#  Double Hopf bifurcation analysis in the memory-based diffusion
system111Partially supported by the National Natural Science Foundation of
China (Nos.11971143, 12071105), and Natural Science Foundation of Zhejiang
Province of China (No.LY19A010010).

Yongli Song1, Yahong Peng2222Corresponding author, Email:
<EMAIL_ADDRESS>, Tonghua Zhang3
1 School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
2 Department of Mathematics, Donghua University, Shanghai, 201620, China
3 Department of Mathematics, Swinburne University of Technology, Hawthorn,
VIC3122, Australia

###### Abstract

In this paper, we derive the algorithm for calculating the normal form of the
double Hopf bifurcation that appears in a memory-based diffusion system via
taking memory-based diffusion coefficient and the memory delay as the
perturbation parameters. Using the obtained theoretical results, we study the
dynamical classification near the double Hopf bifurcation point in a predator-
prey system with Holling type II functional response. We show the existence of
different kinds of stable spatially inhomogeneous periodic solutions, the
transition from one kind to the other as well as the coexistence of two types
of periodic solutions with different spatial profiles by varying the memory-
based diffusion coefficient and the memory delay.

Keywords: Memory-based diffusion; delay; stability; double Hopf bifurcation;
normal form
MSC2010: 35B10; 35B32; 35K57; 37G05

##  1 Introduction

The complex internal mechanism of memory-driven movement is still poorly
understood although it has been known by many biologists that the spatial
memory has an important influence on the animal movement, and results in
complex mathematical and computational challenges [11, 10]. In this regard,
mathematical models may provide deep insights into the theoretical mechanism
behind the biological phenomenon. For example, considering the spatial
memories decay over time and the fact that animal movement is affected by the
population at the past time [11], Shi et al. [36] introduce a time delay (also
known as “memory delay”) into the advection term of the classic reaction-
diffusion-advection equation and propose a memory-based diffusion equation to
model the dynamics of animal movement with memory. From the theoretical
analysis in [36] authors found that the stability of a spatially homogeneous
steady state depends on the reaction term and the relationship of the
coefficients of random diffusion and the directional diffusion, but not the
memory delay. Since then many researchers have shown their interests in
modeling and investigating the dynamics for the single-species model with the
memory [35, 41, 1, 47, 37, 42, 30].

More recently, Song et al. [40] consider a two species model, where authors
assume that the prey, such as plants or “drunk” animals are considered as
resource so that they have no or negligible memory or cognition. Thus they
only introduce a spatial memory into the predator, and propose the following
predator-prey system

$\begin{cases}\frac{\partial u(x,t)}{\partial
t}=d_{11}u_{xx}(x,t)+f(u(x,t),v(x,t)),~{}0<x<\ell\pi,t>0,\vspace{0.1cm}\\\
\frac{\partial v(x,t)}{\partial
t}=d_{22}v_{xx}(x,t)-d_{21}(v(x,t)u_{x}(x,t-\tau))_{x}+g(u(x,t),v(x,t)),~{}0<x<\ell\pi,t>0,\vspace{0.1cm}\\\
u_{x}(0,t)=u_{x}(\ell\pi,t)=v_{x}(0,t)=v_{x}(\ell\pi,t)=0,~{}t\geq
0,\end{cases}$ (1.1)

where $u(x,t)$ and $v(x,t)$ are the density of the prey and predator,
respectively, at the space $x$ and time $t$, $d_{11}$ and $d_{22}$ are the
random diffusion coefficients of the prey and predator respectively, $d_{21}$
is the memory-based diffusion coefficient of the predator, $\tau$ is the time
delay representing the averaged memory period of the predator, $f$ and $g$ are
biological birth/death of prey and predator respectively. The effects of the
memory-based diffusion coefficient $d_{21}$ and time delay $\tau$ on the
stability of the positive constant equilibrium of system (1.1) has been
investigated in [40] and it has been shown that unlike the classic prey-taxis
model, memory-based prey-taxis destabilizes the positive constant equilibrium,
which is a new mechanism for spatiotemporal pattern formation. The spatially
inhomogeneous Hopf bifurcation, double Hopf bifurcation and stability switches
are also found in [40]. The algorithm for computing the normal form to
investigate the spatially inhomogeneous Hopf bifurcation are developed in
[39].

In this paper, we are interested in the complex dynamics due to the
interaction of two spatially inhomogeneous Hopf bifurcations. For this
purpose, we shall first develop the algorithm for computing the normal form of
double Hopf bifurcation for system (1.1). Recall that the standard Hopf
bifurcation happens when the equilibrium loses its stability with a pair of
purely imaginary eigenvalues at the bifurcation value. For this Hopf
bifurcation, there exists a family of the periodic solutions with small
amplitudes near the neighbourhood of the equilibrium when the bifurcation
parameter is taken in the unilateral neighbourhood of the bifurcation value.
However, the interaction of Hopf bifurcations may result in more complex
dynamics like quasi-periodic solution or invariant torus [21]. The interaction
of two Hopf bifurcations, which have a pair of purely imaginary eigenvalues
$\pm i\omega_{1}$ and $\pm i\omega_{2}$, $\omega_{j}>0,j=1,2$, respectively,
is interpreted in the framework of the double Hopf bifurcation also known as
Hopf-Hopf bifurcation. When the ratio of $\omega_{1}/\omega_{2}$ is
irrational, the bifurcation is said to be non-resonant, otherwise resonant.
The resonant double Hopf bifurcation is distinguished into two cases: weakly
and strongly resonant double Hopf bifurcations. The bifurcation is said to be
strongly resonant if there exit two positive integers $m_{1},m_{2}$ so that

$m_{1}\omega_{1}=m_{2}\omega_{2},~{}m_{1}+m_{2}\leq 4,$ (1.2)

and to be weakly resonant if there are no $m_{1},m_{2}$ satisfying
$m_{1}+m_{2}\leq 4$ such that the condition (1.2) holds. The weakly resonant
double Hopf bifurcation is often codimension-two, but the strongly resonant
case is often codimension-three and it is more difficult to analyze the
related dynamics [26, 25].

The two most popular approaches used to investigate the bifurcations are the
rigorous centre manifold reduction and normal form theory [15, 17, 5, 21, 13,
12, 48] and the method of multiple scales [28, 26, 25, 29]. For other methods
to investigate the bifurcations such as the method of small parameters or the
theory of averaging, please refer to [16, 34] and references therein. The
complex dynamics arising form double Hopf bifurcation has been recently
studied by many authors for various dynamical systems, refering to [20, 44,
22, 33, 49] for ordinary differential equations, to [3, 7, 6, 46, 18, 27, 14,
31, 2, 45] for delay differential equations. More recently, based on the
theory of normal forms for partial functional differential equations developed
by Faria [12], the double Hopf bifurcation in the reaction-diffusion system
with delay has attracted the attention of the researchers [8, 9, 4, 24]. The
solutions of these systems involve not only the time but also the space, thus
the investigation of the bifurcation phenomenon is more difficult [50, 32].
The idea in [12] has also been successfully used to calculate the normal form
of the Turing-Hopf bifurcation in the reaction-diffusion system with/without
delay [43, 38, 19]. Unfortunately, the procedure of calculating the normal
forms of the double Hopf bifurcation for the classical reaction-diffusion
system with delay can not apply to the system (1.1), where the delay is
involved the diffusion term not in the reaction term and the diffusion terms
are not linear.

Motivated by the recent results, particularly the aforementioned, on the
double Hopf bifurcation for the reaction-diffusion systems with delays, in
this paper, we investigate the dynamics associated with the double Hopf
bifurcation arising from (1.1) and our results can be summarized as follows:

  1. 1.

We derive the algorithm for computing the normal form of the double Hopf
bifurcation induced by the memory-based diffusion coefficient and memory delay
for the memory-based diffusion system (1.1). The explicit relationship between
the second and third terms of the normal form and those in (1.1) near the
positive equilibrium are completely established;

  2. 2.

For system (1.1) with Holling-type II functional response, the dynamical
classification near the double Hopf bifurcation point are determined for two
cases: (i) the interaction of two Hopf bifurcations with the same spatial
mode-$2$; and (ii) the interaction of two Hopf bifurcations with the spatial
mode-$1$ and mode-$2$;

  3. 3.

We find different kinds of stable spatially inhomogeneous periodic solutions
and the transition from one to another near the neighbourhood of the double
Hopf bifurcation point. Especially, for case 2(i), we find the stable quasi-
periodic solution like a “bird” with the spatial mode-1, and for case 2(ii) we
find the bistability region of two kinds of stable periodic solutions with
spatial mode-$1$ and mode-$2$.

The the rest of the paper is organized as follows. In Section 2, we derive the
algorithm for computing the normal form associated with the double Hopf
bifurcation for (1.1). In Section 3, we study the dynamical classification
near the double Hopf bifurcation point for (1.1) with Holling type-II
functional response by employing the theoretical results developed in Section
2. We conclude our study with a short discussion in Section 4. Finally, we
give all detailed calculations used in Section 2 in the Appendices. Throughout
the paper, $\mathbb{N}$ represents the set of all positive integers, and
$\mathbb{N}_{0}=\mathbb{N}\cup{0}$ represents the set of all nonnegative
integers.

##  2 Normal forms for the double Hopf bifurcation and Hopf bifurcation

###  2.1 Basic assumptions

We assume that system (1.1) has a positive constant equilibrium
$E_{*}(u_{*},v_{*})$ and $f,g\in\mathbb{C}^{3}$ near the neighbourhood of
$E^{*}$ for the calculation of the normal form. Then the characteristic
equation of the linearized system of (1.1) at the positive equilibrium
$E_{*}(u_{*},v_{*})$ is

$\prod\limits_{n\in\mathbb{N}_{0}}\Gamma_{n}(\lambda)=0,$ (2.1)

where $\Gamma_{n}(\lambda)=\mathrm{det}\left(\mathcal{M}_{n}(\lambda)\right)$
with characteristic matrix

$\mathcal{M}_{n}(\lambda)=\lambda
I_{2}+(n/\ell)^{2}D_{1}+(n/\ell)^{2}e^{-\lambda\tau}D_{2}-A,$ (2.2)

where $I_{2}$ is a $2\times 2$ identity matrix,

$D_{1}=\left(\begin{array}[]{cc}d_{11}&0\vspace{0.1cm}\\\
0&d_{22}\end{array}\right),~{}D_{2}=\left(\begin{array}[]{cc}0&0\vspace{0.1cm}\\\
-d_{21}v_{*}&0\end{array}\right),~{}A=(a_{ij})_{2\times 2},$ (2.3)

with

$a_{11}=\frac{\partial f(u_{*},v_{*})}{\partial u},a_{12}=\frac{\partial
f(u_{*},v_{*})}{\partial v},a_{21}=\frac{\partial g(u_{*},v_{*})}{\partial
u},a_{22}=\frac{\partial g(u_{*},v_{*})}{\partial v}$

From (2.2), we have

$\Gamma_{n}(\lambda)=\lambda^{2}-T_{n}\lambda+\widetilde{J}_{n}(\tau)=0,$
(2.4)

where

$T_{n}=Tr(A)-Tr(D_{1})(n/\ell)^{2},$ (2.5)
$\widetilde{J}_{n}(\tau)=d_{11}d_{22}(n/\ell)^{4}-\left(d_{11}a_{22}+d_{22}a_{11}+d_{21}v_{*}a_{12}e^{-\lambda\tau}\right)(n/\ell)^{2}+Det(A).$

Assume that at $(\tau,d_{21})=(\tau_{c},d_{21}^{c})$, Eq.(2.1) has two pairs
of purely imaginary roots $\pm i\omega_{1}$ and $\pm i\omega_{2}$,
respectively, for $n=n_{1}$ and $n=n_{2}$ with $n_{1}\leq n_{2}$ and the
corresponding transversality condition holds, and all other eigenvalues have
negative real parts. We are interested in the cases of weak resonance and non-
resonance, which are codimension-two bifurcation problem, i.e. the strong
resonance condition (1.2) doses not hold for these two $\omega_{1}$ and
$\omega_{2}$.

