\phi$.

######  Remark 2.5.

Provided that an infinite-dimensional system $\frac{du}{dt}=Lu+R(u,\mu)$
satisfies the assump-tions in center manifold Lemma 2.1, then the reduced
system satisfies Hypothesis 2.4, so the normal form Lemma 2.2 can be employed.

##  3 Existence of Breathing Pulses

In this section, we study whether breathing pulses arise when $G_{1}$ and
$G_{2}$ exhibit different linearity. First, we recall the analysis of
existence and stability in [9, 12, 31]. Therein, the explicit leading-order
expressions of the pulse solution and the corresponding eigenfunction are
given. Then, we discuss the dynamic behavior when $G_{1}$ and $G_{2}$ are
linear or nonlinear separately. In order to avoid the complexity of the
discussion and the stacking of formulas, we make some simplifications. At
last, a concrete example which ever appeared in [12] is raised, whose
existence of breathing pulse can be interpreted theoretically within the
framework of our previous analysis.

###  3.1 The Existence and Stability of the Pinned Pulse

In order to obtain the leading order expressions of the pinned pulse solution
and its eigenfunction, which play an important role in analyzing the normal
form, we first present the basic mechanism of the existence and stability.

Consider the system (1.1), owing to the strong spatial localization of
impurities, it can be analyzed in two spatial scales, $x$ versus
$\xi:=\frac{x}{\varepsilon^{2}}$. More specifically, the spatial domain is
divided into three parts $I_{s}^{-}:=(\\-\infty,-\varepsilon)$,
$I_{f}:=[-\varepsilon,\varepsilon]$ and $I_{s}^{+}:=(\varepsilon,\infty)$. In
$I_{s}^{\pm}$, the system (1.1) can be regarded as a semi-coupled linear
system, whose solution can be uniquely determined due to the exponential decay
at infinity. In $I_{f}$, impurities are dominant, and the impact of $-\mu
U_{1}$ and $-bU_{1}-\mu U_{2}$ are least. Hence, the leading-order accumulated
changes of $\frac{dU_{1}}{dx}$ and $\frac{dU_{2}}{dx}$ can be calculated from
the value of $\alpha G_{1}$ and $\beta G_{2}$. Through equalizing these
accumulated changes in these two different scales, we derive the continuous
solution we expect. The following stability analysis is completed in a similar
way because of the same structure of the eigenvalue system and the negative
essential spectrum. Actually, two different approaches to obtain the matching
conditions of existence and stability are given in [12, 31, 9]. One calculates
the fast and slow manifolds separately, while the other gives the explicit
expressions of solutions by method of variation of constant.

To system (1.1), if

$\left\\{\begin{aligned} 2\sqrt{\mu}C_{1}&=\alpha
G_{1}\left(C_{1},C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right),\\\
2\left(\sqrt{\frac{\mu}{D}}C_{2}+\frac{bC_{1}}{(-1+D)\sqrt{\mu}}\right)&=\frac{\beta}{D}G_{2}\left(C_{1},C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\end{aligned}\right.$
(3.1)

admits non-degenerate solutions $C_{1}$ and $C_{2}$, then (1.1) exists a
nontrivial pinned pulse solution
$\Gamma_{p}(x)=(\Gamma_{1,p}(x),\Gamma_{2,p}(x))^{T}=(U_{1,p}(x)+O(\varepsilon),U_{2,p}(x)+O(\varepsilon))^{T}$
with

$U_{1,p}(x)=\begin{cases}C_{1}e^{\sqrt{\mu}x},&x\in I_{s}^{-},\\\ C_{1},&x\in
I_{f},\\\ C_{1}e^{-\sqrt{\mu}x},&x\in I_{s}^{+},\end{cases}\
U_{2,p}(x)=\begin{cases}C_{2}e^{\sqrt{\frac{\mu}{D}}x}+\frac{bC_{1}}{(-1+D)\mu}e^{\sqrt{\mu}x},&x\in
I_{s}^{-},\\\ C_{2}+\frac{bC_{1}}{(-1+D)\mu},&x\in I_{f},\\\
C_{2}e^{-\sqrt{\frac{\mu}{D}}x}+\frac{bC_{1}}{(-1+D)\mu}e^{-\sqrt{\mu}x},&x\in
I_{s}^{+}.\end{cases}$ (3.2)

As for the spectral (and nonlinear) stability of the above pinned pulse
solution, we linearize (1.1) around $\Gamma_{p}$ and yield the eigenvalue
problem

$\left\\{\begin{aligned}
0&=\frac{d^{2}P_{1}}{dx^{2}}-(\mu+\lambda)P_{1}+\frac{\alpha}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\left(\frac{\partial
G_{1}}{\partial U_{1}}(U_{1,p},U_{2,p})P_{1}+\frac{\partial G_{1}}{\partial
U_{2}}(U_{1,p},U_{2,p})P_{2}\right),\\\
0&=D\frac{d^{2}P_{2}}{dx^{2}}-bP_{1}-(\mu+\lambda)P_{2}+\frac{\beta}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\left(\frac{\partial
G_{2}}{\partial U_{1}}(U_{1,p},U_{2,p})P_{1}+\frac{\partial G_{2}}{\partial
U_{2}}(U_{1,p},U_{2,p})P_{2}\right).\end{aligned}\right.$

With $\sigma_{ess}=(-\infty,-\mu]$, it does not yield instabilities.
Consequently, the stability is fully determined by the locations of
eigenvalues. Hence, if

$\left\\{\begin{aligned} &\alpha\left(\frac{\partial G_{1}}{\partial
U_{1}}\left(C_{1},C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)C_{3}+\frac{\partial
G_{1}}{\partial
U_{2}}\left(C_{1},C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)(C_{4}+\frac{bC_{3}}{(-1+D)(\mu+\lambda)})\right)\\\
&=2\sqrt{\mu+\lambda}C_{3},\\\ &\frac{\beta}{D}\left(\frac{\partial
G_{2}}{\partial
U_{1}}\left(C_{1},C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)C_{3}+\frac{\partial
G_{2}}{\partial
U_{2}}\left(C_{1},C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)(C_{4}+\frac{bC_{3}}{(-1+D)(\mu+\lambda)})\right)\\\
&=2\left(\sqrt{\frac{\mu+\lambda}{D}}C_{4}+\frac{bC_{3}}{(-1+D)\sqrt{\mu+\lambda}}\right)\end{aligned}\right.$
(3.3)

admits solutions $\lambda$ ($\lambda\in\mathbb{C}/(-\infty,-\mu]$) which are
all with negative real parts, then this pinned pulse solution is stable. We
denote this eigenfunction as
$(P_{1}(x)+O(\varepsilon),P_{2}(x)+O(\varepsilon))^{T}$, whose expression can
be given as

$P_{1}(x)=\begin{cases}C_{3}e^{\sqrt{\mu+\lambda}x},&x\in I_{s}^{-},\\\
C_{3},&x\in I_{f},\\\ C_{3}e^{-\sqrt{\mu+\lambda}x},&x\in
I_{s}^{+},\end{cases}\
P_{2}(x)=\begin{cases}C_{4}e^{\sqrt{\frac{\mu+\lambda}{D}}x}+\frac{bC_{3}}{(-1+D)(\mu+\lambda)}e^{\sqrt{\mu+\lambda}x},&x\in
I_{s}^{-},\\\ C_{4}+\frac{bC_{3}}{(-1+D)(\mu+\lambda)},&x\in I_{f},\\\
C_{4}e^{-\sqrt{\frac{\mu+\lambda}{D}}x}+\frac{bC_{3}}{(-1+D)(\mu+\lambda)}e^{-\sqrt{\mu+\lambda}x},&x\in
I_{s}^{+}.\end{cases}$ (3.4)

###  3.2 $G_{1}$ and $G_{2}$ are Linear

We can cut through our analysis from one of the simplest perspectives, i.e.,
we assume that $G_{1}$ and $G_{2}$ are linear.

In this case, the existence condition can be verified easily because equations
(3.1) are quadratic equations. For the stability condition, i.e., the
solutions of equations (LABEL:stability), through analysis, it can be
simplified as:

$\frac{2(\mu+\lambda)^{\frac{3}{2}}}{\alpha
G_{13}\sqrt{D}}-\frac{G_{12}(\mu+\lambda)}{G_{13}\sqrt{D}}+\frac{b}{D+\sqrt{D}}=\frac{\beta
G_{22}(\mu+\lambda)^{\frac{1}{2}}}{2D}+\frac{\beta G_{22}(\mu+\lambda)}{\alpha
G_{13}D}-\frac{\beta G_{12}G_{23}(\mu+\lambda)^{\frac{1}{2}}}{2DG_{13}},$
(3.5)

where $G_{ij}$ represents the $j$th coefficient of $G_{i}$ and
$G_{i}=G_{i1}+G_{i2}U_{1}+G_{i3}U_{2}$ with $i=1,2$ and $j=1,2,3$. Denote
$t:=\sqrt{\mu+\lambda}$, then $\rm{Re}$ $t>0$ and (3.5) is equal to

$2t^{3}-(\alpha G_{12}+\frac{\beta
G_{23}}{\sqrt{D}})t^{2}+\frac{\alpha\beta}{2\sqrt{D}}(G_{12}G_{23}-G_{13}G_{22})t+\frac{b\alpha
G_{13}}{\sqrt{D}+1}=0.$ (3.6)

Denote $A:=-(\alpha G_{12}+\frac{\beta G_{23}}{\sqrt{D}})$,
$B:=\frac{\alpha\beta}{2\sqrt{D}}(G_{12}G_{23}-G_{13}G_{22})$ and
$E:=\frac{b\alpha G_{13}}{\sqrt{D}+1}$, we rewrite equation (3.6) as

$H(t):=2t^{3}+At^{2}+Bt+E=0.$ (3.7)

For $B<0$, $H(t)$ has a minimum at $t=\frac{-A+\sqrt{A^{2}-6B}}{6}$. If, in
addition, $E>0$ and $H(\frac{-A+\sqrt{A^{2}-6B}}{6})<0$, then (3.7) has two
real-valued positive solutions. When the value
$H(\frac{-A+\sqrt{A^{2}-6B}}{6})$ becomes positive, these two real-valued
solutions merge and become complex-valued. As a result, we can tune $\mu$ such
that $\rm{Re}\lambda=0$. Hence, this pinned pulse solution can undergo a Hopf
bifurcation, we denote this bifurcation parameter as $\hat{\mu}$. Next, we
discuss the center normal form. We transform the system (1.1) by
$\tilde{U}_{1}=U_{1}-\Gamma_{1,p},\ \tilde{U}_{2}=U_{2}-\Gamma_{2,p},\
\tilde{\mu}=\mu-\hat{\mu}$, to

$\frac{d\tilde{U}}{dt}=L\tilde{U}+R(\tilde{U},\tilde{\mu}),$

where

$\displaystyle\tilde{U}=\begin{pmatrix}\tilde{U}_{1}\\\
\tilde{U}_{2}\end{pmatrix},\ \
R(\tilde{U},\tilde{\mu})=\begin{pmatrix}-\tilde{\mu}\tilde{U}_{1}\\\
-\tilde{\mu}\tilde{U}_{2}\end{pmatrix},$ $\displaystyle
L=\begin{pmatrix}\frac{d^{2}}{dx^{2}}-\hat{\mu}+\frac{\alpha}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\frac{\partial
G_{1}}{\partial
U_{1}}(\Gamma_{1,p},\Gamma_{2,p})&\frac{\alpha}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\frac{\partial
G_{1}}{\partial U_{2}}(\Gamma_{1,p},\Gamma_{2,p})\\\
-b+\frac{\beta}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\frac{\partial
G_{2}}{\partial
U_{1}}(\Gamma_{1,p},\Gamma_{2,p})&D\frac{d^{2}}{dx^{2}}-\hat{\mu}+\frac{\beta}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\frac{\partial
G_{2}}{\partial U_{2}}(\Gamma_{1,p},\Gamma_{2,p})\\\ \end{pmatrix}.$

Since $G_{1}$ and $G_{2}$ are linear, actually,

$L=\begin{pmatrix}\frac{d^{2}}{dx^{2}}-\hat{\mu}+\frac{\alpha
G_{12}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})&\frac{\alpha
G_{13}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\\\ -b+\frac{\beta
G_{22}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})&D\frac{d^{2}}{dx^{2}}-\hat{\mu}+\frac{\beta
G_{23}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\\\ \end{pmatrix}.$

We can observe that $R$ is linear with respect to $\tilde{U}$ and
$\tilde{\mu}$ respectively, which indicates that the $\Psi$ we calculated by
Lemma 2.1 is linear with $\mu$-dependent coefficients. Therefore, there is no
way for the center manifold to own a periodic orbit. Besides, the derivative
of $R$ with respect to $\tilde{U}$ is $-\tilde{\mu}$, which decides the
in(stability) of $\Gamma_{p}$. Here, we use
$\mathcal{X}=\left(L^{2}(\mathbb{R}^{2})\right)^{2}$,
$\mathcal{Y}=\mathcal{Z}=\left(H^{2}(\mathbb{R}^{2})\right)^{2}$. And we get
the conclusions:

######  Remark 3.1.

The differential equation (1.1) with linear $G_{1}$ and linear $G_{2}$ has
precisely one equili-brium $\Gamma_{p}(\mu)$, which is stable when
$\mu>\hat{\mu}$ and unstable when $\mu<\hat{\mu}$. And, there exists no
breathing phenomenon near the pinned pulse solution.