###  2.2 Normal form for the double Hopf bifurcation

In what follows, we set $\tau=\tau_{c}+\mu_{1},d_{21}=d_{21}^{c}+\mu_{2}$ such
that $\left(\mu_{1},\mu_{2}\right)=(0,0)$ is the double Hopf bifurcation value
for Eq.(1.1). In this section, $D_{1}$ is defined by (2.3) and

$D_{2}^{c}=\left(\begin{array}[]{cc}0&0\vspace{0.1cm}\\\
-d_{21}^{c}v_{*}&0\end{array}\right).$

Define the real-valued Sobolev space

$\mathscr{X}=\left\\{U=\left(U_{1},U_{2}\right)^{T}\in\left(W^{2,2}(0,\ell\pi)\right)^{2},\frac{\partial
U_{1}}{\partial x}=0,\frac{\partial U_{2}}{\partial
x}=0,~{}x=0,\ell\pi\right\\},$

and let $\mathscr{C}:=C\left([-1,0];\mathscr{X}\right)$ be the Banach space of
continuous mappings from $[-1,0]$ to $\mathscr{X}$ .

Translating $E_{*}$ to the origin by setting

$U(x,t)=\left(U_{1}(x,t),U_{2}(x,t)\right)^{T}=(u(x,t),v(x,t))^{T}-(u_{*},v_{*})^{T},$

normalizing the delay by the time-scaling $t\rightarrow t/\tau$, and then for
simplification of notation, writing $U(t)$ for $U(x,t)$ and
$U_{t}\in\mathscr{C}$ for $U_{t}(\theta)=U(x,t+\theta),-1\leq\theta\leq 0$,
(1.1) becomes

$\frac{dU(t)}{dt}=d(\mu)\Delta(U_{t})+L(\mu)(U_{t})+F(U_{t},\mu),$ (2.6)

where for $\varphi=(\varphi^{(1)},\varphi^{(2)})^{T}\in\mathscr{C}$,
$d(\mu)\Delta,L(\mu):\mathscr{C}\rightarrow\mathscr{X}$,
$F:\mathscr{C}\times\mathbb{R}^{2}\rightarrow\mathscr{X}$ are given,
respectively, by

$d(\mu)\Delta(\varphi)=d_{0}\Delta(\varphi)+F^{d}(\varphi,\mu),~{}L(\mu)(\varphi)=(\tau_{c}+\mu_{1})A\varphi(0),$

and

$F(\varphi,\mu)=(\tau_{c}+\mu_{1})\left(\begin{array}[]{c}f\left(\varphi^{(1)}(0)+u_{*},\varphi^{(2)}0)+v_{*}\right)\vspace{0.2cm}\\\
g\left(\varphi^{(1)}(0)+u_{*},\varphi^{(2)}(0)+v_{*}\right)\end{array}\right)-L(\mu)(\varphi),$
(2.7)

where

$d_{0}\Delta(\varphi)=\tau_{c}D_{1}\varphi_{xx}(0)+\tau_{c}D_{2}^{c}\varphi_{xx}(-1),$
$\begin{array}[]{lll}&&F^{d}(\varphi,\mu)\vspace{0.2cm}\\\
&=&\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
-d_{21}^{c}\left(\varphi^{(1)}_{x}(-1)\varphi^{(2)}_{x}(0)+\varphi^{(1)}_{xx}(-1)\varphi^{(2)}(0)\right)\end{array}\right)\vspace{0.3cm}\\\
&&+\mu_{1}\left(\begin{array}[]{c}d_{11}\varphi^{(1)}_{xx}(0)\vspace{0.2cm}\\\
-d_{21}^{c}v_{*}\varphi^{(1)}_{xx}(-1)+d_{22}\varphi^{(2)}_{xx}(0)\end{array}\right)-\tau_{c}\mu_{2}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
v_{*}\varphi^{(1)}_{xx}(-1)\end{array}\right)\vspace{0.3cm}\\\
&&-\left(d_{21}^{c}\mu_{1}+\tau_{c}\mu_{2}\right)\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(1)}_{x}(-1)\varphi^{(2)}_{x}(0)+\varphi^{(1)}_{xx}(-1)\varphi^{(2)}(0)\end{array}\right)\vspace{0.3cm}\\\
&&-\mu_{1}\mu_{2}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
v_{*}\varphi^{(1)}_{xx}(-1)\end{array}\right)-\mu_{1}\mu_{2}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(1)}_{x}(-1)\varphi^{(2)}_{x}(0)+\varphi^{(1)}_{xx}(-1)\varphi^{(2)}(0)\end{array}\right).\end{array}$
(2.8)

Noticing that $\mu_{1},\mu_{2}$ are perturbation parameters and treated as
variables in the calculation of normal forms, we denote
$L_{0}(\varphi)=\tau_{c}A\varphi(0)$ and rewrite (2.6) as the following linear
form from nonlinear terms

$\frac{dU(t)}{dt}=d_{0}\Delta(U_{t})+L_{0}(U_{t})+\widetilde{F}(U_{t},\mu_{1},\mu_{2}),$
(2.9)

where for $\varphi=(\varphi_{1},\varphi_{2})^{T}\in\mathscr{C}$,

$\widetilde{F}(\varphi,\mu_{1},\mu_{2})=\mu_{1}A\varphi(0)+F(\varphi,\mu)+F^{d}(\varphi,\mu).$
(2.10)

The characteristic equation for the linearized system

$\frac{dU(t)}{dt}=d_{0}\Delta(U_{t})+L_{0}(U_{t})$ (2.11)

is

$\prod\limits_{n\in\mathbb{N}_{0}}\widetilde{\Gamma}_{n}(\lambda)=0,$ (2.12)

where
$\widetilde{\Gamma}_{n}(\lambda)=\mathrm{det}\left(\widetilde{\mathcal{M}}_{n}(\lambda)\right)$
with

$\widetilde{\mathcal{M}}_{n}(\lambda)=\lambda
I_{2}+\tau_{c}(n/\ell)^{2}D_{1}+\tau_{c}(n/\ell)^{2}e^{-\lambda}D_{2}^{c}-\tau_{c}A.$
(2.13)

Comparing (2.13) with (2.2), we know that Eq.(2.12) has two pairs of purely
imaginary roots $\pm i\omega_{1c}$ and $\pm i\omega_{2c}$ for $n=n_{1}$ and
$n=n_{2}$, respectively, and all other eigenvalues have negative real parts,
where $\omega_{jc}=\tau_{c}\omega_{j},j=1,2$.

It is well known that the eigenvalue problem

$-\gamma^{\prime\prime}=\mu\gamma,~{}~{}x\in(0,\ell\pi);~{}~{}\gamma^{\prime}(0)=\gamma^{\prime}(\ell\pi)=0$

has eigenvalues $\mu_{n}=(n/\ell)^{2},n\in\mathbb{N}_{0}$, with corresponding
normalized eigenfunctions

$\gamma_{n}(x)=\frac{\cos\left(\frac{nx}{\ell}\right)}{\|\cos\left(\frac{nx}{\ell}\right)\|_{2,2}}=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{\ell\pi}},&~{}\mathrm{for}~{}n=0,\vspace{0.2cm}\\\
\frac{\sqrt{2}}{\sqrt{\ell\pi}}\cos\left(\frac{nx}{\ell}\right),&~{}\mathrm{for}~{}n\neq
0,\end{array}\right.$

where the norm $\|\cdot\|_{2,2}$ is induced by the inner product
$\left[\cdot,\cdot\right]$ as follows

$\left[u,v\right]=\int_{0}^{\ell\pi}u^{T}vdx,~{}\mbox{for}~{}u,v\in\mathscr{X}.$

Let $\beta_{n}^{(j)}=\gamma_{n}(x)e_{j},j=1,2$, where $e_{j}$ is the unit
coordinate vector of $\mathbb{R}^{2}$, and
$\mathscr{B}_{n}=\mbox{span}\left\\{\left[v(\cdot),\beta_{n}^{(j)}\right]\beta_{n}^{(j)}|~{}v\in\mathscr{C},j=1,2\right\\}$.
Then it is easy to verify that

$L_{0}(\mathscr{B}_{n})\subset\mbox{span}\left\\{\beta_{n}^{(1)},\beta_{n}^{(2)}\right\\},n\in\mathbb{N}_{0}.$

Assume that $z_{t}(\theta)\in C=C\left([-1,0],\mathbb{R}^{2}\right)$ and

$z_{t}^{T}(\theta)\left(\begin{array}[]{c}\beta_{n}^{(1)}\vspace{0.1cm}\\\
\beta_{n}^{(2)}\end{array}\right)\in\mathscr{B}_{n}.$

Then, on $\mathscr{B}_{n}$, the linearized equation (2.11) is equivalent to
the following functional differential equation (FDE) in $C$:

$\dot{z}(t)=L_{0}^{d}\left(z_{t}(\theta)\right)+L_{0}(z_{t}(\theta)),$ (2.14)

where

$L_{0}^{d}\left(z_{t}(\theta)\right)=\tau_{c}\left(\begin{array}[]{cc}-d_{11}(n/\ell)^{2}&0\\\
0&-d_{22}(n/\ell)^{2}\end{array}\right)z_{t}(0)+\tau_{c}\left(\begin{array}[]{cc}0&0\\\
d_{21}^{c}v_{*}(n/\ell)^{2}&0\end{array}\right)z_{t}(-1).$

The characteristic equation of linear system (2.14) is the same as given in
(2.12).

Define $\eta_{n}(\theta)\in BV([-1,0],\mathbb{R}^{2})$ such that

$\int^{0}_{-1}d\eta_{n}(\theta)\varphi(\theta)=L_{0}^{d}(\varphi(\theta))+L_{0}(\varphi(\theta)),~{}~{}\varphi\in
C,$

and use the adjoint bilinear form on $C^{*}\times C$,
$C^{*}=C([0,1],\mathbb{R}^{2*})$, where $\mathbb{R}^{2*}$ is the 2-dimensional
space of row vectors, as follows

$\langle\psi(s),\varphi(\theta)\rangle_{n}=\psi(0)\varphi(0)-\int^{0}_{-1}\int^{\theta}_{0}\psi(\xi-\theta)d\eta_{n}(\theta)\varphi(\xi)d\xi,\
\ \ \textmd{for}\ \psi\in C^{*},\varphi\in C.$

Let $\Lambda=\\{i\omega_{1c},-i\omega_{1c},i\omega_{2c},-i\omega_{2c}\\}$.
Denote the generalized eigenspace of (2.14) associated with $\Lambda$ by
$P_{n_{j}}$ and the corresponding adjoint space by $P_{n_{j}}^{*}$. Then, by
the adjoint theory of functional differential equation [15], $C$ can be
decomposed as $C=P_{n_{j}}\oplus Q_{n_{j}}$, $j=1,2,$ where
$Q_{n_{j}}=\\{\varphi\in C:\langle\psi,\varphi\rangle=0,\forall\psi\in
P_{n_{j}}^{*}\\}$. Choose the bases $\Phi_{n_{j}}(\theta)$ and
$\Psi_{n_{j}}(s)$ of $P_{n_{j}}$ and $P_{n_{j}}^{*}$, respectively, as follows

$\Phi_{n_{j}}(\theta)=\left(\phi_{n_{j}}(\theta),\overline{\phi}_{n_{j}}(\theta)\right),~{}\Psi_{n_{j}}(s)=\mbox{col}\left(\psi_{n_{j}}^{T}(s),\overline{\psi}_{n_{j}}^{T}(s)\right),$

such that $\langle\Psi_{n_{j}},\Phi_{n_{j}}\rangle_{n_{j}}=I_{2},$ where

$\phi_{n_{j}}(\theta)=\left(\begin{array}[]{c}\phi^{(1)}_{n_{j}}(\theta)\vspace{0.2cm}\\\
\phi^{(2)}_{n_{j}}(\theta)\end{array}\right)=\phi_{n_{j}}(0)e^{i\omega_{jc}\theta},~{}~{}\psi_{n_{j}}(s)=\left(\begin{array}[]{c}\psi^{(1)}_{n_{j}}(s)\vspace{0.2cm}\\\
\psi^{(2)}_{n_{j}}(s)\end{array}\right)=\psi_{n_{j}}(0)e^{-i\omega_{jc}s},$

and

$\phi_{n_{j}}(0)=\left(\begin{array}[]{c}1\vspace{0.2cm}\\\
\frac{i\omega_{j}+(n_{j}/\ell)^{2}d_{11}-a_{11}}{a_{12}}\end{array}\right),~{}~{}\psi_{n_{j}}(0)=\alpha_{j}\left(\begin{array}[]{c}1\vspace{0.2cm}\\\
\frac{a_{12}}{i\omega_{j}+(n_{j}/\ell)^{2}d_{22}-a_{22}}\end{array}\right),$

with

$\alpha_{j}=\frac{i\omega_{j}+(n_{j}/\ell)^{2}d_{22}-a_{22}}{i\omega_{j}+(n_{j}/\ell)^{2}d_{11}-a_{11}+i\omega_{j}+(n_{j}/\ell)^{2}d_{22}-a_{22}+\tau_{c}a_{12}d_{21}^{c}v_{*}(n_{j}/\ell)^{2}e^{-i\omega_{jc}}}.$

Using the decomposition $C=P_{n_{j}}\oplus Q_{n_{j}}$, the phase space
$\mathscr{C}$ for (2.6) can be decomposed as

$\mathscr{C}=\mathcal{P}\oplus\mathcal{Q},~{}\mathcal{P}=\mbox{Im}\pi,~{}\mathcal{Q}=\mbox{Ker}\pi,$

where $\pi:\mathscr{C}\rightarrow\mathcal{P}$ is the projection operator
defined by

$\begin{array}[]{lll}\pi(\phi)&=&\Phi_{n_{1}}(\theta)\left\langle\Psi_{n_{1}}(\theta),\left(\begin{array}[]{c}\left[\phi(\cdot),\beta_{n_{1}}^{(1)}\right]\vspace{0.2cm}\\\
\left[\phi(\cdot),\beta_{n_{1}}^{(2)}\right]\end{array}\right)\right\rangle_{n_{1}}\gamma_{n_{1}}(x)\vspace{0.2cm}\\\
&&+\Phi_{n_{2}}(\theta)\left\langle\Psi_{n_{2}}(\theta),\left(\begin{array}[]{c}\left[\phi(\cdot),\beta_{n_{2}}^{(1)}\right]\vspace{0.2cm}\\\
\left[\phi(\cdot),\beta_{n_{2}}^{(2)}\right]\end{array}\right)\right\rangle_{n_{2}}\gamma_{n_{2}}(x).\end{array}$

In the following, for simplification of notations, we use
$\phi_{n_{j}},\overline{\phi}_{n_{j}},\psi_{n_{j}}$ and
$\overline{\psi}_{n_{j}}$ for
$\phi_{n_{j}}(\theta),\overline{\phi}_{n_{j}}(\theta),\psi_{n_{j}}(\theta)$
and $\overline{\psi}_{n_{j}}(\theta)$, respectively. In addition, notice that
for $n_{1}=n_{2}$, $\phi_{n_{1}}(\theta)$ looks like $\phi_{n_{2}}(\theta)$
but they are actually different since $\omega_{1c}\neq\omega_{2c}$. Thus, in
this case, one can not replace $\phi_{n_{1}}(\theta)$ by
$\phi_{n_{2}}(\theta)$ or conversely.

Following [12] and [43], we define
$\mathscr{C}_{0}^{1}=\left\\{\phi\in\mathscr{C}:\dot{\phi}\in\mathscr{C},\phi(0)\in\mbox{dom}(d\Delta)\right\\}$
and let

$z_{x}=\left(z_{1}(t)\gamma_{n_{1}}(x),z_{2}(t)\gamma_{n_{1}}(x),z_{3}(t)\gamma_{n_{2}}(x),z_{4}(t)\gamma_{n_{2}}(x)\right)^{T}$

and

$\Phi(\theta)=\left(\phi_{n_{1}}(\theta),\overline{\phi}_{n_{1}}(\theta),\phi_{n_{2}}(\theta),\overline{\phi}_{n_{2}}(\theta)\right).$

For $\varphi(\theta)\in\mathscr{C}_{0}^{1}$, we have the following
decomposition

$\varphi(\theta)=\Phi(\theta)z_{x}+w,~{}~{}w=(w^{(1)},w^{(2)})^{T}\in\mathscr{C}_{0}^{1}\cap\mbox{Ker}\pi:=\mathscr{Q}^{1}.$

For simplicity of notations, we write

$\left(\begin{array}[]{c}\left[\widetilde{F},\beta_{\nu}^{(1)}\right]\vspace{0.2cm}\\\
\left[\widetilde{F},\beta_{\nu}^{(2)}\right]\end{array}\right)_{\nu=n_{1}}^{\nu=n_{2}}~{}~{}\mathrm{for}~{}~{}\mathrm{col}\left(\left(\begin{array}[]{c}\left[\widetilde{F},\beta_{n_{1}}^{(1)}\right]\vspace{0.2cm}\\\
\left[\widetilde{F},\beta_{n_{1}}^{(2)}\right]\end{array}\right),\left(\begin{array}[]{c}\left[\widetilde{F},\beta_{n_{2}}^{(1)}\right]\vspace{0.2cm}\\\
\left[\widetilde{F},\beta_{n_{2}}^{(2)}\right]\end{array}\right)\right)$

and let $z=(z_{1}(t)~{}z_{2}(t)~{}z_{3}(t)~{}z_{4}(t))^{T}.$ Following the
notations in [12], we define

$X_{0}(\theta)=\left\\{\begin{array}[]{ll}0,&-1\leq\theta<0,\vspace{0.2cm}\\\
1,&\theta=0.\end{array}\right.$

and then

$\begin{array}[]{lll}&&\pi\left(X_{0}(\theta)\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right)\right)\vspace{0.2cm}\\\
&=&\Phi_{n_{1}}(\theta)\Psi_{n_{1}}(0)\left(\begin{array}[]{c}\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{n_{1}}^{(1)}\right]\vspace{0.2cm}\\\
\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{n_{1}}^{(2)}\right]\end{array}\right)\gamma_{n_{1}}(x)\vspace{0.2cm}\\\
&&+\Phi_{n_{2}}(\theta)\Psi_{n_{2}}(0)\left(\begin{array}[]{c}\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{n_{2}}^{(1)}\right]\vspace{0.2cm}\\\
\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{n_{2}}^{(2)}\right]\end{array}\right)\gamma_{n_{2}}(x).\end{array}$
(2.15)

.

Then system (2.9) is decomposed as a system of abstract ODEs on
$\mathbb{R}^{4}\times\mbox{Ker}\pi$:

$\begin{cases}\dot{z}=Bz+\Psi(0)\left(\begin{array}[]{c}\left[\widetilde{F}\left(\Phi(\theta)z_{x}+w,\mu\right),\beta_{\nu}^{(1)}\right]\vspace{0.1cm}\\\
\left[\widetilde{F}\left(\Phi(\theta)z_{x}+w,\mu\right),\beta_{\nu}^{(2)}\right]\end{array}\right)^{\nu=n_{2}}_{\nu=n_{1}},\vspace{0.1cm}\\\
\dot{w}=A_{\mathcal{Q}^{1}}w+(I-\pi)X_{0}(\theta)\widetilde{F}\left(\Phi(\theta)z_{x}+w,\mu\right),\end{cases}$
(2.16)

where

$\Psi(0)=\mbox{diag}\left\\{\Psi_{n_{1}}(0),\Psi_{n_{2}}(0)\right\\}~{}~{}B=\mbox{diag}\left\\{i\omega_{1c},-i\omega_{1c},i\omega_{2c},-i\omega_{2c}\right\\},$

$A_{\mathcal{Q}^{1}}:\mathcal{Q}^{1}\rightarrow\mbox{Ker}\pi$ is defined by

$A_{\mathscr{Q}^{1}}w=\dot{w}+X_{0}(\theta)\left(L_{0}(w)+L_{0}^{d}(w)-\dot{w}(0)\right).$