######  Remark 3.2.

When $D=1$, the expressions for (3.2) and (3.4) fail, the corresponding pulse
solution and eigenfunction become

$U_{2,p}(x)=\begin{cases}C_{2}e^{\sqrt{\mu}x}+\frac{bC_{1}}{4\mu}e^{\sqrt{\mu}x}\left(-1+2\sqrt{\mu}x\right),&x\in
I_{s}^{-},\\\ C_{2}-\frac{bC_{1}}{4\mu},&x\in I_{f},\\\
C_{2}e^{-\sqrt{\mu}x}+\frac{bC_{1}}{4\mu}e^{-\sqrt{\mu}x}\left(-1-2\sqrt{\mu}x\right),&x\in
I_{s}^{+},\end{cases}$

and

$P_{2}(x)=\begin{cases}C_{4}e^{\sqrt{\mu+\lambda}x}+\frac{bC_{3}}{4(\mu+\lambda)}e^{\sqrt{\mu+\lambda}x}\left(-1+2\sqrt{\mu+\lambda}x\right),&x\in
I_{s}^{-},\\\ C_{4}-\frac{bC_{3}}{4(\mu+\lambda)},&x\in I_{f},\\\
C_{4}e^{-\sqrt{\mu+\lambda}x}+\frac{bC_{3}}{4(\mu+\lambda)}e^{-\sqrt{\mu+\lambda}x}\left(-1-2\sqrt{\mu+\lambda}x\right),&x\in
I_{s}^{+}.\end{cases}$

Here $C_{1}$, $C_{2}$ satisfy

$\left\\{\begin{aligned} 2\sqrt{\mu}C_{1}&=\alpha
G_{1}\left(C_{1},C_{2}-\frac{bC_{1}}{4\mu}\right),\\\
2\left(\sqrt{\mu}C_{2}+\frac{bC_{1}}{4\sqrt{\mu}}\right)&=\beta
G_{2}\left(C_{1},C_{2}-\frac{bC_{1}}{4\mu}\right).\end{aligned}\right.$

$C_{3}$, $C_{4}$ satisfy

$\left\\{\begin{aligned} &2\sqrt{\mu+\lambda}C_{3}\\\
&=\alpha\left(\frac{\partial G_{1}}{\partial
U_{1}}\left(C_{1},C_{2}-\frac{bC_{1}}{4\mu}\right)C_{3}+\frac{\partial
G_{1}}{\partial
U_{2}}\left(C_{1},C_{2}-\frac{bC_{1}}{4\mu}\right)(C_{4}-\frac{bC_{3}}{4(\mu+\lambda)})\right),\\\
&2\left(\sqrt{\mu+\lambda}C_{4}+\frac{bC_{3}}{4\sqrt{\mu+\lambda}}\right)\\\
&=\frac{\beta}{D}\left(\frac{\partial G_{2}}{\partial
U_{1}}\left(C_{1},C_{2}-\frac{bC_{1}}{4\mu}\right)C_{3}+\frac{\partial
G_{2}}{\partial
U_{2}}\left(C_{1},C_{2}-\frac{bC_{1}}{4\mu}\right)(C_{4}-\frac{bC_{3}}{4(\mu+\lambda)})\right),\end{aligned}\right.$

which yields

$\frac{2}{\alpha
G_{13}}(\mu+\lambda)^{\frac{3}{2}}-\frac{G_{12}}{G_{13}}(\mu+\lambda)+\frac{b}{2}=\frac{\beta
G_{22}}{2}(\mu+\lambda)^{\frac{1}{2}}+\frac{\beta G_{23}}{\alpha
G_{13}}(\mu+\lambda)-\frac{\beta
G_{12}G_{23}}{2G_{13}}(\mu+\lambda)^{\frac{1}{2}}.$

It equals to equation (3.5) when $D=1$. Hence, we get the same conclusion as
in Remark 3.1.

###  3.3 $G_{1}$ is Nonlinear and $G_{2}$ is Linear

Next, we examine the sub-simple situation, i.e., $G_{1}$ is nonlinear and
$G_{2}$ is linear. We further denote
$G_{1}=G_{11}+G_{12}U_{1}+G_{13}U_{2}+G_{14}U_{1}^{2}+G_{15}U_{2}^{2}+G_{16}U_{1}U_{2}+...$.
In order to reveal the essence, we shift our focus on the coupled case when
there is only one nonlinear term, i.e., $G_{1}=G_{11}+G_{15}U_{2}^{2}$. In
order not to bring in cumbersome formulas and complex expressions from
$G_{2}$, we discuss from three situations.

####  3.3.1 $G_{2}=G_{21}$

First, we assume that $G_{2}$ is a constant function with $G_{2}=G_{21}$, then
(1.1) becomes

$\left\\{\begin{aligned} \frac{\partial U_{1}}{\partial
t}&=\frac{\partial^{2}U_{1}}{\partial x^{2}}-\mu
U_{1}+\frac{\alpha}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(G_{11}+G_{15}U_{2}^{2}),\\\
\frac{\partial U_{2}}{\partial t}&=D\frac{\partial^{2}U_{2}}{\partial
x^{2}}-bU_{1}-\mu
U_{2}+\frac{\beta}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(G_{21}).\end{aligned}\right.$
(3.8)

As before, system (3.8) admits the stationary pinned pulse solution like (3.2)
if

$\left\\{\begin{aligned}
2\sqrt{\mu}C_{1}&=\alpha\left(G_{11}+G_{15}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})^{2}\right),\\\
\frac{\beta
G_{21}}{D}&=2\left(\sqrt{\frac{\mu}{D}}C_{2}+\frac{bC_{1}}{(-1+D)\sqrt{\mu}}\right)\end{aligned}\right.$
(3.9)

admits non-degenerate solutions $C_{1}$ and $C_{2}$, i.e.,
$\Delta=4\mu+\frac{4\alpha G_{15}b}{(1+\sqrt{D})\mu}\left(\frac{\beta
G_{21}}{\sqrt{D}}-\frac{\alpha bG_{11}}{\mu(1+\sqrt{D})}\right)>0$. Here, we
assume that $D\neq 1$. Unless otherwise stated, we adhere to this assumption
in the following.

Then, under this assumption of existence, consider solutions of the eigenvalue
equation

$\left\\{\begin{aligned}
0&=\frac{d^{2}P_{1}}{dx^{2}}-(\mu+\lambda)P_{1}+\frac{\alpha}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(2G_{15}\Gamma_{2,p}(x)P_{2}),\\\
0&=D\frac{d^{2}P_{2}}{dx^{2}}-bP_{1}-(\mu+\lambda)P_{2},\end{aligned}\right.$
(3.10)

i.e., the solutions of

$\left\\{\begin{aligned} 2\sqrt{\mu+\lambda}C_{3}&=2\alpha
G_{15}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\left(C_{4}+\frac{bC_{3}}{(-1+D)(\mu+\lambda)}\right),\\\
0&=2\left(\sqrt{\frac{\mu+\lambda}{D}}C_{4}+\frac{bC_{3}}{(-1+D)\sqrt{\mu+\lambda}}\right).\end{aligned}\right.$
(3.11)

We solve it to get

$\displaystyle\mu+\lambda$
$\displaystyle=\left|\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\frac{-\alpha
bG_{15}}{1+\sqrt{D}}\right|^{\frac{2}{3}},$ $\displaystyle\mu+\lambda$
$\displaystyle=\left|\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\frac{-\alpha
bG_{15}}{1+\sqrt{D}}\right|^{\frac{2}{3}}(-\frac{1}{2}+\frac{\sqrt{3}}{2}i),$
$\displaystyle\mu+\lambda$
$\displaystyle=\left|\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\frac{-\alpha
bG_{15}}{1+\sqrt{D}}\right|^{\frac{2}{3}}(-\frac{1}{2}-\frac{\sqrt{3}}{2}i),$

where $\left|\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\frac{-\alpha
bG_{15}}{1+\sqrt{D}}\right|^{\frac{2}{3}}$ is positive and real. This outcome
implies that Hopf bifurcation is impossible in this case regardless of $\mu$
values.

######  Remark 3.3.

There exists no Hopf bifurcation in system (3.8).

####  3.3.2 $G_{2}=G_{22}U_{1}$

Next, we consider the case that $G_{2}$ is simple linear in $U_{1}$, i.e.,

$\left\\{\begin{aligned} \frac{\partial U_{1}}{\partial
t}&=\frac{\partial^{2}U_{1}}{\partial x^{2}}-\mu
U_{1}+\frac{\alpha}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(G_{11}+G_{15}U_{2}^{2}),\\\
\frac{\partial U_{2}}{\partial t}&=D\frac{\partial^{2}U_{2}}{\partial
x^{2}}-bU_{1}-\mu
U_{2}+\frac{\beta}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(G_{22}U_{1}).\end{aligned}\right.$
(3.12)

Provided that

$\displaystyle(A1)\ \Delta=4\mu-4\alpha^{2}G_{15}G_{11}\left(\frac{\beta
G_{22}}{2\sqrt{\mu D}}-\frac{b}{\mu(1+\sqrt{D})}\right)^{2}>0,$

then system (3.12) exists a unique pinned pulse solution.