Consider the formal Taylor expansion

$\widetilde{F}(\varphi,\mu)=\sum\limits_{j\geq
2}\frac{1}{j!}\widetilde{F}_{j}(\varphi,\mu),~{}~{}F(\varphi,\mu)=\sum\limits_{j\geq
2}\frac{1}{j!}F_{j}(\varphi,\mu)$

and

$F^{d}(\varphi,\mu)=\frac{1}{2}F^{d}_{2}(\varphi,\mu)+\frac{1}{3!}F^{d}_{3}(\varphi,\mu)+\frac{1}{4!}F^{d}_{4}(\varphi,\mu).$

From (2.8), we have

$F^{d}_{2}(\varphi,\mu)=F^{d(0,0)}_{2}(\varphi)+\mu_{1}F^{d(1,0)}_{2}(\varphi)+\mu_{2}F^{d(0,1)}_{2}(\varphi),$
(2.17)

$F^{d}_{3}(\varphi,\mu)=\mu_{1}F^{d(1,0)}_{3}(\varphi)+\mu_{2}F^{d(0,1)}_{3}(\varphi)+\mu_{1}\mu_{2}F^{d(1,1)}_{3}(\varphi),$
(2.18)

with

$\left\\{\begin{array}[]{l}F^{d(0,0)}_{2}(\varphi)=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(1)}_{x}(-1)\varphi^{(2)}_{x}(0)+\varphi^{(1)}_{xx}(-1)\varphi^{(2)}(0)\end{array}\right),\vspace{0.2cm}\\\
F^{d(1,0)}_{2}(\varphi)=2D_{1}\varphi_{xx}(0)+2D_{2}^{c}\varphi_{xx}(-1),\vspace{0.2cm}\\\
F^{d(0,1)}_{2}(\varphi)=\frac{2\tau_{c}}{d_{21}^{c}}D_{2}^{c}\varphi_{xx}(-1),\vspace{0.2cm}\\\
F^{d(1,0)}_{3}(\varphi)=-6d_{21}^{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(1)}_{x}(-1)\varphi^{(2)}_{x}(0)+\varphi^{(1)}_{xx}(-1)\varphi^{(2)}(0)\end{array}\right),\vspace{0.2cm}\\\
F^{d(0,1)}_{3}(\varphi)=-6\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(1)}_{x}(-1)\varphi^{(2)}_{x}(0)+\varphi^{(1)}_{xx}(-1)\varphi^{(2)}(0)\end{array}\right),\vspace{0.2cm}\\\
F^{d(1,1)}_{3}(\varphi)=-6\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
v_{*}\varphi^{(1)}_{xx}(-1)\end{array}\right),\end{array}\right.$ (2.19)

and

$F^{d}_{4}(\varphi,\mu)=-24\mu_{1}\mu_{2}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(1)}_{x}(-1)\varphi^{(2)}_{x}(0)+\varphi^{(1)}_{xx}(-1)\varphi^{(2)}(0)\end{array}\right).$

From (2.10), we have

$\widetilde{F}_{2}(\varphi,\mu)=2\mu_{1}A\varphi(0)+F_{2}(\varphi,\mu)+F_{2}^{d}(\varphi,\mu),$
(2.20)

and

$\widetilde{F}_{3}(\varphi,\mu)=F_{3}(\varphi,\mu)+F_{3}^{d}(\varphi,\mu).$
(2.21)

Then (2.16) is written as

$\left\\{\begin{array}[]{l}\dot{z}=Bz+\sum\limits_{j\geq
2}\frac{1}{j!}f^{1}_{j}(z,w,\mu),\vspace{0.2cm}\\\
\dot{w}=A_{\mathscr{Q}^{1}}w+\sum\limits_{j\geq
2}\frac{1}{j!}f^{2}_{j}(z,w,\mu),\end{array}\right.$

where

$f^{1}_{j}(z,w,\mu)=\Psi(0)\left(\begin{array}[]{c}\left[\widetilde{F}_{j}\left(\Phi(\theta)z_{x}+w,\mu\right),\beta_{\nu}^{(1)}\right]\vspace{0.1cm}\\\
\left[\widetilde{F}_{j}\left(\Phi(\theta)z_{x}+w,\mu\right),\beta_{\nu}^{(2)}\right]\end{array}\right)^{\nu=n_{2}}_{\nu=n_{1}},$
(2.22)
$f^{2}_{j}(z,w,\mu)=(I-\pi)X_{0}(\theta)\widetilde{F}_{j}\left(\Phi(\theta)z_{x}+w,\mu\right).$
(2.23)

In terms of the normal form theory of partial functional differential
equations [12], after a recursive transformation of variables of the form

$(z,w)=(\widetilde{z},\widetilde{w})+\frac{1}{j!}\left(U_{j}^{1}(\widetilde{z},\mu),U_{j}^{2}(\widetilde{z},\mu)(\theta)\right),j\geq
2,$ (2.24)

where $z,\widetilde{z}\in\mathbb{R}^{4},w,\widetilde{w}\in\mathscr{Q}^{1}$ and
$U_{j}^{1}:\mathbb{R}^{6}\rightarrow\mathbb{R}^{4},U_{j}^{2}:\mathbb{R}^{6}\rightarrow\mathscr{Q}^{1}$
are homogeneous polynomials of degree $j$ in $\widetilde{z}$ and $\mu,$ the
flow on the local center manifold for (2.9) is written as

$\dot{z}=Bz+\sum\limits_{j\geq 2}\frac{1}{j!}g_{j}^{1}(z,0,\mu),$ (2.25)

which is the normal form as in the usual sense for ODEs.

Following [23] and [18], we have

$g_{2}^{1}(z,0,\mu)=\mbox{Proj}_{\mbox{Ker}(M_{2}^{1})}f_{2}^{1}(z,0,\mu),$

and

$g_{3}^{1}(z,0,\mu)=\mbox{Proj}_{\mbox{Ker}(M_{3}^{1})}\widetilde{f}_{3}^{1}(z,0,\mu)=\mbox{Proj}_{S}\widetilde{f}_{3}^{1}(z,0,0)+O(|\mu|^{2}|z|),$
(2.26)

where $\widetilde{f}_{3}^{1}(z,0,\mu)$ is the terms of order $3$ in $(z,\mu)$
obtained after performing the change of variables (2.24) of order $2$ and is
determined by (2.36),

$\mbox{Ker}\left(M_{2}^{1}\right)=\mbox{Span}\left\\{\mu_{i}z_{1}e_{1},\mu_{i}z_{2}e_{2},\mu_{i}z_{3}e_{3},\mu_{i}z_{4}e_{4},i=1,2\right\\},$
$\mbox{Ker}\left(M_{3}^{1}\right)=\mbox{Span}\left\\{\begin{array}[]{l}\mu_{1}\mu_{2}z_{1}e_{1},\mu_{i}^{2}z_{1}e_{1},z_{1}^{2}z_{2}e_{1},z_{1}z_{3}z_{4}e_{1},\mu_{1}\mu_{2}z_{2}e_{2},\mu_{i}^{2}z_{2}e_{2},\vspace{0.3cm}\\\
z_{1}z_{2}^{2}e_{2},z_{2}z_{3}z_{4}e_{2},\mu_{1}\mu_{2}z_{3}e_{3},\mu_{i}^{2}z_{3}e_{3},z_{3}^{2}z_{4}e_{3},z_{1}z_{2}z_{3}e_{3},\vspace{0.3cm}\\\
\mu_{1}\mu_{2}z_{4}e_{4},\mu_{i}^{2}z_{4}e_{4},z_{3}z_{4}^{2}e_{4},z_{1}z_{2}z_{4}e_{4},~{}~{}i=1,2\end{array}\right\\},$

and

$S=\mbox{Span}\left\\{z_{1}^{2}z_{2}e_{1},z_{1}z_{3}z_{4}e_{1},z_{1}z_{2}^{2}e_{2},z_{2}z_{3}z_{4}e_{2},z_{3}^{2}z_{4}e_{3},z_{1}z_{2}z_{3}e_{3},z_{3}z_{4}^{2}e_{4},z_{1}z_{2}z_{4}e_{4}\right\\}.$

For convenience, in what follows we set

$\mathcal{H}\left(\alpha
z_{1}^{q_{1}}z_{2}^{q_{2}}z_{3}^{q_{3}}z_{4}^{q_{4}}\mu_{1}^{\iota_{1}}\mu_{2}^{\iota_{2}}\right)=\left(\begin{array}[]{c}\alpha
z_{1}^{q_{1}}z_{2}^{q_{2}}z_{3}^{q_{3}}z_{4}^{q_{4}}\mu_{1}^{\iota_{1}}\mu_{2}^{\iota_{2}}\vspace{0.2cm}\\\
\overline{\alpha}z_{1}^{q_{2}}z_{2}^{q_{1}}z_{3}^{q_{4}}z_{4}^{q_{3}}\mu_{1}^{\iota_{1}}\mu_{2}^{\iota_{2}}\end{array}\right),~{}\alpha\in\mathbb{C}.$

####  2.2.1 Calculation of $\bf{g_{2}^{1}(z,0,\mu)}$

It follows from (2.22) that

$f^{1}_{2}(z,0,\mu)=\Psi(0)\left(\begin{array}[]{c}\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},\mu\right),\beta_{\nu}^{(1)}\right]\vspace{0.1cm}\\\
\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},\mu\right),\beta_{\nu}^{(2)}\right]\end{array}\right)^{\nu=n_{2}}_{\nu=n_{1}}.$
(2.27)

From (2.20), we have

$\widetilde{F}_{2}(\Phi(\theta)z_{x},\mu)=2\mu_{1}A\left(\Phi(0)z_{x}\right)+F_{2}(\Phi(\theta)z_{x},\mu)+F_{2}^{d}(\Phi(\theta)z_{x},\mu).$
(2.28)

From (2.17)- (2.19), (2.27) and (2.28), and noticing that

$\left[\gamma_{n_{i}}(x),\gamma_{n_{j}}(x)\right]=\left\\{\begin{array}[]{ll}1,&i=j,\vspace{0.3cm}\\\
0,&i\not=j,\end{array}\right.$

then for $n_{1}\not=n_{2}$, we have

$\left(\begin{array}[]{c}\left[2\mu_{1}A\left(\Phi(0)z_{x}\right),\beta_{\nu}^{(1)}\right]\vspace{0.3cm}\\\
\left[2\mu_{1}A\left(\Phi(0)z_{x}\right),\beta_{\nu}^{(2)}\right]\end{array}\right)=\left\\{\begin{array}[]{ll}2\mu_{1}A\left(\Phi_{n_{1}}(0)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\right),&\nu=n_{1},\vspace{0.3cm}\\\
2\mu_{1}A\left(\Phi_{n_{2}}(0)\left(\begin{array}[]{c}z_{3}\\\
z_{4}\end{array}\right)\right),&\nu=n_{2},\end{array}\right.$ (2.29)
$\begin{array}[]{lll}&&\left(\begin{array}[]{c}\left[\mu_{1}F^{d(1,0)}_{2}\left(\Phi(\theta)z_{x}\right),\beta_{\nu}^{(1)}\right]\vspace{0.3cm}\\\
\left[\mu_{1}F^{d(1,0)}_{2}\left(\Phi(\theta)z_{x}\right),\beta_{\nu}^{(2)}\right]\end{array}\right)\vspace{0.3cm}\\\
&=&\left\\{\begin{array}[]{ll}-2(n_{1}/\ell)^{2}\mu_{1}\left(D_{1}\left(\Phi_{n_{1}}(0)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\right)+D_{2}^{c}\left(\Phi_{n_{1}}(-1)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\right)\right),&\nu=n_{1},\vspace{0.3cm}\\\
-2(n_{2}/\ell)^{2}\mu_{1}\left(D_{1}\left(\Phi_{n_{2}}(0)\left(\begin{array}[]{c}z_{3}\\\
z_{4}\end{array}\right)\right)+D_{2}^{c}\left(\Phi_{n_{2}}(-1)\left(\begin{array}[]{c}z_{3}\\\
z_{4}\end{array}\right)\right)\right),&\nu=n_{2},\end{array}\right.\end{array}$
(2.30)

and

$\begin{array}[]{lll}&&\left(\begin{array}[]{c}\left[\mu_{2}F^{d(0,1)}_{2}\left(\Phi(\theta)z_{x}\right),\beta_{\nu}^{(1)}\right]\vspace{0.3cm}\\\
\left[\mu_{2}F^{d(0,1)}_{2}\left(\Phi(\theta)z_{x}\right),\beta_{\nu}^{(2)}\right]\end{array}\right)\vspace{0.3cm}\\\
&=&\left\\{\begin{array}[]{ll}-\frac{2(n_{1}/\ell)^{2}\tau_{c}}{d_{21}^{c}}\mu_{2}D_{2}^{c}\left(\Phi_{n_{1}}(-1)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)\right),&\nu=n_{1},\vspace{0.3cm}\\\
-\frac{2(n_{2}/\ell)^{2}\tau_{c}}{d_{21}^{c}}\mu_{2}D_{2}^{c}\left(\Phi_{n_{2}}(-1)\left(\begin{array}[]{c}z_{3}\\\
z_{4}\end{array}\right)\right),&\nu=n_{2}.\end{array}\right.\end{array}$
(2.31)

By (2.7), it is easy to verify that for all $\mu\in\mathbb{R}^{2}$,

$F_{2}(\Phi(\theta)z_{x},\mu)=F_{2}(\Phi(\theta)z_{x},0).$ (2.32)

This, together with (2.29), (2.30) and (2.31), yields to

$\begin{array}[]{lll}g^{1}_{2}(z,0,\mu)&=&\mbox{Proj}_{\tiny{\mbox{Ker}(M_{2}^{1})}}f_{2}^{1}(z,0,\mu)\vspace{0.2cm}\\\
&=&\left(\begin{array}[]{c}\mathcal{H}\left(\left(B^{(1)}_{1}\mu_{1}+B^{(2)}_{1}\mu_{2}\right)z_{1}\right)\vspace{0.2cm}\\\
\mathcal{H}\left(\left(B^{(1)}_{3}\mu_{1}+B^{(2)}_{3}\mu_{2}\right)z_{3}\right)\end{array}\right),\end{array}$
(2.33)

where

$\begin{array}[]{l}B^{(1)}_{1}=2\psi_{n_{1}}^{T}(0)\left(A\phi_{n_{1}}(0)-(n_{1}/\ell)^{2}\left(D_{1}\phi_{n_{1}}(0)+D_{2}^{c}\phi_{n_{1}}(-1)\right)\right)=2i\omega_{1}\psi_{n_{1}}^{T}(0)\phi_{n_{1}}(0),\vspace{0.2cm}\\\
B^{(1)}_{3}=2\psi_{n_{2}}^{T}(0)\left(A\phi_{n_{2}}(0)-(n_{2}/\ell)^{2}\left(D_{1}\phi_{n_{2}}(0)+D_{2}^{c}\phi_{n_{2}}(-1)\right)\right)=2i\omega_{2}\psi_{n_{2}}^{T}(0)\phi_{n_{2}}(0),\vspace{0.2cm}\\\
B^{(2)}_{1}=-\frac{2(n_{1}/\ell)^{2}\tau_{c}}{d_{21}^{c}}\psi_{n_{1}}^{T}(0)\left(D_{2}^{c}\phi_{n_{1}}(-1)\right),~{}~{}~{}~{}B^{(2)}_{3}=-\frac{2(n_{2}/\ell)^{2}\tau_{c}}{d_{21}^{c}}\psi_{n_{2}}^{T}(0)\left(D_{2}^{c}\phi_{n_{2}}(-1)\right).\end{array}$

For $n_{1}=n_{2}$, noticing the fact that
$\left[\Phi(\theta)z_{x},\gamma_{n_{j}}(x)\right]=\Phi(\theta)z,j=1,2,$ and
using the similar calculations as above, it is easy to obtain the same
$g^{1}_{2}(z,0,\mu)$ as in (2.33).

####  2.2.2 Calculation of $\bf{g_{3}^{1}(z,0,\mu)}$

In this subsection, we calculate the third term $g^{1}_{3}(z,0,0)$ in terms of
(2.26). Notice that $\frac{1}{3!}\widetilde{f}_{3}^{1}$ in (2.26) is the term
of order 3 obtained after the changes of variables in previous step. Denote

$f^{(1,1)}_{2}(z,w,0)=\Psi(0)\left(\begin{array}[]{c}\left[F_{2}\left(\Phi(\theta)z_{x}+w,0\right),\beta_{\nu}^{(1)}\right]\vspace{0.2cm}\\\
\left[F_{2}\left(\Phi(\theta)z_{x}+w,0\right),\beta_{\nu}^{(2)}\right]\end{array}\right)^{\nu=n_{2}}_{\nu=n_{1}},$
(2.34)
$f^{(1,2)}_{2}(z,w,0)=\Psi(0)\left(\begin{array}[]{c}\left[F^{d}_{2}\left(\Phi(\theta)z_{x}+w,0\right),\beta_{\nu}^{(1)}\right]\vspace{0.2cm}\\\
\left[F^{d}_{2}\left(\Phi(\theta)z_{x}+w,0\right),\beta_{\nu}^{(2)}\right]\end{array}\right)^{\nu=n_{2}}_{\nu=n_{1}}.$
(2.35)

In addition, it follows from (2.33) that $g_{2}^{1}(z,0,0)=(0,0,0,0)^{T}$.
Then $\widetilde{f}_{3}^{1}(z,0,0)$ is determined by

$\begin{array}[]{lll}&&\widetilde{f}_{3}^{1}(z,0,0)\vspace{0.3cm}\\\
&=&f_{3}^{1}(z,0,0)+\frac{3}{2}\left[\left(D_{z}f_{2}^{1}(z,0,0)\right)U_{2}^{1}(z,0)+\left(D_{w}f_{2}^{(1,1)}(z,0,0)\right)U_{2}^{2}(z,0)(\theta)\right.\vspace{0.3cm}\\\
&&\left.+\left(D_{w,w_{x},w_{xx}}f^{(1,2)}_{2}(z,0,0)\right)U_{2}^{(2,d)}(z,0)(\theta)\right],\end{array}$
(2.36)

where $f_{2}^{1}(z,0,0)=f_{2}^{(1,1)}(z,0,0)+f_{2}^{(1,2)}(z,0,0)$,

$D_{w,w_{x},w_{xx}}f^{(1,2)}_{2}(z,0,0)=\left(D_{w}f^{(1,2)}_{2}(z,0,0),D_{w_{x}}f^{(1,2)}_{2}(z,0,0),D_{w_{xx}}f^{(1,2)}_{2}(z,0,0)\right),$
$U_{2}^{1}(z,0)=\left(M_{2}^{1}\right)^{-1}\mbox{Proj}_{Im\left(M_{2}^{1}\right)}f_{2}^{1}(z,0,0),~{}~{}U_{2}^{2}(z,0)(\theta)=\left(M_{2}^{2}\right)^{-1}f_{2}^{2}(z,0,0),$

and

$U_{2}^{(2,d)}(z,0)(\theta)=\mbox{col}\left(U_{2}^{2}(z,0)(\theta),U_{2x}^{2}(z,0)(\theta),U_{2xx}^{2}(z,0)(\theta)\right).$
(2.37)

Next, we compute $\mbox{Proj}_{S}\widetilde{f}_{3}^{1}(z,0,0)$ step by step
according to (2.36). The calculation is divided into the following four steps.

Step 1: The calculation of $\bf{\mbox{Proj}_{S}f_{3}^{1}(z,0,0)}$
From (2.18) and (2.21), we have
$\widetilde{F}_{3}(\Phi(\theta)z_{x},0)=F_{3}(\Phi(\theta)z_{x},0)$, which can
be written as

$\widetilde{F}_{3}(\Phi(\theta)z_{x},0)=F_{3}(\Phi(\theta)z_{x},0)=\sum_{q_{1}+q_{2}+q_{3}+q_{4}=3}A_{q_{1}q_{2}q_{3}q_{4}}\gamma_{n_{1}}^{q_{1}+q_{2}}(x)\gamma_{n_{2}}^{q_{3}+q_{4}}(x)z_{1}^{q_{1}}z_{2}^{q_{2}}z_{3}^{q_{3}}z_{4}^{q_{4}},$
(2.38)

where $q_{1},q_{2},q_{3},q_{4}\in\mathbb{N}_{0}$, and

$A_{q_{1}q_{2}q_{3}q_{4}}=\left(A^{(1)}_{q_{1}q_{2}q_{3}q_{4}},A^{(2)}_{q_{1}q_{2}q_{3}q_{4}}\right)^{T}.$
(2.39)