Under this assumption, the eigenvalues can be calculated by

$(\mu+\lambda)^{\frac{3}{2}}=\alpha
G_{15}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\left(\frac{\beta
G_{22}}{2\sqrt{D}}(\mu+\lambda)^{\frac{1}{2}}-\frac{b}{1+\sqrt{D}}\right).$
(3.13)

Similarly, denote $B_{1}:=-\frac{\alpha\beta
G_{15}G_{22}}{2\sqrt{D}}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)$,
$E_{1}:=\frac{\alpha
bG_{15}}{1+\sqrt{D}}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)$, we rewrite
(3.13) as

$H_{1}(t):=t^{3}+B_{1}t+E_{1}=0.$

If $E_{1}>0$, $B_{1}<0$ and $H_{1}(\sqrt{\frac{-B_{1}}{3}})<0$, then there
exist two real-valued positive solutions. These two real-valued solutions
merge and become complex-valued when $H_{1}(\sqrt{\frac{-B_{1}}{3}})>0$.
Hence, if

$\displaystyle(A2)\ \frac{\alpha
bG_{15}}{1+\sqrt{D}}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)>0;$
$\displaystyle(A3)\ \alpha\beta
G_{15}G_{22}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)>0;$
$\displaystyle(A4)\ \left(\frac{\alpha\beta
G_{15}G_{22}}{3}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\right)^{\frac{3}{2}}+\frac{\alpha
bG_{15}}{1+\sqrt{D}}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)$
$\displaystyle\ \ \ \ \ -\alpha\beta
G_{15}G_{22}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\left(\frac{\alpha\beta
G_{15}G_{22}}{3}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\right)^{\frac{1}{2}}>0,\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $

then there exist a pair of conjugate imaginary roots. According to the
Shengjin formula [17] of the cubic equation, this pair of conjugate imaginary
solutions can be calculated to be

$\displaystyle t_{1,2}$
$\displaystyle=\frac{1}{6}\left(\sqrt[3]{\frac{3}{2}\left(9C_{1}+\sqrt{81C_{1}^{2}+12B_{1}^{3}}\right)}+\sqrt[3]{\frac{3}{2}\left(9C_{1}-\sqrt{81C_{1}^{2}+12B_{1}^{3}}\right)}\right)$
$\displaystyle\pm\frac{\sqrt{3}}{6}\left(\sqrt[3]{\frac{3}{2}\left(9C_{1}+\sqrt{81C_{1}^{2}+12B_{1}^{3}}\right)}-\sqrt[3]{\frac{3}{2}\left(9C_{1}-\sqrt{81C_{1}^{2}+12B_{1}^{3}}\right)}\right)i.$

Denote

$\displaystyle n_{r}:$
$\displaystyle=\frac{1}{6}\left(\sqrt[3]{\frac{3}{2}\left(9C_{1}+\sqrt{81C_{1}^{2}+12B_{1}^{3}}\right)}+\sqrt[3]{\frac{3}{2}\left(9C_{1}-\sqrt{81C_{1}^{2}+12B_{1}^{3}}\right)}\right),$
$\displaystyle n_{i}:$
$\displaystyle=\frac{\sqrt{3}}{6}\left(\sqrt[3]{\frac{3}{2}\left(9C_{1}+\sqrt{81C_{1}^{2}+12B_{1}^{3}}\right)}-\sqrt[3]{\frac{3}{2}\left(9C_{1}-\sqrt{81C_{1}^{2}+12B_{1}^{3}}\right)}\right),$

then setting $\mu=n_{r}^{2}-n_{i}^{2}$ can give $\lambda=\pm 2n_{r}n_{i}i$.
This equals to the condition

$\displaystyle(A5)\ 9\mu=2\sqrt[3]{\frac{3}{2}\left(\frac{-9\alpha
bG_{15}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)}{1+\sqrt{D}}+\sqrt{\Delta}\right)}\sqrt[3]{\frac{3}{2}\left(\frac{-9\alpha
bG_{15}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)}{1+\sqrt{D}}-\sqrt{\Delta}\right)}$
$\displaystyle\ -\frac{1}{2}\sqrt[3]{\left(\frac{-27\alpha
bG_{15}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)}{2+2\sqrt{D}}+\frac{3}{2}\sqrt{\Delta}\right)^{2}}-\frac{1}{2}\sqrt[3]{\left(\frac{-27\alpha
bG_{15}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)}{2+2\sqrt{D}}-\frac{3}{2}\sqrt{\Delta}\right)^{2}}$
$\displaystyle\ admits\ positive\ roots\ \mu,\ where$ $\displaystyle\
\Delta=81\left(\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\frac{\alpha
bG_{15}}{1+\sqrt{D}}\right)^{2}-12\left(\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\frac{\alpha\beta
G_{22}G_{15}}{2\sqrt{D}}\right)^{3}.$

As a result, there is

######  Theorem 3.4.

Assume that (A1), (A2), (A3), (A4) and (A5) hold, then there exists a Hopf
bifurcation for system (3.12), where we denote this bifurcation parameter
value by $\hat{\mu}$.

Based on this, we proceed to study the existence of breathing pulses. First,
we transform the system (3.12) and get

$\frac{d\tilde{U}}{dt}=L\tilde{U}+R(\tilde{U},\tilde{\mu}),$ (3.14)

where

$\displaystyle\tilde{U}=\begin{pmatrix}\tilde{U}_{1}\\\
\tilde{U}_{2}\end{pmatrix},\ \
R(\tilde{U},\tilde{\mu})=\begin{pmatrix}-\tilde{\mu}\tilde{U}_{1}+\frac{\alpha
G_{15}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{U}_{2}^{2}\\\
-\tilde{\mu}\tilde{U}_{2}\end{pmatrix},\
L=\begin{pmatrix}\frac{d^{2}}{dx^{2}}-\hat{\mu}&\frac{2\alpha
G_{15}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\Gamma_{2,p}\\\
-b+\frac{\beta
G_{22}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})&D\frac{d^{2}}{dx^{2}}-\hat{\mu}\\\
\end{pmatrix}.$ (3.15)

We still use the function space
$\mathcal{X}=\left(L^{2}(\mathbb{R}^{2})\right)^{2}$,
$\mathcal{Y}=\mathcal{Z}=\left(H^{2}(\mathbb{R}^{2})\right)^{2}$, and
$L:\mathcal{Z}=\mathcal{Y}\rightarrow\mathcal{X}$. $L$ is a closed operator in
$\mathcal{X}$ with domain $\mathcal{Z}$. Since the operator $L$ is real, and
the fact that its point spectrum is given as $\lambda=\pm 2n_{r}n_{i}i$, we
know the associated spectral subspace $\mathcal{E}_{0}$ is two-dimensional
spanned by the eigenfunctions ${P,\bar{P}}$. And $P=(P_{1},P_{2})^{T}$
satisfies

$\left\\{\begin{aligned} 0&=\frac{\partial^{2}P_{1}}{\partial
x^{2}}-(n_{r}^{2}-n_{i}^{2}+2n_{r}n_{i}i)P_{1}+\frac{\alpha}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(2G_{15}\Gamma_{2,p}^{2}P_{2}),\\\
0&=D\frac{\partial^{2}P_{2}}{\partial
x^{2}}-bP_{1}-(n_{r}^{2}-n_{i}^{2}+2n_{r}n_{i}i)P_{2}+\frac{\beta}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(G_{22}P_{1}).\end{aligned}\right.$

According to the parameter-dependent center manifolds Lemma 2.1, we have

$\tilde{U}=U_{0}+\Psi(U_{0},\tilde{\mu}),\ U_{0}\in\mathcal{E}_{0},\
\Psi(U_{0},\tilde{\mu})\in\mathcal{Z}_{h}=\mathcal{Y}_{h}.$

Then, by applying the normal form Lemma 2.3, we find
$U_{0}=V_{0}+\Phi_{\tilde{\mu}}(V_{0}),\
V_{0}\in\mathcal{O}_{1}\subset\mathcal{E}_{0}$, which transforms equation
(3.14) into the normal form

$\frac{dV_{0}}{dt}=LV_{0}+N_{\tilde{\mu}}V_{0}+\rho(V_{0},\tilde{\mu}),$

We therefore have

$\tilde{U}=V_{0}+\tilde{\Psi}(V_{0},\tilde{\mu}),\ \
\tilde{\Psi}(V_{0},\tilde{\mu})=\Phi_{\tilde{\mu}}(V_{0})+\Psi(V_{0}+\Phi_{\tilde{\mu}}(V_{0}),\tilde{\mu}).$
(3.16)

Since $V_{0}(t)\in\mathcal{E}_{0}$, it is convenient to write
$V_{0}(t)=A(t)P+\,\overline{\\!{A(t)P}},\ A(t)\in\mathbb{C}$, then equation
(3.3.2) can be transformed into an “amplitude equation”
$\frac{dA}{dt}=i2n_{r}n_{i}A+N_{\tilde{\mu}}(A,\bar{A})+\rho(A,\bar{A},\tilde{\mu})$,
which can be simplified further by Lemma 2.4 to be

$\frac{dA}{dt}=i2n_{r}n_{i}A+AQ(|A|^{2},\tilde{\mu})+\rho(A,\bar{A},\tilde{\mu}).$
(3.17)

Here,
$Q(|A|^{2},\tilde{\mu})=a\tilde{\mu}+b|A|^{2}+O((|\tilde{\mu}|+|A|^{2})^{2})$.
Hence, equation (3.17) equals to

$\frac{dA}{dt}=i2n_{r}n_{i}A+a\tilde{\mu}A+b|A|^{2}A+O(|A|(|\tilde{\mu}|+|A|^{2})^{2}).$

We neglect the higher order terms $\rho$, which has no impact on our final
analysis. Then, by introducing polar coordinates $A=re^{i\phi}$, we obtain

$\frac{dr}{dt}+ir\frac{d\phi}{dt}=i2n_{r}n_{i}r+rQ(r^{2},\tilde{\mu}),$

whose real and imaginary parts satisfy

$\displaystyle\frac{dr}{dt}$
$\displaystyle=a_{r}\tilde{\mu}r+b_{r}r^{3}+O(r(|\tilde{\mu}|+r^{2})^{2}),$
(3.18) $\displaystyle\frac{d\phi}{dt}$
$\displaystyle=2n_{r}n_{i}+a_{i}\tilde{\mu}+b_{i}r^{2}+O((|\tilde{\mu}|+r^{2})^{2}),$

in which the real coefficients $a_{r}$ and $b_{r}$ represent the real parts of
$a$ and $b$, whereas $a_{i}$ and $b_{i}$ represent the imaginary parts of $a$
and $b$. The key property of system (3.18) is that the radial equation for $r$
decouples, which corresponds to a pitchfork bifurcation occurs at
$\tilde{\mu}=0$. If the bifurcation here is supercritical, then there exists
an oscillating solution near $\Gamma_{p}$, which is called a breathing pulse.
Therefore, figuring out the values of coefficients $a$ and $b$ should be the
next step. For this, differentiate (3.16) with respect to $t$ and replace
$\frac{dU}{dt}$ and $\frac{dV_{0}}{dt}$ respectively. There is

$D_{V_{0}}\tilde{\Psi}(V_{0},\tilde{\mu})LV_{0}-L\tilde{\Psi}(V_{0},\tilde{\mu})+N_{\tilde{\mu}}V_{0}=Q(V_{0},\tilde{\mu}),$
(3.19)

where

$Q(V_{0},\tilde{\mu})=\Pi_{p}\left(R(V_{0}+\tilde{\Psi}(V_{0},\tilde{\mu}),\tilde{\mu})-D_{V_{0}}\tilde{\Psi}(V_{0},\tilde{\mu})N_{\tilde{\mu}}V_{0}\right).$

Here $\Pi_{p}$ represents the linear map that associates to a map of class
$\mathcal{C}^{p}$ the polynomial of degree $p$ in its Taylor expansion. We
then write the Taylor expansion of $R$ and $\tilde{\Psi}$ as follows:

$\displaystyle R(\tilde{U},\tilde{\mu})$ $\displaystyle=\sum_{1\leq q+l\leq
p}R_{ql}(\tilde{U}^{(q)},\tilde{\mu}^{(l)})+o((\|\tilde{U}\|+\|\tilde{\mu}\|)^{p}),$
$\displaystyle\tilde{\Psi}(V_{0},\tilde{\mu})$ $\displaystyle=\sum_{1\leq
q+l\leq
p}\tilde{\Psi}_{ql}(V_{0}^{(q)},\tilde{\mu}^{(l)})+o((\|V_{0}\|+\|\tilde{\mu}\|)^{p}),$

with

$R_{ql}(\tilde{U}^{(q)},\tilde{\mu}^{(l)})=\tilde{U}^{q}\tilde{\mu}^{l}R_{ql},\
\tilde{\Psi}_{ql}(V_{0}^{(q)},\tilde{\mu}^{(l)})=\tilde{\mu}^{l}\sum_{q_{1}+q_{2}=q}A^{q_{1}}\bar{A}^{q_{2}}\Psi_{q_{1}q_{2}l}.$

By identifying in (3.19) the terms of order $O(\mu)$, $O(A^{2})$, and
$O(A\bar{A})$, we obtain

$\displaystyle-L\Psi_{001}$ $\displaystyle=R_{01},$
$\displaystyle(i4n_{r}n_{i}i-L)\Psi_{200}$ $\displaystyle=R_{20}(P,P),$
$\displaystyle-L\Psi_{110}$ $\displaystyle=2R_{20}(P,\bar{P}).$

Since $\sigma_{pt}=\\{\pm 2n_{r}n_{i}i\\}$, the operators $L$ and
$(i4n_{r}n_{i}i-L)$ on the left sides are invertible, so that $\Psi_{001}$,
$\Psi_{200}$ and $\Psi_{110}$ can be uniquely determined. Similarly, we
identify the terms of order $O(\mu A)$ and $O(A^{2}\bar{A})$ to find

$\displaystyle(L-i2n_{r}n_{i})\Psi_{101}$ $\displaystyle=aP-
R_{11}(P)-2R_{20}(P,\Psi_{001}),$ (3.20)
$\displaystyle(L-i2n_{r}n_{i})\Psi_{210}$
$\displaystyle=bP-2R_{20}(P,\Psi_{110})-2R_{20}(\bar{P},\Psi_{200})-3R_{30}(P,P,\bar{P}).$
(3.21)

We already known that $i2n_{r}n_{i}$ is a simple isolated eigenvalue of $L$,
the range of $(L-i2n_{r}n_{i})$ is of codimension 1, therefore we can solve
the above two equations if and only if the right-hand sides satisfy one
solvability condition. According to the Fredholm alternative theorem (P.28 of
[28]), it demands that the right-hand sides of (3.20) and (3.21) are
orthogonal to the kernel of the adjoint operator $(L^{*}+i2n_{r}n_{i})$. It is
easily to find that the kernel of the adjoint operator $(L^{*}+i2n_{r}n_{i})$
is one-dimensional just as the kernel of $(L-i2n_{r}n_{i})$. Moreover, $L^{*}$
is real. So we introduce the eigenfunction $P^{*}$ by
$L^{*}P^{*}=-i2n_{r}n_{i}P^{*}$, which gives

$\displaystyle a$ $\displaystyle=\frac{\langle
R_{11}(P)+2R_{20}(P,\Psi_{001}),P^{*}\rangle}{\langle P,P^{*}\rangle},$
$\displaystyle b$ $\displaystyle=\frac{\langle
2R_{20}(P,\Psi_{110})+2R_{20}(\bar{P},\Psi_{200})+3R_{30}(P,P,\bar{P}),P^{*}\rangle}{\langle
P,P^{*}\rangle}.$

According to (3.15), $R_{01}=0,\ R_{11}=-1,\
R_{20}(U,V)=\begin{pmatrix}\frac{\alpha
G_{15}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})U_{2}V_{2}\\\
0\end{pmatrix},\ R_{30}=0$, then

$\displaystyle a$ $\displaystyle=\frac{\langle-P,P^{*}\rangle}{\langle
P,P^{*}\rangle}=-1,$ $\displaystyle b$
$\displaystyle=\frac{\int_{\mathbb{R}}^{\ }\left(4|P_{2}|^{2}(\frac{\alpha
G_{15}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(-L)^{-1}_{11}I(\frac{x}{\varepsilon^{2}})+2(\frac{\alpha
G_{15}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(2i\omega_{H}-L)^{-1}_{11}I(\frac{x}{\varepsilon^{2}})P_{2}^{2}\right)\overline{P_{1}^{*}}dx}{\langle
P,P^{*}\rangle},$

where subscript $\ {}_{11}$ represents the location in this operator matrix
and $P^{*}$ satisfies $(L^{*}+i2n_{r}n_{i})P^{*}=0$. Similarly, $P^{*}$ can be
solved by using the method as before. And

$P_{1}^{*}(x)=\begin{cases}C_{5}e^{\sqrt{\hat{\mu}-i2n_{r}n_{i}}x}+\frac{bC_{6}e^{\sqrt{\frac{\hat{\mu}-i2n_{r}n_{i}}{D}}x}}{(-1+\frac{1}{D})(\hat{\mu}-i2n_{r}n_{i})},&x\in
I_{s}^{-},\\\
C_{5}+\frac{bC_{6}}{(-1+\frac{1}{D})(\hat{\mu}-i2n_{r}n_{i})},&x\in I_{f},\\\
C_{5}e^{-\sqrt{\hat{\mu}-i2n_{r}n_{i}}x}+\frac{bC_{6}e^{-\sqrt{\frac{\hat{\mu}-i2n_{r}n_{i}}{D}}x}}{(-1+\frac{1}{D})(\hat{\mu}-i2n_{r}n_{i})},&x\in
I_{s}^{+},\end{cases}\
P_{2}^{*}(x)=\begin{cases}C_{6}e^{\sqrt{\frac{\hat{\mu}-i2n_{r}n_{i}}{D}}x},&x\in
I_{s}^{-},\\\ C_{6},&x\in I_{f},\\\
C_{6}e^{-\sqrt{\frac{\hat{\mu}-i2n_{r}n_{i}}{D}}x},&x\in
I_{s}^{+}.\end{cases}$

Here, $C_{5}$ and $C_{6}$ are likewise required to satisfy existence
constraints of the solution. Then we divide the calculation of $b$ into six
parts as $b=\frac{I_{1}+I_{2}+I_{3}}{L_{1}+L_{2}+L_{3}},$ where

$\displaystyle I_{1}$
$\displaystyle=\int_{-\infty}^{-\varepsilon}\left(4|P_{2}|^{2}(\frac{\alpha
G_{15}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(-L)^{-1}_{11}I(\frac{x}{\varepsilon^{2}})+2(\frac{\alpha
G_{15}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(i4n_{r}n_{i}-L)^{-1}_{11}I(\frac{x}{\varepsilon^{2}})P_{2}^{2}\right)\overline{P_{1}^{*}}dx;$
$\displaystyle I_{2}$
$\displaystyle=\int_{-\varepsilon}^{+\varepsilon}\left(4|P_{2}|^{2}(\frac{\alpha
G_{15}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(-L)^{-1}_{11}I(\frac{x}{\varepsilon^{2}})+2(\frac{\alpha
G_{15}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(i4n_{r}n_{i}-L)^{-1}_{11}I(\frac{x}{\varepsilon^{2}})P_{2}^{2}\right)\overline{P_{1}^{*}}dx;$
$\displaystyle I_{3}$
$\displaystyle=\int_{+\varepsilon}^{+\infty}\left(4|P_{2}|^{2}(\frac{\alpha
G_{15}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(-L)^{-1}_{11}I(\frac{x}{\varepsilon^{2}})+2(\frac{\alpha
G_{15}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(i4n_{r}n_{i}-L)^{-1}_{11}I(\frac{x}{\varepsilon^{2}})P_{2}^{2}\right)\overline{P_{1}^{*}}dx;$
$\displaystyle L_{1}$
$\displaystyle=\int_{-\infty}^{-\varepsilon}P_{1}(x)\overline{P_{1}^{*}}(x)+P_{2}(x)\overline{P_{2}^{*}}(x)dx,$
$\displaystyle L_{2}$
$\displaystyle=\int_{-\varepsilon}^{+\varepsilon}P_{1}(x)\overline{P_{1}^{*}}(x)+P_{2}(x)\overline{P_{2}^{*}}(x)dx,$
$\displaystyle L_{3}$
$\displaystyle=\int_{+\varepsilon}^{+\infty}P_{1}(x)\overline{P_{1}^{*}}(x)+P_{2}(x)\overline{P_{2}^{*}}(x)dx.$

For $I_{1}$, the operator $(-L)^{-1}_{11}$ and $(2i\omega_{H}-L)^{-1}_{11}$
are bounded, the function $P$ and $P^{*}$ are bounded,
$I(\frac{x}{\varepsilon^{2}})$ is exponentially small. Hence, $I_{1}=0$ to
leading order. Likewise, we also have $I_{3}=0$ to leading order. For $I_{2}$,
something different happens. $I(\frac{x}{\varepsilon^{2}})$ is not
exponentially small. However, to leading order,
$P_{2}=C_{4}+\frac{bC_{3}}{(-1+D)(\hat{\mu}+i2n_{r}n_{i})}$ and
$\overline{P_{1}^{*}}=C_{5}+\frac{bC_{6}}{(-1+\frac{1}{D})(\hat{\mu}-i2n_{r}n_{i})}$,
which facilitates our calculation. For $L_{2}$, $P$ and $P^{*}$ are constants,
so we have $L_{2}=0$ to leading order. For $L_{1}$ and $L_{3}$, since the
integration interval is symmetric and the integration function is an even
function, we get $L_{1}=L_{3}$ to leading order. In summary,
$b=\frac{I_{2}}{2L_{1}}$.

######  Theorem 3.5.

Assume that the real part of $b$ is not equal to 0, then for the system (3.12)
a supercritical (resp. subcritical) Hopf bifurcation occurs at $\mu=\hat{\mu}$
when $b_{r}<0$ (resp., $b_{r}>0$). More precisely, for $\mu$ sufficiently
close to $\hat{\mu}$:

  1. (i)

If $b_{r}>0$ (resp., $b_{r}<0$), the differential equation has precisely one
equilibrium $U(\mu)$ for $\mu<\hat{\mu}$ (resp., $\mu>\hat{\mu}$). This
equilibrium is stable when $b_{r}<0$ and unstable when $b_{r}>0$;

  2. (ii)

If $b_{r}>0$ (resp., $b_{r}<0$), the differential equation possesses for
$\mu>\hat{\mu}$ (resp., $\mu<\hat{\mu}$) an equilibrium $U(\mu)$ and a unique
periodic orbit $U^{*}(\mu)=O(|\mu-\hat{\mu}|^{\frac{1}{2}})$ which surrounds
this equilibrium. The periodic orbit is stable when $b_{r}<0$ and unstable
when $b_{r}>0$. Moreover, there exists a stable oscillating solution
$\Gamma_{osc}$ near $\Gamma_{p}$, which is given by

$\Gamma_{osc}=\Gamma_{p}+\sqrt{\bigg{|}\frac{\mu-\hat{\mu}}{b_{r}}\bigg{|}}e^{i2n_{r}n_{i}t}P_{\hat{\mu}}+c.c.+O(|\mu|).$

###### Proof.

The weakly nonlinear stability of the pinned pulse solution $\Gamma_{p}$ near
a Hopf bifurcation is determined by

$\displaystyle\frac{dr}{dt}$
$\displaystyle=-\tilde{\mu}r+b_{r}r^{3}+O(r(|\tilde{\mu}|+r^{2})^{2}),$
$\displaystyle\frac{d\phi}{dt}$
$\displaystyle=2n_{r}n_{i}+O(|\tilde{\mu}|+r^{2}),$

where $O$-terms originate from the nonlinearity of $R$. The trivial
equilibrium $r=0$ of the first radial equation is stable when $\tilde{\mu}>0$
and unstable when $\tilde{\mu}<0$. Moreover, the radial equation has a
nontrivial leading order equilibrium if and only if
$r^{2}=\frac{\tilde{\mu}}{b_{r}}$ exists a positive solution
$r^{*}=O(|\tilde{\mu}|^{\frac{1}{2}})$. $r^{*}$ has opposite stability to
$r=0$, i.e., $r^{*}$ is stable if $\tilde{\mu}<0$, while $r^{*}$ is unstable
when $\tilde{\mu}>0$. Moreover, according to (3.16),
$U=A(t)P+\,\overline{\\!{A(t)P}}+\tilde{\Psi}(A(t)P+\,\overline{\\!{A(t)P}},\tilde{\mu})$,
which yields the expansion expression for $\Gamma_{osc}$ as