It follows from (2.22) and (2.38) that

$f_{3}^{1}(z,0,0)=\Psi(0)\left(\begin{array}[]{c}\sum\limits_{q_{1}+q_{2}+q_{3}+q_{4}=3}A_{q_{1}q_{2}q_{3}q_{4}}\int_{0}^{\ell\pi}\gamma_{n_{1}}^{q_{1}+q_{2}+1}(x)\gamma^{q_{3}+q_{4}}_{n_{2}}(x)dxz_{1}^{q_{1}}z_{2}^{q_{2}}z_{3}^{q_{3}}z_{4}^{q_{4}}\vspace{0.3cm}\\\
\sum\limits_{q_{1}+q_{2}+q_{3}+q_{4}=3}A_{q_{1}q_{2}q_{3}q_{4}}\int_{0}^{\ell\pi}\gamma_{n_{1}}^{q_{1}+q_{2}}(x)\gamma^{q_{3}+q_{4}+1}_{n_{2}}(x)dxz_{1}^{q_{1}}z_{2}^{q_{2}}z_{3}^{q_{3}}z_{4}^{q_{4}}\end{array}\right),$

which, together with the fact that

$\int_{0}^{\ell\pi}\gamma_{n_{1}}^{2}(x)\gamma_{n_{2}}^{2}(x)dx=\left\\{\begin{array}[]{cc}\frac{3}{2\ell\pi},&n_{2}=n_{1},\\\
\frac{1}{\ell\pi},&n_{2}\not=n_{1},\end{array}\right.$

implies that

$\mbox{Proj}_{S}f_{3}^{1}(z,0,0)=\left(\begin{array}[]{c}\mathcal{H}\left(C_{11}z_{1}^{2}z_{2}+C_{12}z_{1}z_{3}z_{4}\right)\vspace{0.2cm}\\\
\mathcal{H}\left(C_{31}z_{3}^{2}z_{4}+C_{32}z_{1}z_{2}z_{3}\right)\end{array}\right),$
(2.40)

where

$\begin{array}[]{l}C_{11}=\frac{3}{2\ell\pi}\psi_{n_{1}}^{T}(0)A_{2100},~{}~{}C_{31}=\frac{3}{2\ell\pi}\psi_{n_{2}}^{T}(0)A_{0021},\vspace{0.2cm}\\\
C_{12}=\left\\{\begin{array}[]{cc}\frac{3}{2\ell\pi}\psi_{n_{1}}^{T}(0)A_{1011},&n_{2}=n_{1},\\\
\frac{1}{\ell\pi}\psi_{n_{1}}^{T}(0)A_{1011},&n_{2}\not=n_{1},\end{array}\right.~{}~{}C_{32}=\left\\{\begin{array}[]{cc}\frac{3}{2\ell\pi}\psi_{n_{2}}^{T}(0)A_{1110},&n_{2}=n_{1},\\\
\frac{1}{\ell\pi}\psi_{n_{2}}^{T}(0)A_{1110},&n_{2}\not=n_{1}.\end{array}\right.\end{array}$

Step 2: The calculation of
$\bf{\mbox{Proj}_{S}\left(\left(D_{z}f_{2}^{1}\right)(z,0,0)U_{2}^{1}(z,0)\right)}$

From (2.20), we have

$\widetilde{F}_{2}(\Phi(\theta)z_{x},0)=F_{2}(\Phi(\theta)z_{x},0)+F^{d(0,0)}_{2}\left(\Phi(\theta)z_{x}\right)$
(2.41)

By (2.32), we write

$\begin{array}[]{lll}&&F_{2}(\Phi(\theta)z_{x}+w,\mu)=F_{2}(\Phi(\theta)z_{x}+w,0)\vspace{0.2cm}\\\
&=&\sum\limits_{q_{1}+q_{2}+q_{3}+q_{4}=2}A_{q_{1}q_{2}q_{3}q_{4}}\gamma_{n_{1}}^{q_{1}+q_{2}}(x)\gamma_{n_{2}}^{q_{3}+q_{4}}(x)z_{1}^{q_{1}}z_{2}^{q_{2}}z_{3}^{q_{3}}z_{4}^{q_{4}}\vspace{0.2cm}\\\
&&+\mathcal{S}_{2}(\Phi(\theta)z_{x},w)+O\left(|w|^{2}\right),\end{array}$
(2.42)

where $q_{1},q_{2},q_{3},q_{4}\in\mathbb{N}_{0}$ and
$\mathcal{S}_{2}(\Phi(\theta)z_{x},w)$ is the second cross terms of $\Phi
z_{x}$ and $w$. In addition, by (2.17), we write

$\begin{array}[]{lll}&&F_{2}^{d}\left(\Phi(\theta)z_{x},0\right)=F^{d(0,0)}_{2}\left(\Phi(\theta)z_{x}\right)\vspace{0.2cm}\\\
&=&\sum_{q_{1}+q_{2}+q_{3}+q_{4}=2}A^{(d,1)}_{q_{1}q_{2}q_{3}q_{4}}\left((-n_{1}/\ell)\xi_{n_{1}}(x)\right)^{q_{1}+q_{2}}\left((-n_{2}/\ell)\xi_{n_{2}}(x)\right)^{q_{3}+q_{4}}z_{1}^{q_{1}}z_{2}^{q_{2}}z_{3}^{q_{3}}z_{4}^{q_{4}}\vspace{0.2cm}\\\
&&-(n_{1}/\ell)^{2}A^{(d,2)}_{2000}\gamma_{n_{1}}^{2}(x)z_{1}^{2}-(n_{1}/\ell)^{2}A^{(d,2)}_{0200}\gamma^{2}_{n_{1}}(x)z_{2}^{2}-(n_{2}/\ell)^{2}A^{(d,2)}_{0020}\gamma_{n_{2}}^{2}(x)z_{3}^{2}\vspace{0.2cm}\\\
&&-(n_{2}/\ell)^{2}A^{(d,2)}_{0002}\gamma_{n_{2}}^{2}(x)z_{4}^{2}-(n_{1}/\ell)^{2}A^{(d,2)}_{1100}\gamma_{n_{1}}^{2}(x)z_{1}z_{2}-(n_{2}/\ell)^{2}A^{(d,2)}_{0011}\gamma_{n_{2}}^{2}(x)z_{3}z_{4}\vspace{0.2cm}\\\
&&-\gamma_{n_{1}}(x)\gamma_{n_{2}}(x)\left(\left((n_{1}/\ell)^{2}A^{(d,2)}_{1010}+(n_{2}/\ell)^{2}A^{(d,3)}_{1010}\right)z_{1}z_{3}+\left((n_{1}/\ell)^{2}A^{(d,2)}_{1001}+(n_{2}/\ell)^{2}A^{(d,3)}_{1001}\right)z_{1}z_{4}\right.\vspace{0.2cm}\\\
&&+\left((n_{1}/\ell)^{2}A^{(d,2)}_{0110}+(n_{2}/\ell)^{2}A^{(d,3)}_{0110}\right)z_{2}z_{3}+\left((n_{1}/\ell)^{2}A^{(d,2)}_{0101}+(n_{2}/\ell)^{2}A^{(d,3)}_{0101}\right)z_{2}z_{4},\end{array}$
(2.43)

where
$\xi_{n_{j}}=\frac{\sqrt{2}}{\sqrt{\ell\pi}}\sin\left(\frac{n_{j}x}{\ell}\right)$
with $j=1,2$, and $A_{i_{1}i_{2}i_{3}i_{4}}^{(d,j)}$ with $j=1,2,3$ are given
in Appendix A.

It is easy to verify that

$\int_{0}^{\ell\pi}\gamma_{n_{1}}^{3}(x)dx=\int_{0}^{\ell\pi}\gamma_{n_{2}}^{3}(x)dx=\int_{0}^{\ell\pi}\gamma_{n_{1}}(x)\gamma^{2}_{n_{2}}(x)dx=0,$
$\int_{0}^{\ell\pi}\xi_{n_{j}}^{2}(x)\gamma_{n_{j}}(x)dx=0,j=1,2,\int_{0}^{\ell\pi}\xi_{n_{2}}^{2}(x)\gamma_{n_{1}}(x)dx=\int_{0}^{\ell\pi}\xi_{n_{1}}(x)\xi_{n_{2}}(x)\gamma_{n_{2}}(x)dx=0,$

$\int_{0}^{\ell\pi}\gamma_{n_{1}}^{2}(x)\gamma_{n_{2}}(x)dx=\int_{0}^{\ell\pi}\xi_{n_{1}}(x)\xi_{n_{2}}(x)\gamma_{n_{1}}(x)dx=\left\\{\begin{array}[]{cc}\frac{1}{\sqrt{2\ell\pi}},&n_{2}=2n_{1},\\\
0,&n_{2}\not=2n_{1},\end{array}\right.$

and

$\int_{0}^{\ell\pi}\xi_{n_{1}}^{2}(x)\gamma_{n_{2}}(x)dx=\left\\{\begin{array}[]{cc}-\frac{1}{\sqrt{2\ell\pi}},&n_{2}=2n_{1},\\\
0,&n_{2}\not=2n_{1}.\end{array}\right.$

Then, by (2.41) and a direct calculation, we have

$\begin{array}[]{lll}&&f^{1}_{2}(z,0,0)=\Psi(0)\left(\begin{array}[]{c}\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{\nu}^{(1)}\right]\vspace{0.1cm}\\\
\left[\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right),\beta_{\nu}^{(2)}\right]\end{array}\right)^{\nu=n_{2}}_{\nu=n_{1}}\vspace{0.3cm}\\\
&=&\left\\{\begin{array}[]{cc}\frac{1}{\sqrt{2\ell\pi}}\Psi(0)\left(\begin{array}[]{c}\widetilde{A}_{1010}z_{1}z_{3}+\widetilde{A}_{1001}z_{1}z_{4}+\widetilde{A}_{0110}z_{2}z_{3}+\widetilde{A}_{0101}z_{2}z_{4}\vspace{0.2cm}\\\
\widetilde{A}_{2000}z_{1}^{2}+\widetilde{A}_{0200}z_{2}^{2}+\widetilde{A}_{1100}z_{1}z_{2}\end{array}\right),&n_{2}=2n_{1},\vspace{0.2cm}\\\
(0,0,0,0)^{T},&n_{2}\not=2n_{1},\vspace{0.2cm}\\\
\end{array}\right.\end{array}$

where

$\left\\{\begin{array}[]{l}\widetilde{A}_{j_{1}j_{2}j_{3}j_{4}}=A_{j_{1}j_{2}j_{3}j_{4}}+\frac{n_{1}n_{2}}{\ell^{2}}A_{j_{1}j_{2}j_{3}j_{4}}^{(d,1)}-\frac{n_{1}^{2}}{\ell^{2}}A_{j_{1}j_{2}j_{3}j_{4}}^{(d,2)}-\frac{n_{2}^{2}}{\ell^{2}}A_{j_{1}j_{2}j_{3}j_{4}}^{(d,3)},\vspace{0.2cm}\\\
j_{1},j_{2},j_{3},j_{4}=0,1,\quad j_{1}+j_{2}=1,\quad
j_{3}+j_{4}=1,\end{array}\right.$ (2.44)
$\left\\{\begin{array}[]{l}\widetilde{A}_{j_{1}j_{2}j_{3}j_{4}}=A_{j_{1}j_{2}j_{3}j_{4}}-\frac{n_{1}^{2}}{\ell^{2}}\left(A_{j_{1}j_{2}j_{3}j_{4}}^{(d,1)}+A_{j_{1}j_{2}j_{3}j_{4}}^{(d,2)}\right),\vspace{0.2cm}\\\
j_{1},j_{2}=0,1,2,\quad j_{1}+j_{2}=2,\quad j_{3}=j_{4}=0.\end{array}\right.$
(2.45)

Then, for $n_{2}\not=2n_{1}$, $U_{2}^{1}(z,0)=(0,0,0,0)^{T}$, and for
$n_{2}=2n_{1}$,