$\Gamma_{osc}=\sqrt{\bigg{|}\frac{\tilde{\mu}}{b_{r}}\bigg{|}}e^{i\omega_{H}t}P+c.c.+O(|\tilde{\mu}|).$

Back to the original system, i.e, reversing the transformation
$\left\\{\begin{aligned} \tilde{U}_{1}&=U_{1}-\Gamma_{1,p}\\\
\tilde{U}_{2}&=U_{2}-\Gamma_{2,p}\\\
\tilde{\mu}&=\mu-\hat{\mu}\end{aligned}\right.$, we come to the conclusion. ∎

######  Remark 3.6.

Similar to Remark 3.2, when $D=1$, the expressions need to be modified. Not
only this, $P^{*}_{1}$ needs to be rewritten as

$P_{1}^{*}(x)=\begin{cases}C_{5}e^{\sqrt{\hat{\mu}-i2n_{r}n_{i}}x}+\frac{bC_{6}}{4(\hat{\mu}-i2n_{r}n_{i})}e^{\sqrt{\hat{\mu}-i2n_{r}n_{i}}x}(-1+2\sqrt{\hat{\mu}-i2n_{r}n_{i}}x),&x\in
I_{s}^{-},\\\ C_{5}-\frac{bC_{6}}{4(\hat{\mu}-i2n_{r}n_{i})},&x\in I_{f},\\\
C_{5}e^{-\sqrt{\hat{\mu}-i2n_{r}n_{i}}x}+\frac{bC_{6}}{4(\hat{\mu}-i2n_{r}n_{i})}e^{-\sqrt{\hat{\mu}-i2n_{r}n_{i}}x}(-1-2\sqrt{\hat{\mu}-i2n_{r}n_{i}}x),&x\in
I_{s}^{+},\end{cases}$ (3.22)

which leads to the change of all calculations.

####  3.3.3 $G_{2}=G_{23}U_{2}$

When $G_{2}$ is simple linear in $U_{2}$, i.e., $G_{2}=G_{23}U_{2}$, the
system admits the pinned pulse solution, provided that

$\displaystyle(A6)\
\Delta=4\mu-4\alpha^{2}G_{11}G_{15}\left(\frac{2b}{(-1+D)\sqrt{\mu}}\frac{\sqrt{D}-D}{2\sqrt{\mu
D}-\beta G_{23}}\right)^{2}>0.$

Correspondingly, we can obtain a univariate cubic equation about
$\sqrt{\mu+\lambda}$ analogous to (3.13). In order to ensure the existence of
Hopf bifurcation, we impose the following assumptions:

$\displaystyle(A7)\ \Delta>0;$ $\displaystyle(A8)\ \beta
G_{23}-\sqrt[3]{-(\beta
G_{23})^{3}+3\sqrt{D}(-B+\sqrt{\Delta})}-\sqrt[3]{-(\beta
G_{23})^{3}+3\sqrt{D}(-B-\sqrt{\Delta})}\leq 0;$ $\displaystyle(A9)\
36D\mu=-\frac{1}{2}\sqrt[3]{\left(-(\beta
G_{23})^{3}+3\sqrt{D}(-B+\sqrt{\Delta})\right)^{2}}-\frac{1}{2}\sqrt[3]{\left(-(\beta
G_{23})^{3}+3\sqrt{D}(-B-\sqrt{\Delta})\right)^{2}}$ $\displaystyle+(\beta
G_{23})^{2}+\beta G_{23}\left(\sqrt[3]{-(\beta
G_{23})^{3}+3\sqrt{D}(-B+\sqrt{\Delta})}+\sqrt[3]{-(\beta
G_{23})^{3}+3\sqrt{D}(-B-\sqrt{\Delta})}\right)$
$\displaystyle+2\sqrt[3]{-(\beta
G_{23})^{3}+3\sqrt{D}(-B+\sqrt{\Delta})}\sqrt[3]{-(\beta
G_{23})^{3}+3\sqrt{D}(-B-\sqrt{\Delta})}\ admits\ positive\ roots\ \mu\ with$
$\displaystyle\Delta=\left(\frac{-2\alpha
b\sqrt{D}G_{15}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})}{1+\sqrt{D}}\right)\left(324D\left(\frac{-2\alpha
b\sqrt{D}G_{15}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})}{1+\sqrt{D}}\right)+12(\beta
G_{23})^{3}\right).$

######  Theorem 3.7.

Assume that (A6), (A7), (A8) and (A9) hold, then the system considered here
exists Hopf bifurcation when $\mu=\hat{\mu}$.

Now, there are

$\displaystyle\ \
R(\tilde{U},\tilde{\mu})=\begin{pmatrix}-\tilde{\mu}\tilde{U}_{1}+\frac{\alpha
G_{15}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{U}_{2}^{2}\\\
-\tilde{\mu}\tilde{U}_{2}\end{pmatrix},\
L=\begin{pmatrix}\frac{d^{2}}{dx^{2}}-\hat{\mu}&\frac{2\alpha
G_{15}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\Gamma_{2,p}\\\
-b&D\frac{d^{2}}{dx^{2}}-\hat{\mu}+\frac{\beta
G_{23}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\\\ \end{pmatrix}.$

Hence, $a$ still equals to -1 and the calculation of $b$ can be simplified to
the same expression $b=\frac{I_{2}}{2L_{1}}$.

######  Theorem 3.8.

Assume that the real part of $b$ is not equal to 0, then we get the same
conclusions as Theorem 3.5 with $\hat{\mu}$ satisfies Theorem 3.7.

Besides, $\sigma_{+}=\varnothing$ gives the following remark according to
Lemma 2.2:

######  Remark 3.9.

The local center manifold $\mathcal{M}_{0}(\mu)=V_{0}+\tilde{\Psi}(V_{0},\mu)$
considered in this subsection is locally attracting.

######  Remark 3.10.

Remark 3.3, Theorem 3.5 and 3.8 deal with three simple situations, which
involve only one nonlinear term and the simplest linear terms. For other
nonlinear $G_{1}$ and linear $G_{2}$, we can also ascertain the conditions for
the occurrence of breathing pulses by using the same analytical approach.

###  3.4 $G_{2}$ is Nonlinear and $G_{1}$ is Linear

Then, we consider the case when $G_{2}$ is nonlinear and $G_{1}$ is linear.
Since we are interested in mutually coupled systems, we may wish
$G_{1}=G_{11}+G_{13}U_{2}$ and
$G_{2}=G_{21}+G_{22}U_{1}+G_{23}U_{2}+G_{24}U_{1}^{2}+G_{25}U_{2}^{2}+G_{26}U_{1}U_{2}+...$.
Similarly, we study three simple situations.

####  3.4.1 $G_{2}=G_{24}U_{1}^{2}$

First, we assume $G_{2}$ is a function about $U_{1}^{2}$, i.e.,
$G_{2}=G_{24}U_{1}^{2}$. Then the pinned pulse solution can be uniquely
determined by the existence of non-degenerate solutions, i.e., the assumption

$\displaystyle(A10)\ \sqrt{\frac{D}{\mu}}\left(2\mu+\frac{\alpha
G_{13}b}{(\sqrt{D}+1)\sqrt{\mu}}\right)^{2}>2\alpha^{2}\beta
G_{11}G_{13}G_{24}$

holds. Moreover, we propose hypothesises:

$\displaystyle(A11)\ 27D\sqrt{D}b^{2}>2\alpha G_{13}(\beta
G_{24}C_{1})^{3}(1+\sqrt{D})^{2};$ $\displaystyle(A12)\ \alpha bG_{13}>0;$
$\displaystyle(A13)\ 72D\mu=-\sqrt[3]{\left(-3\sqrt{D}\left(\frac{18\alpha
DbG_{13}}{1+\sqrt{D}}+\sqrt{\Delta}\right)\right)^{2}}-\sqrt[3]{\left(-3\sqrt{D}\left(\frac{18\alpha
DbG_{13}}{1+\sqrt{D}}-\sqrt{\Delta}\right)\right)^{2}}$ $\displaystyle\
+4\sqrt[3]{-3\sqrt{D}\left(\frac{18\alpha
DbG_{13}}{1+\sqrt{D}}+\sqrt{\Delta}\right)}\sqrt[3]{-3\sqrt{D}\left(\frac{18\alpha
DbG_{13}}{1+\sqrt{D}}-\sqrt{\Delta}\right)}\ admits\ positive\ roots\ \mu,\ \
\ \ $ $\displaystyle\ where\ \Delta=\left(\frac{18\alpha
DbG_{13}}{1+\sqrt{D}}\right)^{2}-24\sqrt{D}\left(\alpha\beta
G_{13}G_{24}C_{1}\right)^{3}.$

Therefore, there is

######  Theorem 3.11.

Assume that (A10), (A11), (A12) and (A13) hold, then the system considered
above exists parameter $\hat{\mu}$ such that Hopf bifurcation takes place.

Here,

$\displaystyle
R(\tilde{U},\tilde{\mu})=\begin{pmatrix}-\tilde{\mu}\tilde{U}_{1}\\\
-\tilde{\mu}\tilde{U}_{2}+\frac{\beta
G_{24}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{U}_{1}^{2}\end{pmatrix},\
L=\begin{pmatrix}\frac{d^{2}}{dx^{2}}-\hat{\mu}&\frac{\alpha
G_{13}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\\\ -b+\frac{2\beta
G_{24}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\Gamma_{1,p}&D\frac{d^{2}}{dx^{2}}-\hat{\mu}\end{pmatrix}.$

Hence, we derive

$\displaystyle a$ $\displaystyle=-1,$ $\displaystyle b$
$\displaystyle=\frac{\int_{\mathbb{R}}^{\ }\left(4|P_{1}|^{2}(\frac{\beta
G_{24}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{1}}(-L)^{-1}_{22}I(\frac{x}{\varepsilon^{2}})+2(\frac{\beta
G_{24}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{1}}(i4n_{r}n_{i}-L)^{-1}_{22}I(\frac{x}{\varepsilon^{2}})P_{1}^{2}\right)\overline{P_{2}^{*}}dx}{\langle
P,P^{*}\rangle}.$

Now, the operator $(-L)^{-1}_{22}$ and $(i4n_{r}n_{i}-L)^{-1}_{22}$ are
bounded, and we can obtain $b=\frac{I_{2}}{2L_{1}}$ by the similar analysis as
before.

######  Theorem 3.12.

Assume that the real part of above $b$ is not equal to 0, then for the system
above, we get the same conclusions as in Theorem 3.5.

####  3.4.2 $G_{2}=G_{25}U_{2}^{2}$

Then, we assume $G_{2}$ is a function about $U_{2}^{2}$, i.e.,
$G_{2}=G_{25}U_{2}^{2}$. For this system, its pinned pulse solution must
satisfy

$\displaystyle(A14)\
\left(\frac{4\sqrt{\mu}G_{11}}{\alpha}+\frac{4\sqrt{D}G_{13}\mu}{\alpha\beta
G_{25}}+\frac{2b(G_{13})^{2}(D-\sqrt{D})}{\beta\sqrt{\mu}(-1+D)G_{25}}\right)^{2}>\frac{16\mu}{\alpha^{2}}\left((G_{11})^{2}+\frac{2\sqrt{D\mu}G_{11}G_{13}}{\beta
G_{25}}\right).$

Also, we propose hypothesises:

$\displaystyle(A15)\
27D\left(\frac{b(D-\sqrt{D})}{-1+D}\right)^{2}>\frac{8b(D-\sqrt{D})}{\alpha
G_{13}(-1+D)}\left(\beta
G_{25}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\right)^{3};$
$\displaystyle(A16)\ 2\beta
G_{25}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\leq$ $\displaystyle\ \alpha
G_{13}\left(\sqrt[3]{-\left(\frac{2\beta G_{25}}{\alpha
G_{13}}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})\right)^{3}+\frac{3\sqrt{D}}{\alpha
G_{13}}\left(\frac{18bD}{\alpha
G_{13}(\sqrt{D}+1)}+\sqrt{\Delta}\right)}\right.\ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle\ \left.+\sqrt[3]{-\left(\frac{2\beta G_{25}}{\alpha
G_{13}}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})\right)^{3}+\frac{3\sqrt{D}}{\alpha
G_{13}}\left(\frac{18bD}{\alpha
G_{13}(\sqrt{D}+1)}-\sqrt{\Delta}\right)}\right),$ $\displaystyle\ where\
\Delta=\left(\frac{18bD}{\alpha
G_{13}(\sqrt{D}+1)}\right)^{2}-\frac{96b\sqrt{D}}{\sqrt{D}+1}\left(\frac{\beta
G_{25}}{\alpha G_{13}}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})\right)^{3};$
$\displaystyle(A17)\
\frac{\alpha^{2}(G_{13})^{2}}{36D}\mu=-\frac{1}{2}\sqrt[3]{\left(-\left(\frac{2\beta
G_{25}}{\alpha
G_{13}}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})\right)^{3}+\frac{3\sqrt{D}}{\alpha
G_{13}}\left(\frac{18bD}{\alpha
G_{13}(\sqrt{D}+1)}+\sqrt{\Delta}\right)\right)^{2}}$ $\displaystyle\
-\frac{1}{2}\sqrt[3]{\left(-\left(\frac{2\beta G_{25}}{\alpha
G_{13}}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})\right)^{3}+\frac{3\sqrt{D}}{\alpha
G_{13}}\left(\frac{18bD}{\alpha
G_{13}(\sqrt{D}+1)}-\sqrt{\Delta}\right)\right)^{2}}$ $\displaystyle\
+\sqrt{-\left(\frac{2\beta G_{25}}{\alpha
G_{13}}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})\right)^{3}+\frac{3\sqrt{D}}{\alpha
G_{13}}\left(\frac{18bD}{\alpha G_{13}(\sqrt{D}+1)}+\sqrt{\Delta}\right)}$
$\displaystyle\ \cdot\sqrt{-\left(\frac{2\beta G_{25}}{\alpha
G_{13}}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})\right)^{3}+\frac{3\sqrt{D}}{\alpha
G_{13}}\left(\frac{18bD}{\alpha G_{13}(\sqrt{D}+1)}-\sqrt{\Delta}\right)}$
$\displaystyle\ \cdot\left(2+2\left(\frac{\beta G_{25}}{\alpha
G_{13}}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})\right)\right)+4\left(\frac{\beta
G_{25}}{\alpha G_{13}}(C_{2}+\frac{bC_{1}}{(-1+D)\mu})\right)^{2}\ admits\
positive\ roots\ \mu.$

Hence, we derive

######  Theorem 3.13.

Assume that (A14), (A15), (A16) and (A17) hold, then the Hopf bifurcation
takes place with the bifurcation parameter $\mu=\hat{\mu}$.

In this case, there are

$\begin{aligned} \ \
R(\tilde{U},\tilde{\mu})=\begin{pmatrix}-\tilde{\mu}\tilde{U}_{1}\\\
-\tilde{\mu}\tilde{U}_{2}+\frac{\beta
G_{25}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{U}_{2}^{2}\end{pmatrix},\
L=\begin{pmatrix}\frac{d^{2}}{dx^{2}}-\hat{\mu}&\frac{\alpha
G_{13}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\\\
-b&D\frac{d^{2}}{dx^{2}}-\hat{\mu}+\frac{2\beta
G_{25}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\Gamma_{2,p}\end{pmatrix}\end{aligned},$

which give

$\displaystyle a$ $\displaystyle=-1,$ $\displaystyle b$
$\displaystyle=\frac{\int_{\mathbb{R}}^{\ }\left(4|P_{2}|^{2}(\frac{\beta
G_{24}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(-L)^{-1}_{22}I(\frac{x}{\varepsilon^{2}})+2(\frac{\beta
G_{24}}{\varepsilon^{2}})^{2}I(\frac{x}{\varepsilon^{2}})\overline{P_{2}}(i4n_{r}n_{i}-L)^{-1}_{22}I(\frac{x}{\varepsilon^{2}})P_{2}^{2}\right)\overline{P_{2}^{*}}dx}{\langle
P,P^{*}\rangle}.$

######  Theorem 3.14.

Assume that the real part of above $b$ is not equal to 0, then for this case,
we get the same conclusions as Theorem 3.5.

####  3.4.3 $G_{2}=G_{26}U_{1}U_{2}$

Finally, we consider the case when $G_{2}=G_{26}U_{1}U_{2}$, whose pulse
solution can be given if

$\left\\{\begin{aligned}
2\sqrt{\mu}C_{1}&=\alpha\left(G_{11}+G_{13}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\right),\\\
2\left(\sqrt{\frac{\mu}{D}}C_{2}+\frac{b\sqrt{\mu}}{(-1+D)\mu}C_{1}\right)&=\frac{\beta
G_{26}C_{1}}{D}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)\end{aligned}\right.$
(3.23)

admits non-degenerate solutions $C_{1}$ and $C_{2}$. Correspondingly, the
existence condition of eigenvalue function is that

$\left\\{\begin{aligned}
2\sqrt{\mu+\lambda}C_{3}&=\alpha\left(G_{13}\left(C_{4}+\frac{bC_{3}}{(-1+D)(\mu+\lambda)}\right)\right),\\\
\frac{\beta
G_{26}C_{3}}{D}\left(C_{2}+\frac{bC_{1}}{(-1+D)\mu}\right)&+\frac{\beta
G_{26}C_{1}}{D}\left(C_{4}+\frac{bC_{3}}{(-1+D)(\mu+\lambda)}\right)=2\sqrt{\frac{\mu+\lambda}{D}}C_{4}+\frac{2bC_{3}}{(-1+D)\sqrt{\mu+\lambda}}\end{aligned}\right.$
(3.24)

admits non-degenerate solutions. Assuming that the existence assumptions about
(3.23) and (3.24) hold, then there exists a parameter $\hat{\mu}$ such that
this system undergoes a Hopf bifurcation. We omit the exact expression here
due to its verbosity.

Note that in this case

$R(\tilde{U},\tilde{\mu})=\begin{pmatrix}-\tilde{\mu}\tilde{U}_{1}\\\
-\tilde{\mu}\tilde{U}_{2}+\frac{\beta
G_{26}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{U}_{1}\tilde{U}_{2}\end{pmatrix},\
L=\begin{pmatrix}\frac{d^{2}}{dx^{2}}-\hat{\mu}&\frac{\alpha
G_{13}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\\\ -b+\frac{\beta
G_{26}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\Gamma_{2,p}&D\frac{d^{2}}{dx^{2}}-\hat{\mu}+\frac{\beta
G_{26}}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\Gamma_{1,p}\\\
\end{pmatrix},$

i.e.,

$R_{01}=0,\ R_{11}=-1,\ R_{20}(U,V)=\binom{0}{\frac{\beta
G_{26}}{2\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(U_{1}V_{2}+V_{1}U_{2})},\
R_{30}=0.$

In this case, we can still calculate $a$ and $b$. However, the calculation of
$b$ requires many lengthy equalities so we will not delve into more detail
here.

######  Remark 3.15.

Theorem 3.12 and 3.14 study the situations where $G_{2}$ exhibits the simplest
nonlinearity. When $G_{2}$ involves more nonlinear terms, we can still analyze
in a similar way.

Also, $\sigma_{+}=\varnothing$ gives the same remark:

######  Remark 3.16.

The local center manifold $\mathcal{M}_{0}(\mu)=V_{0}+\tilde{\Psi}(V_{0},\mu)$
considered in this subsection is locally attracting.

######  Remark 3.17.

In the subsection 3.3 and 3.4, we use the assumption that the real part of $b$
is nonzero, which can be removed if we consider the higher order terms like
$O(A|A|^{4})$ etc.

###  3.5 A concrete example

To illustrate the above Remark 3.10, we present the computation of a specific
example, which was previously proposed in [12] to analyze the Hopf
bifurcation. The system we are considering is

$\left\\{\begin{aligned} \frac{\partial U_{1}}{\partial
t}&=\frac{\partial^{2}U_{1}}{\partial x^{2}}-\mu
U_{1}+\frac{2}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(U_{2}+1+\nu
U_{1}^{3}),\\\ \frac{\partial U_{2}}{\partial
t}&=4\frac{\partial^{2}U_{2}}{\partial x^{2}}-\frac{\sqrt{3}}{3}U_{1}-\mu
U_{2}+\frac{2}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(U_{1}+2).\end{aligned}\right.$
(3.25)

For this system, if polynomial equations

$\left\\{\begin{aligned}
\sqrt{\mu}\tilde{C}_{1}&=\tilde{C_{2}}+\frac{\tilde{C_{1}}}{3\sqrt{3}\mu}+1+\nu\tilde{C_{1}}^{3},\\\
\tilde{C_{1}}+2&=4\left(\frac{\sqrt{\mu}}{2}\tilde{C_{2}}+\frac{\tilde{C_{1}}}{3\sqrt{3\mu}}\right)\end{aligned}\right.$
(3.26)

admit non-degenerate solutions $\tilde{C_{1}}$ and $\tilde{C_{2}}$. Its pinned
one-pulse solution, to leading order, can be given as

$\tilde{U}_{1,p}(x)=\begin{cases}\tilde{C}_{1}e^{\sqrt{\mu}x},&x\in
I_{s}^{-},\\\ \tilde{C}_{1},&x\in I_{f},\\\
\tilde{C}_{1}e^{-\sqrt{\mu}x},&x\in I_{s}^{+},\end{cases}\
\tilde{U}_{2,p}(x)=\begin{cases}\tilde{C}_{2}e^{\frac{\sqrt{\mu}x}{2}}+\frac{\tilde{C_{1}}}{3\sqrt{3}\mu}e^{\sqrt{\mu}x},&x\in
I_{s}^{-},\\\ \tilde{C_{2}}+\frac{\tilde{C_{1}}}{3\sqrt{3}\mu},&x\in I_{f},\\\
\tilde{C}_{2}e^{\frac{-\sqrt{\mu}x}{2}}+\frac{\tilde{C_{1}}}{3\sqrt{3}\mu}e^{-\sqrt{\mu}x},&x\in
I_{s}^{+}.\end{cases}$

By calculation, equations (3.26) equal to

$\nu\sqrt{\mu}\tilde{C_{1}}^{3}+\frac{3\sqrt{3\mu}-2-6\sqrt{3\mu}\mu}{6\sqrt{3\mu}}\tilde{C_{1}}+1+\sqrt{\mu}=0.$
(3.27)

If we consider the small nonlinearity as perturbations to the linear $G_{1}$,
we may as well assume $\nu\ll 1$. Then (3.27) can be solved as:
$\tilde{C_{1}}=\frac{-6\sqrt{3}\mu-6\sqrt{3\mu}}{-6\sqrt{3\mu}\mu+3\sqrt{3\mu}-2}$,
$\tilde{C_{2}}=\frac{6\sqrt{3\mu}\mu+3\sqrt{3}\mu-4\sqrt{\mu}-2}{6\sqrt{3}\mu^{2}-3\sqrt{3}\mu+2\sqrt{\mu}}$,
to leading order. The eigenvalue problem about this pinned 1-pulse solution
$\tilde{\Gamma}_{p}(x)=(\tilde{\Gamma}_{1,p},\tilde{\Gamma}_{2,p})^{T}=(\tilde{U}_{1,p}(x)+O(\varepsilon),\tilde{U}_{2,p}(x)+O(\varepsilon))^{T}$
is