$\begin{array}[]{lll}&&U_{2}^{1}(z,0)\vspace{0.2cm}\\\
&=&\left(M_{2}^{1}\right)^{-1}\mbox{Proj}_{\mbox{Im}M_{2}^{1}}f_{2}^{1}(z,0,0)\vspace{0.2cm}\\\
&=&\frac{1}{i\omega_{1c}\sqrt{2\ell\pi}}\left(\begin{array}[]{c}\psi_{n_{1}}^{T}(0)\left(\frac{\omega_{1c}}{\omega_{2c}}\widetilde{A}_{1010}z_{1}z_{3}-\frac{\omega_{1c}}{\omega_{2c}}\widetilde{A}_{1001}z_{1}z_{4}+\frac{\omega_{1c}}{\omega_{2c}-2\omega_{1c}}\widetilde{A}_{0110}z_{2}z_{3}-\frac{\omega_{1c}}{\omega_{2c}+2\omega_{1c}}\widetilde{A}_{0101}z_{2}z_{4}\right)\vspace{0.1cm}\\\
\overline{\psi}_{n_{1}}^{T}(0)\left(\frac{\omega_{1c}}{\omega_{2c}+2\omega_{1c}}\widetilde{A}_{1010}z_{1}z_{3}-\frac{\omega_{1c}}{\omega_{2c}-2\omega_{1c}}\widetilde{A}_{1001}z_{1}z_{4}+\frac{\omega_{1c}}{\omega_{2c}}\widetilde{A}_{0110}z_{2}z_{3}-\frac{\omega_{1c}}{\omega_{2c}}\widetilde{A}_{0101}z_{2}z_{4}\right)\vspace{0.4cm}\\\
\psi_{n_{2}}^{T}(0)\left(-\frac{\omega_{1c}}{\omega_{2c}-2\omega_{1c}}\widetilde{A}_{2000}z_{1}^{2}-\frac{\omega_{1c}}{\omega_{2c}+2\omega_{1c}}\widetilde{A}_{0200}z_{2}^{2}-\frac{\omega_{1c}}{\omega_{2c}}\widetilde{A}_{1100}z_{1}z_{2}\right)\vspace{0.1cm}\\\
\overline{\psi}_{n_{2}}^{T}(0)\left(\frac{\omega_{1c}}{\omega_{2c}+2\omega_{1c}}\widetilde{A}_{2000}z_{1}^{2}+\frac{\omega_{1c}}{\omega_{2c}-2\omega_{1c}}\widetilde{A}_{0200}z_{2}^{2}+\frac{\omega_{1c}}{\omega_{2c}}\widetilde{A}_{1100}z_{1}z_{2}\right)\end{array}\right).\end{array}$

Hence,

$\mbox{Proj}_{S}\left[\left(D_{z}f_{2}^{1}\right)(z,0,0)U_{2}^{1}(z,0)\right]=\left(\begin{array}[]{c}\mathcal{H}\left(D_{11}z_{1}^{2}z_{2}+D_{12}z_{1}z_{3}z_{4}\right)\vspace{0.1cm}\\\
\mathcal{H}\left(D_{31}z_{3}^{2}z_{4}+D_{32}z_{1}z_{2}z_{3}\right)\vspace{0.1cm}\\\
\end{array}\right),$ (2.46)

where for $n_{2}\not=2n_{1}$, $D_{11}=D_{12}=D_{31}=D_{32}=0$, and for
$n_{2}=2n_{1}$, $D_{31}=0$,

$\begin{array}[]{lll}D_{11}&=&\frac{1}{2i\ell\pi}\left(-\frac{1}{\omega_{2c}}\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{1010}\right)\left(\psi_{n_{2}}^{T}(0)\widetilde{A}_{1100}\right)+\frac{1}{\omega_{2c}}\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{1001}\right)\left(\overline{\psi}_{n_{2}}^{T}(0)\widetilde{A}_{1100}\right)\right.\vspace{0.2cm}\\\
&&\left.-\frac{1}{\omega_{2c}-2\omega_{1c}}\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{0110}\right)\left(\psi_{n_{2}}^{T}(0)\widetilde{A}_{2000}\right)+\frac{1}{\omega_{2c}+2\omega_{1c}}\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{0101}\right)\left(\overline{\psi}_{n_{2}}^{T}(0)\widetilde{A}_{2000}\right)\right),\end{array}$
$\begin{array}[]{lll}D_{12}&=&\frac{1}{2i\ell\pi}\left(-\frac{1}{\omega_{2c}}\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{1010}\right)\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{1001}\right)+\frac{1}{\omega_{2c}}\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{1001}\right)\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{1010}\right)\right.\vspace{0.2cm}\\\
&&\left.-\frac{1}{\omega_{2c}-2\omega_{1c}}\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{0110}\right)\left(\overline{\psi}_{n_{1}}^{T}(0)\widetilde{A}_{1001}\right)+\frac{1}{\omega_{2c}+2\omega_{1c}}\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{0101}\right)\left(\overline{\psi}_{n_{1}}^{T}(0)\widetilde{A}_{1010}\right)\right),\end{array}$
$\begin{array}[]{lll}D_{32}&=&\frac{1}{2i\ell\pi}\left(\frac{2}{\omega_{2c}-2\omega_{1c}}\left(\psi_{n_{2}}^{T}(0)\widetilde{A}_{2000}\right)\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{0110}\right)+\frac{1}{\omega_{2c}}\left(\psi_{n_{2}}^{T}(0)\widetilde{A}_{1100}\right)\left(\psi_{n_{1}}^{T}(0)\widetilde{A}_{1010}\right)\right.\vspace{0.2cm}\\\
&&\left.+\frac{1}{\omega_{2c}}\left(\psi_{n_{2}}^{T}(0)\widetilde{A}_{1100}\right)\left(\overline{\psi}_{n_{1}}^{T}(0)\widetilde{A}_{0110}\right)+\frac{2}{\omega_{2c}+2\omega_{1c}}\left(\psi_{n_{2}}^{T}(0)\widetilde{A}_{0200}\right)\left(\overline{\psi}_{n_{1}}^{T}(0)\widetilde{A}_{1010}\right)\right).\end{array}$

Step 3: The calculation of
$\bf{\mbox{Proj}_{S}\left(\left(D_{w}f_{2}^{(1,1)}(z,0,0)\right)U_{2}^{2}(z,0)(\theta)\right)}$

Let

$U_{2}^{2}(z,0)(\theta)\triangleq
h(\theta,z)=\sum\limits_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x),$
(2.47)

where

$h_{n}(\theta,z)=\sum\limits_{q_{1}+q_{2}+q_{3}+q_{4}=2}h_{n,q_{1}q_{2}q_{3}q_{4}}(\theta)z_{1}^{q_{1}}z_{2}^{q_{2}}z_{3}^{q_{3}}z_{4}^{q_{4}}$

with

$h_{n,q_{1}q_{2}q_{3}q_{4}}(\theta)=\left(h_{n,q_{1}q_{2}q_{3}q_{4}}^{(1)}(\theta)\quad
h_{n,q_{1}q_{2}q_{3}q_{4}}^{(2)}(\theta)\right)^{T}.$

By (2.47), we have

$\left\\{\begin{array}[]{l}U_{2x}^{2}(z,0)(\theta)=h_{x}(\theta,z)=-\sum\limits_{n\in\mathbb{N}_{0}}(n/\ell)h_{n}(\theta,z)\xi_{n}(x),\vspace{0.2cm}\\\
U_{2xx}^{2}(z,0)(\theta)=h_{xx}(\theta,z)=-\sum\limits_{n\in\mathbb{N}_{0}}(n/\ell)^{2}h_{n}(\theta,z)\gamma_{n}(x).\end{array}\right.$
(2.48)

Then, from (2.34) and (2.47), we obtain

$\begin{array}[]{lll}&&\left(D_{w}f^{(1,1)}_{2}(z,0,0)\right)U_{2}^{2}(z,0)\vspace{0.2cm}\\\
&=&\Psi(0)\left(\begin{array}[]{c}\left[\left.D_{w}F_{2}\left(\Phi(\theta)z_{x}+w,0\right)\right|_{w=0}\left(\sum\limits_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{\nu}^{(1)}\right]\vspace{0.2cm}\\\
\left[\left.D_{w}F_{2}\left(\Phi(\theta)z_{x}+w,0\right)\right|_{w=0}\left(\sum\limits_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{\nu}^{(2)}\right]\end{array}\right)_{\nu=n_{1}}^{\nu=n_{2}}.\\\
\end{array}$

By (2.42) and a straightforward computation, we obtain

$\left.D_{w}F_{2}\left(\Phi(\theta)z_{x}+w,0\right)\right|_{w=0}\left(\sum\limits_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x)\right)=\mathcal{S}_{2}\left(\Phi(\theta)z_{x},\sum\limits_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x)\right)$

and

$\begin{array}[]{lll}&&\left(\begin{array}[]{c}\left[\mathcal{S}_{2}\left(\Phi(\theta)z_{x},\sum\limits_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{\nu}^{(1)}\right]\vspace{0.2cm}\\\
\left[\mathcal{S}_{2}\left(\Phi(\theta)z_{x},\sum\limits_{n\in\mathbb{N}_{0}}h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{\nu}^{(2)}\right]\end{array}\right)\vspace{0.2cm}\\\
&=&\sum\limits_{n\in\mathbb{N}_{0}}b_{n_{1},n,\nu}\left(\mathcal{S}_{2}\left(\phi_{n_{1}}(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}\left(\overline{\phi}_{n_{1}}(\theta)z_{2},h_{n}(\theta,z)\right)\right)\vspace{0.2cm}\\\
&&+\sum\limits_{n\in\mathbb{N}_{0}}b_{n_{2},n,\nu}\left(\mathcal{S}_{2}\left(\phi_{n_{2}}(\theta)z_{3},h_{n}(\theta,z)\right)+\mathcal{S}_{2}\left(\overline{\phi}_{n_{2}}(\theta)z_{4},h_{n}(\theta,z)\right)\right),\end{array}$

where $n=0,1,2,\cdots,\nu=n_{1},n_{2},$

$b_{n_{j},n,\nu}=\int_{0}^{\ell\pi}\gamma_{n_{j}}(x)\gamma_{n}(x)\gamma_{\nu}(x)dx=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{\ell\pi}},&n=0,\nu=n_{j},\vspace{0.2cm}\\\
\frac{1}{\sqrt{2\ell\pi}},&n=2n_{j},\nu=n_{j},\vspace{0.2cm}\\\
\frac{1}{\sqrt{2\ell\pi}},&n=n_{1}+n_{2},\nu=n_{j+(-1)^{j+1}},\vspace{0.2cm}\\\
\frac{1}{\sqrt{2\ell\pi}},&n=n_{2}-n_{1},\nu=n_{j+(-1)^{j+1}},n_{1}<n_{2},\vspace{0.2cm}\\\
0,&\mbox{otherwise}.\end{array}\right.$ (2.49)