$\left\\{\begin{aligned}
0&=\frac{d^{2}p_{1}}{dx^{2}}-(\mu+\lambda)p_{1}+\frac{2}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(p_{2}+3\nu\tilde{\Gamma}_{1,p}^{2}p_{1}),\\\
0&=4\frac{d^{2}p_{2}}{dx^{2}}-\frac{\sqrt{3}}{3}p_{1}-(\mu+\lambda)p_{2}+\frac{2}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(p_{1}),\end{aligned}\right.$
(3.28)

and its eigenfunction

$\tilde{P_{1}}(x)=\begin{cases}\tilde{C_{3}}e^{\sqrt{\mu+\lambda}x},&x\in
I_{s}^{-},\\\ \tilde{C_{3}},&x\in I_{f},\\\
\tilde{C_{3}}e^{-\sqrt{\mu+\lambda}x},&x\in I_{s}^{+},\end{cases}\
\tilde{P_{2}}(x)=\begin{cases}\tilde{C}_{4}e^{\frac{\sqrt{\mu+\lambda}x}{2}}+\frac{\tilde{C_{3}}}{3\sqrt{3}(\mu+\lambda)}e^{\sqrt{\mu+\lambda}x},&x\in
I_{s}^{-},\\\ \tilde{C}_{4}+\frac{\tilde{C_{3}}}{3\sqrt{3}(\mu+\lambda)},&x\in
I_{f},\\\
\tilde{C}_{4}e^{\frac{-\sqrt{\mu+\lambda}x}{2}}+\frac{\tilde{C_{3}}}{3\sqrt{3}(\mu+\lambda)}e^{-\sqrt{\mu+\lambda}x},&x\in
I_{s}^{+},\end{cases}$

must satisfy

$\left\\{\begin{aligned}
\sqrt{\mu+\lambda}\tilde{C_{3}}&=\tilde{C_{4}}+\frac{\tilde{C_{3}}}{3\sqrt{3}(\mu+\lambda)}+3\nu\tilde{C_{1}}^{2}\tilde{C_{3}},\\\
\tilde{C_{3}}&=4\left(\frac{\sqrt{\mu+\lambda}\tilde{C_{4}}}{2}+\frac{\tilde{C_{3}}}{3\sqrt{3}\sqrt{\mu+\lambda}}\right).\end{aligned}\right.$
(3.29)

To leading order, the equation (3.29) is equivalent to
$(\sqrt{\mu+\lambda})^{3}-\frac{\sqrt{\mu+\lambda}}{2}+\frac{1}{3\sqrt{3}}=0$.
Its solutions are

$\displaystyle t_{1,2}$ $\displaystyle=\frac{(1\pm
i\sqrt{3})(\frac{1}{2}(-\sqrt{\frac{3}{2}}+\sqrt{3}))^{\frac{1}{3}}}{2\cdot
3^{\frac{2}{3}}}+\frac{(1\mp i\sqrt{3})}{2\cdot
2^{\frac{2}{3}}(3(-\sqrt{\frac{3}{2}}+\sqrt{3}))^{\frac{1}{3}}},$
$\displaystyle t_{3}$
$\displaystyle=-(\frac{1}{2}(-\sqrt{\frac{3}{2}}+\sqrt{3}))^{\frac{1}{3}}\cdot\frac{1}{3^{\frac{2}{3}}}-\frac{1}{2^{\frac{2}{3}}(3(-\sqrt{\frac{3}{2}}+\sqrt{3}))^{\frac{1}{3}}}.$

Therefore, when
$\hat{\mu}=\frac{1}{12}(4-\sqrt[3]{3+2\sqrt{2}}-\sqrt[3]{3-2\sqrt{2}})$, Hopf
bifurcation takes place and $\sigma_{pt}=\sigma_{0}:=\\{\pm
i\omega_{H}\\}=\\{\pm\frac{\sqrt{3}}{12}(\sqrt[3]{3+2\sqrt{2}}-\sqrt[3]{3-2\sqrt{2}})i\\}$.
We denote the eigenfunction at the Hopf bifurcation as
$(\hat{P}_{1},\hat{P}_{2})^{T}$. Then, in order to unfold this bifurcation, we
transform system (3.25) to

$\displaystyle\frac{d\begin{pmatrix}\tilde{U}_{1}\\\
\tilde{U}_{2}\end{pmatrix}}{dt}$
$\displaystyle=\begin{pmatrix}\frac{d^{2}}{dx^{2}}-\hat{\mu}+\frac{2}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(3\nu\tilde{\Gamma}_{1,p}^{2})&\frac{2}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\\\
-\frac{\sqrt{3}}{3}+\frac{2}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})&4\frac{d^{2}}{dx^{2}}-\hat{\mu}\end{pmatrix}\begin{pmatrix}\tilde{U}_{1}\\\
\tilde{U}_{2}\end{pmatrix}+\begin{pmatrix}-\tilde{\mu}\tilde{U}_{1}+\frac{2}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})(\nu\tilde{U}_{1}^{3}+3\nu\tilde{U}_{1}^{2}\tilde{\Gamma}_{1,p})\\\
-\tilde{\mu}\tilde{U}_{2}\end{pmatrix},$ (3.30)

whose $R_{01}=0,\ R_{11}=-1,\
R_{20}(U,V)=\binom{\frac{6\nu}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{\Gamma}_{1,p}U_{1}V_{1}}{0},\
R_{30}(U,V,W)=\binom{\frac{2\nu}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})U_{1}V_{1}W_{1}}{0}$.
Therefore, to leading order, we have

$-L\Psi_{001}=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\displaystyle(2i\omega_{H}-L)\Psi_{200}$
$\displaystyle=\binom{\frac{6\nu}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{U}_{1,p}\hat{P}_{1}^{2}}{0},$
$\displaystyle-L\Psi_{110}$
$\displaystyle=\binom{\frac{12\nu}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{U}_{1,p}\hat{P}_{1}\overline{\hat{P}}_{1}}{0}.$

Owing to the fact that $2i\omega_{H}$ and 0 do not belong to $\sigma(L)$, we
can derive the leading order expressions for $\Psi_{200}$ and $\Psi_{110}$ as

$\Psi_{200,1}(x)=\begin{cases}\tilde{C_{7}}e^{\sqrt{\hat{\mu}+2i\omega_{H}}x},&x\in
I_{s}^{-},\\\ \tilde{C_{7}},&x\in I_{f},\\\
\tilde{C_{7}}e^{-\sqrt{\hat{\mu}+2i\omega_{H}}x},&x\in I_{s}^{+},\end{cases}\
\Psi_{200,2}(x)=\begin{cases}\tilde{C}_{8}e^{\frac{\sqrt{\hat{\mu}}x}{2}}+\frac{\sqrt{3}\tilde{C_{7}}e^{\sqrt{\hat{\mu}+2i\omega_{H}}x}}{3(3\hat{\mu}+8i\omega_{H})},&x\in
I_{s}^{-},\\\
\tilde{C}_{8}+\frac{\sqrt{3}\tilde{C_{7}}}{3(3\hat{\mu}+8i\omega_{H})},&x\in
I_{f},\\\
\tilde{C}_{8}e^{\frac{-\sqrt{\hat{\mu}}x}{2}}+\frac{\sqrt{3}\tilde{C_{7}}e^{-\sqrt{\hat{\mu}+2i\omega_{H}}x}}{3(3\hat{\mu}+8i\omega_{H})},&x\in
I_{s}^{+},\end{cases}$
$\Psi_{110,1}(x)=\begin{cases}\tilde{C_{9}}e^{\sqrt{\hat{\mu}}x},&x\in
I_{s}^{-},\\\ \tilde{C_{9}},&x\in I_{f},\\\
\tilde{C_{9}}e^{-\sqrt{\hat{\mu}}x},&x\in I_{s}^{+},\end{cases}\ \ \
\Psi_{110,2}(x)=\begin{cases}\tilde{C}_{10}e^{\frac{\sqrt{\hat{\mu}}x}{2}}+\frac{\sqrt{3}\tilde{C_{7}}}{9\hat{\mu}}e^{\sqrt{\hat{\mu}}x},&x\in
I_{s}^{-},\\\ \tilde{C}_{10}+\frac{\sqrt{3}\tilde{C_{7}}}{9\hat{\mu}},&x\in
I_{f},\\\
\tilde{C}_{10}e^{\frac{-\sqrt{\hat{\mu}}x}{2}}+\frac{\sqrt{3}\tilde{C_{7}}}{9\hat{\mu}}e^{-\sqrt{\hat{\mu}}x},&x\in
I_{s}^{+},\end{cases}$

with $\tilde{C}_{7}$ and $\tilde{C}_{8}$ satisfy

$\left\\{\begin{aligned}
2\sqrt{\hat{\mu}+2i\omega_{H}}\tilde{C}_{7}&=6\nu\tilde{C}_{1}^{2}\tilde{C}_{7}+2\left(+\frac{\sqrt{3}\tilde{C}_{7}}{3(3\hat{\mu}+8i\omega_{H})}\right)+6\nu\tilde{C}_{1}|\tilde{C}_{3}|^{2},\\\
\frac{1}{2}\tilde{C}_{7}&=2\left(\frac{\sqrt{\hat{\mu}}\tilde{C}_{8}}{2}+\frac{\sqrt{3}\tilde{C}_{7}\sqrt{\hat{\mu}+2i\omega_{H}}}{3(3\hat{\mu}+8i\omega_{H})}\right),\end{aligned}\right.$
(3.31)

$\tilde{C}_{9}$ and $\tilde{C}_{10}$ satisfy

$\left\\{\begin{aligned}
2\sqrt{\hat{\mu}}\tilde{C}_{9}&=6\nu\tilde{C}_{1}^{2}\tilde{C}_{9}+2\left(\tilde{C}_{10}+\frac{\sqrt{3}\tilde{C}_{9}}{9\hat{\mu}}\right)+12\nu\tilde{C}_{1}|\tilde{C}_{3}|^{2},\\\
\frac{1}{2}\tilde{C}_{9}&=2\left(\frac{\sqrt{\hat{\mu}}\tilde{C}_{10}}{2}+\frac{\sqrt{3}\tilde{C}_{9}}{9\sqrt{\hat{\mu}}}\right).\end{aligned}\right.$
(3.32)

Therefore, we can derive the expression of $b$ as
$b=\frac{\hat{I}_{2,1}+\hat{I}_{2,2}+\hat{I}_{2,3}}{2\hat{L}},$ where

$\displaystyle\hat{I}_{2,1}=\int_{-\varepsilon}^{\varepsilon}\frac{12\nu}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{C}_{1}\tilde{C}_{3}\tilde{C}_{9}\overline{\left(\tilde{C}_{5}-\frac{4\sqrt{3}\tilde{C}_{6}}{9(\hat{\mu}-i\omega_{H})}\right)}dx,$
$\displaystyle\hat{I}_{2,2}=\int_{-\varepsilon}^{\varepsilon}\frac{12\nu}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}})\tilde{C}_{1}\overline{\tilde{C}_{3}}\tilde{C}_{7}\overline{\left(\tilde{C}_{5}-\frac{4\sqrt{3}\tilde{C}_{6}}{9(\hat{\mu}-i\omega_{H})}\right)}dx,$
$\displaystyle\hat{I}_{2,3}=\int_{-\varepsilon}^{\varepsilon}\frac{12\nu}{\varepsilon^{2}}I(\frac{x}{\varepsilon^{2}}){\tilde{C}_{3}}^{2}\overline{\tilde{C}_{3}}\overline{\left(\tilde{C}_{5}-\frac{4\sqrt{3}\tilde{C}_{6}}{9(\hat{\mu}-i\omega_{H})}\right)}dx,$
$\displaystyle\hat{L}=2\int_{-\infty}^{-\varepsilon}\left(e^{\sqrt{\hat{\mu}+i\omega_{H}}x}\overline{\tilde{C}_{5}e^{\sqrt{\hat{\mu}+i\omega_{H}}x}-\frac{4\sqrt{3}\tilde{C}_{6}e^{\frac{\sqrt{\hat{\mu}+i\omega_{H}}x}{2}}}{9(\hat{\mu}-i\omega_{H})}}\right.$
$\displaystyle\ \ \
\left.+\left(\left(\sqrt{\hat{\mu}+i{\omega_{H}}}-\frac{1}{3\sqrt{3}(\hat{\mu}+i{\omega_{H}})}\right)e^{\frac{\sqrt{\hat{\mu}+i{\omega_{H}}}x}{2}}+\frac{1}{3\sqrt{3}(\hat{\mu}+i{\omega_{H}})}e^{\sqrt{\hat{\mu}+i{\omega_{H}}}x}\right)\overline{\tilde{C}_{6}e^{\frac{\sqrt{\hat{\mu}+i\omega_{H}}x}{2}}}\right)dx.$

When $\mu=0.1<\hat{\mu}$, $\nu=-0.001$, $\varepsilon=0.1$, by numerical
calculation, $a=1,\ b\approx-1.49318+3.7192i$. By Theorem 3.5, system (3.25)
admits a stable periodic solution, whose numerical simulation is shown in
Figure 1(a). This result is consistent with the numerical simulation in Figure
5$(b)$ [12]. If we change $\nu$ to 0.001, then there are $a=1,\ b\approx
1.49318-3.7192i$, which leads to an unstable equilibrium like Figure 1(b). If
we set bifurcation parameter $\mu$ to be 0.2, i.e., $\mu>\hat{\mu}$, then we
have Figure 1(c) when $\nu=0.001$ and Figure 1(d) when $\nu=-0.001$.

(a) $b_{r}>0$, $\mu<\hat{\mu}$ (b) $b_{r}>0$, $\mu<\hat{\mu}$ (c) $b_{r}>0$,
$\mu>\hat{\mu}$ (d) $b_{r}<0$, $\mu>\hat{\mu}$

Figure 1: numerical simulation of stationary solutions around Hopf bifurcation

##  4 Discussion

This paper presents a research inspired by numerical simulations in a linear
reaction–diffusion system with strong spatially localized impurities [12] and
the technical approach of Hopf normal form advocated in [34]. It demonstrates
that introducing a small nonlinearity to the fast variables can stabilize the
stationary pulses that would otherwise blow up, and create new stable
oscillating pulses when the stationary pulses undergo a Hopf bifurcation,
which is exhibited in Remark 3.10 and numerically verified in subsection 3.5.
This confirms once again that the Hopf bifurcation could be the birthplace of
breathing pulses [18]. Besides, subsections 3.3 and 3.4 present a series of
simple nonlinear cases in which the breathing pulses may emerge around the
Hopf bifurcation when specific conditions are satisfied.

Furthermore, the center manifold associated with the Hopf bifurcation can be
expanded not only up to third order as we already emphasized in Remark 3.17.
Thus, our next step is to implement the fifth order expansion around the
generalized Bautin point. Correspondingly, the non-zero assumption for $b_{r}$
will be replaced, and the normal form will be changed to

$\frac{dA}{dt}=i\omega_{H}A+(a_{0}\hat{\mu}+a_{1}\hat{\mu}^{2})A+(b_{0}+b_{1}\hat{\mu})A|A|^{2}+cA|A|^{4}+o(|\hat{\mu}|^{3},|A|^{5}).$

It involves more calculations and more inverse problems, while the
descriptions for the dynamically modulated pulse amplitudes are available,
whose amplitude may be quasiperiodically or even chaotically modulated [35].

Moreover, a detailed unfolding of a Bogdanov–Takens bifurcation [30, 1] or a
Dumortier–Roussarie –Sotomayor bifurcation [15] of a localized pinned pulse
solution is expected to be continued based on our paper and the analysis in
[34]. In this case, the multiple eigenvalue $\lambda=0$ must be analyzed
separately. In turn, these results may serve as a first analytical step
towards understanding the pattern formations [11]. As stated earlier,
oscillating fronts also play an important role in understanding these
phenomena analytically, for example, as shown in [10]. However, whether
explicit expressions for pinned fronts (and, more importantly, their
eigenfunctions) can be given is an important prerequisite. According to [9]
and [31], it can be solved easily and the normal form expansion parameters can
therefore be evaluated. In particular, the analysis procedure is also valid
for multi-pulses in [9], which are composed of piecewise periodic solutions
with explicit expressions. Especially, the evenness of the eigenfunctions can
bring great convenience in calculating the normal form expansions.

## Acknowledgement

J. Li, Q. Yu and Q. Zhang are supported by NSFC 12171174.

## Data available

The data that support the findings of this study are available within the
article.

## References

  * [1] M. Argentina and P. Coullet, Chaotic nucleation of metastable domains, Phys. Rev. E., 56.3 (1997), R2359.
  * [2] R. Bastiaansen, P. Carter, and A. Doelman, Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems. Nonlinearity, 32 (2019), pp. 2759–2814.
  * [3] P.C. Bressloff and S.E. Folias, Front bifurcations in an excitatory neural network, SIAM J. Appl. Math., 65.1 (2004), pp. 131–151.
  * [4] P.C. Bressloff, S.E. Folias, A. Prat and Y.X. Li, Oscillatory waves in inhomogeneous neural media, Phys. Rev. Lett., 91.17 (2003), 178101.
  * [5] P.C. Bressloff and Z.P. Kilpatrick, Two-dimensional bumps in piecewise smooth neural fields with synaptic depression, SIAM J. Appl. Math., 71.2 (2011), pp. 379–408.
  * [6] P. Carter and A. Doelman, Traveling stripes in the Klausmeier model of vegetation pattern formation. SIAM J. Appl. Math., 78.6 (2018), pp. 3213–3237.
  * [7] W. Chen and M.J. Ward, Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model. Eur. J. Appl. Math., 20.2 (2009), pp. 187–214.
  * [8] W. Chen and M. J. Ward, The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model. SIAM J. Appl. Dyn. Syst., 10.2 (2011), pp. 582–666.
  * [9] Y.X. Chen, J. Li, J.H. Shen and Q. Zhang, Pattern formations in nonlinear reaction-diffusion systems with strong localized impurities, J. Differential Equations, 402(2024), pp. 250–289.
  * [10] M. Chirilus, P. van Heijster, H. Ikeda and J.D. Rademacher, Unfolding symmetric Bogdanov–Takens bifurcations for front dynamics in a reaction–diffusion system, J. Nonlinear Sci., 29 (2019), pp. 2911–2953.
  * [11] S. Coombes, P.B. Graben, R. Potthast and J.Wright, Neural fields: theory and applications, Springer, 2014.
  * [12] A. Doelman, P. van Heijster, and J.H. Shen, Pulse dynamics in reaction–diffusion equations with strong spatially localized impurities. Philos. Trans. Roy. Soc. A, 376 (2018), pp. 1–20.
  * [13] A. Doelman, T. J. Kaper, and K. Promislow, Nonlinear asymptotic stability of the semistrong pulse dynamics in a regularized Gierer-Meinhardt model. SIAM J. Math. Anal., 38.6 (2007), pp. 1760–1787.
  * [14] A. Doelman, T.J Kaper, and P.A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity, 10.2 (1997), 523.
  * [15] F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3. Ergodic Theory Dynam. Systems, 7.3 (1987), pp. 375–413.
  * [16] M. Eslami, M. Kanafchian and G. Oppo, Oscillatory and chaotic regimes of patterns and dark cavity solitons in cavities displaying EIT: Static multihead dual chimera states, Chaos Solitons Fractals, 167(2023), 113080.
  * [17] S.J. Fan, A new extracting formula and a new distinguishing means on the one variable cubic equation. Natur. Sci. J. Hainan Teacheres College, 2 (1989), pp. 91–98.
  * [18] W.J. Firth, G.K. Harkness, A. Lord, J.M. McSloy, D. Gomila and P. Colet, Dynamical properties of two-dimensional Kerr cavity solitons. J. Opti. Soc. Amer. B-Opti. Phys., 19 (2002), pp. 747–52.
  * [19] S.E. Folias and P.C. Bressloff, Breathing pulses in an excitatory neural network, SIAM J. Appl. Dyn. Syst., 3.3 (2004), pp. 378–407.
  * [20] S.E. Folias and P.C. Bressloff, Stimulus-Locked Traveling Waves and Breathers in an Excitatory Neural Network, SIAM J. Appl. Math., 65.6 (2005), pp. 2067–2092.
  * [21] A Gierer and H Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), pp. 30–39.
  * [22] P. Gray and S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability, Chem. Eng. Sci., 38.1 (1983), pp. 29–43.
  * [23] D. Gomila, P. Colet, M.A. Matías, G.L. Oppa and M.S. Miguel, Localized structures in nonlinear optical cavities, Topical Problems of Nonlinear Wave Physics. Vol. 5975. SPIE, 2006.
  * [24] S. S. Gopalakrishnan, M. Tlidi,M. Taki and K. Panajotov, Breathing of dissipative light bullets of nonlinear polarization mode in Kerr resonators, Optics Letters, 47.15 (2022), pp. 3652–3655.
  * [25] S.V. Gurevich, S. Amiranashvili and H.G. Purwins, Breathing dissipative solitons in three-component reaction-diffusion system, Phys. Rev. E., 74.6 (2006): 066201.
  * [26] M. Haragus and G. Iooss, Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. London: Springer, 2011.
  * [27] C.K.R.T. Jones, Geometric singular perturbation theory. Dynamical Systems: Lectures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 13–22, 1994 (1995): 44–118.
  * [28] T. Kapitula and K. Promislow, Spectral and dynamical stability of nonlinear waves. New York: Springer, 2013.
  * [29] S. Koga and Y. Kuramoto, Localized patterns in reaction-diffusion systems, Prog. Theor. Phys., 63.1 (1980): 106–121.
  * [30] Y.A. Kuznetsov, I.A. Kuznetsov and Y. Kuznetsov, Elements of applied bifurcation theory, Vol. 112. New York: Springer, 1998.
  * [31] J.Li, J.H. Shen and Q.Zhang. Pinned pulses in nonlinear reaction-diffusion equations with strong localized impurities. Submitted.
  * [32] A.L. Shil’Nikov, L.P. Shil’Nikov and D.V. Turaev, Normal forms and Lorenz attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 3.5 (1993): 1123–1139.
  * [33] D. Turaev and A.G. Vladimirov and S. Zelik, Long-range interaction and synchronization of oscillating dissipative solitons, Phys. Rev. Lett., 108.26 (2012): 263906.
  * [34] F. Veerman, Breathing pulses in singularly perturbed reaction-diffusion systems, Nonlinearity, 28.7 (2015): 2211.
  * [35] F.Veerman and A. Doelman, Pulses in a Gierer-Meinhardt equation with a slow nonlinearity. SIAM J. Appl. Dyn. Syst., 12.1 (2013): 28–60.
  * [36] A.G. Vladimirov and S.V. Fedorov and N.A. Kaliteevskii and G.V. Khodova and N.N. Rosanov, Numerical investigation of laser localized structures, Journal of Optics B: Quantum and Semiclassical Optics, 1.1 (1999): 101.
  * [37] M.J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Sci., 13 (2003): 209–264.
  * [38] M.J. Ward and J. Wei, Hopf bifurcation of spike solutions for the shadow Gierer–Meinhardt model, Eur. J. Appl. Math., 14.6 (2003): 677–711.