Hence,

$\begin{array}[]{lll}&&\left(D_{w}f_{2}^{(1,1)}(z,0,0)\right)U^{2}_{2}(z,0)(\theta)\vspace{0.2cm}\\\
&=&\Psi(0)\left(\begin{array}[]{l}\sum\limits_{n=0,2n_{1}}b_{n_{1},n,n_{1}}\left(\mathcal{S}_{2}\left(\phi_{n_{1}}(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}\left(\overline{\phi}_{n_{1}}(\theta)z_{2},h_{n}(\theta,z)\right)\right)\vspace{0.2cm}\\\
\quad\quad+\sum\limits_{n=n_{1}+n_{2},n_{2}-n_{1}}b_{n_{2},n,n_{1}}\left(\mathcal{S}_{2}\left(\phi_{n_{2}}(\theta)z_{3},h_{n}(\theta,z)\right)+\mathcal{S}_{2}\left(\overline{\phi}_{n_{2}}(\theta)z_{4},h_{n}(\theta,z)\right)\right)\vspace{0.4cm}\\\
\sum\limits_{n=0,2n_{2}}b_{n_{2},n,n_{2}}\left(\mathcal{S}_{2}\left(\phi_{n_{2}}(\theta)z_{3},h_{n}(\theta,z)\right)+\mathcal{S}_{2}\left(\overline{\phi}_{n_{2}}(\theta)z_{4},h_{n}(\theta,z)\right)\right)\vspace{0.2cm}\\\
\quad\quad+\sum\limits_{n=n_{1}+n_{2},n_{2}-n_{1}}b_{n_{1},n,n_{2}}\left(\mathcal{S}_{2}\left(\phi_{n_{1}}(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}\left(\overline{\phi}_{n_{1}}(\theta)z_{2},h_{n}(\theta,z)\right)\right)\end{array}\right).\end{array}$

Then, we have

$\textmd{Proj}_{S}\left(D_{w}f_{2}^{(1,1)}(z,0,0)U^{2}_{2}(z,0)\right)=\left(\begin{array}[]{c}\mathcal{H}\left(E_{11}z_{1}^{2}z_{2}+E_{12}z_{1}z_{3}z_{4}\right)\vspace{0.1cm}\\\
\mathcal{H}\left(E_{31}z_{3}^{2}z_{4}+E_{32}z_{1}z_{2}z_{3}\right)\vspace{0.1cm}\\\
\end{array}\right),$ (2.50)
$\left\\{\begin{array}[]{l}\begin{array}[]{lll}E_{11}&=&\frac{1}{\sqrt{\ell\pi}}\psi_{n_{1}}^{T}(0)\left(\mathcal{S}_{2}(\phi_{n_{1}}(\theta),h_{0,1100}(\theta))+\mathcal{S}_{2}(\overline{\phi}_{n_{1}}(\theta),h_{0,2000}(\theta))\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{1}}^{T}(0)\left(\mathcal{S}_{2}(\phi_{n_{1}}(\theta),h_{2n_{1},1100}(\theta))+\mathcal{S}_{2}(\overline{\phi}_{n_{1}}(\theta),h_{2n_{1},2000}(\theta))\right),\end{array}\vspace{0.3cm}\\\
\begin{array}[]{lll}E_{12}&=&\frac{1}{\sqrt{\ell\pi}}\psi_{n_{1}}^{T}(0)\mathcal{S}_{2}(\phi_{n_{1}}(\theta),h_{0,0011}(\theta))+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{1}}^{T}(0)\mathcal{S}_{2}(\phi_{n_{1}}(\theta),h_{2n_{1},0011}(\theta))\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{1}}^{T}(0)\left(\mathcal{S}_{2}(\phi_{n_{2}}(\theta),h_{n_{1}+n_{2},1001}(\theta))+\mathcal{S}_{2}(\overline{\phi}_{n_{2}}(\theta),h_{n_{1}+n_{2},1010}(\theta))\right),\vspace{0.2cm}\\\
&&+\delta_{n_{1}n_{2}}\psi_{n_{1}}^{T}(0)\left(\mathcal{S}_{2}(\phi_{n_{2}}(\theta),h_{n_{2}-n_{1},1001}(\theta))+\mathcal{S}_{2}(\overline{\phi}_{n_{2}}(\theta),h_{n_{2}-n_{1},1010}(\theta))\right)),\end{array}\vspace{0.3cm}\\\
\begin{array}[]{lll}E_{31}&=&\frac{1}{\sqrt{\ell\pi}}\psi_{n_{2}}^{T}(0)\left(\mathcal{S}_{2}(\phi_{n_{2}}(\theta),h_{0,0011}(\theta))+\mathcal{S}_{2}(\overline{\phi}_{n_{2}}(\theta),h_{0,0020}(\theta))\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{2}}^{T}(0)\left(\mathcal{S}_{2}(\phi_{n_{2}}(\theta),h_{2n_{2},0011}(\theta))+\mathcal{S}_{2}(\overline{\phi}_{n_{2}}(\theta),h_{2n_{2},0020}(\theta))\right)),\end{array}\vspace{0.3cm}\\\
\begin{array}[]{lll}E_{32}&=&\frac{1}{\sqrt{\ell\pi}}\psi_{n_{2}}^{T}(0)\mathcal{S}_{2}(\phi_{n_{2}}(\theta),h_{0,1100}(\theta))+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{2}}^{T}(0)\mathcal{S}_{2}(\phi_{n_{2}}(\theta),h_{2n_{2},1100}(\theta))\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{2}}^{T}(0)\left(\mathcal{S}_{2}(\phi_{n_{1}}(\theta),h_{n_{1}+n_{2},0110}(\theta))+\mathcal{S}_{2}(\overline{\phi}_{n_{1}}(\theta)_{n_{1}+n_{2},1010}(\theta))\right))\vspace{0.2cm}\\\
&&+\delta_{n_{1}n_{2}}\psi_{n_{2}}^{T}(0)\left(\mathcal{S}_{2}(\phi_{n_{1}}(\theta),h_{n_{2}-n_{1},0110}(\theta))+\mathcal{S}_{2}(\overline{\phi}_{n_{1}}(\theta),h_{n_{2}-n_{1},1010}(\theta))\right)),\end{array}\end{array}\right.$
(2.51)

where

$\delta_{n_{1}n_{2}}=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{2\ell\pi}},&n_{1}<n_{2},\vspace{0.2cm}\\\
\frac{1}{\sqrt{\ell\pi}},&n_{1}=n_{2}.\end{array}\right.$

Step 4: The calculation of
$\bf{\mbox{Proj}_{S}\left(\left(D_{w,w_{x},w_{xx}}f^{(1,2)}_{2}(z,0,0)\right)U_{2}^{(2,d)}(z,0)(\theta)\right)}$

The calculation of
$\mbox{Proj}_{S}\left(\left(D_{w,w_{x},w_{xx}}f^{(1,2)}_{2}(z,0,0)\right)U_{2}^{(2,d)}(z,0)(\theta)\right)$
is similar to that in Step 3 but is more tedious. We leave the calculation to
Appendix B.

$\textmd{Proj}_{S}\left(\left(D_{w,w_{x},w_{xx}}f^{(1,2)}_{2}(z,0,0)\right)U_{2}^{(2,d)}(z,0)(\theta)\right)=\left(\begin{array}[]{c}\mathcal{H}\left(E_{11}^{d}z_{1}^{2}z_{2}+E_{12}^{d}z_{1}z_{3}z_{4}\right)\vspace{0.1cm}\\\
\mathcal{H}\left(E_{31}^{d}z_{3}^{2}z_{4}+E_{32}^{d}z_{1}z_{2}z_{3}\right)\vspace{0.1cm}\\\
\end{array}\right),$ (2.52)

where

$\left\\{\begin{array}[]{l}\begin{array}[]{lll}E_{11}^{d}&=&-\frac{1}{\sqrt{\ell\pi}}\frac{n_{1}^{2}}{\ell^{2}}\psi_{n_{1}}^{T}(0)\left(\mathcal{S}_{2}^{(d,1)}\left(\phi_{n_{1}}(\theta),h_{0,1100}(\theta)\right)+\mathcal{S}_{2}^{(d,1)}\left(\overline{\phi}_{n_{1}}(\theta),h_{0,2000}(\theta)\right)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{1}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(1,j)}_{2n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\phi_{n_{1}}(\theta),h_{2n_{1},1100}(\theta)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{1}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(1,j)}_{2n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\overline{\phi}_{n_{1}}(\theta),h_{2n_{1},2000}(\theta)\right),\end{array}\vspace{0.3cm}\\\
\begin{array}[]{lll}E_{12}^{d}&=&-\frac{1}{\sqrt{\ell\pi}}\frac{n_{1}^{2}}{\ell^{2}}\psi_{n_{1}}^{T}(0)\mathcal{S}_{2}^{(d,1)}\left(\phi_{n_{1}}(\theta),h_{0,0011}(\theta)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{1}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(1,j)}_{2n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\phi_{n_{1}}(\theta),h_{2n_{1},0011}(\theta)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{1}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(2,j)}_{n_{2}+n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\phi_{n_{2}}(\theta),h_{(n_{2}+n_{1}),1001}(\theta)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{1}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(2,j)}_{n_{2}+n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\overline{\phi}_{n_{2}}(\theta),h_{(n_{2}+n_{1}),1010}(\theta)\right)\vspace{0.2cm}\\\
&&+\delta_{n_{1}n_{2}}\psi_{n_{1}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(2,j)}_{n_{2}-n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\phi_{n_{2}}(\theta),h_{(n_{2}-n_{1}),1001}(\theta)\right)\vspace{0.2cm}\\\
&&+\delta_{n_{1}n_{2}}\psi_{n_{1}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(2,j)}_{n_{2}-n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\overline{\phi}_{n_{2}}(\theta),h_{(n_{2}-n_{1}),1010}(\theta)\right),\end{array}\end{array}\right.$
(2.53)

and

$\left\\{\begin{array}[]{l}\begin{array}[]{lll}E_{31}^{d}&=&-\frac{1}{\sqrt{\ell\pi}}\frac{n_{2}^{2}}{\ell^{2}}\psi_{n_{2}}^{T}(0)\left(\mathcal{S}_{2}^{(d,1)}\left(\phi_{n_{2}}(\theta),h_{0,0011}(\theta)\right)+\mathcal{S}_{2}^{(d,1)}\left(\overline{\phi}_{n_{2}}(\theta),h_{0,0020}(\theta)\right)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{2}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(2,j)}_{2n_{2}}\mathcal{S}_{2}^{(d,j)}\left(\phi_{n_{2}}(\theta),h_{2n_{2},0011}(\theta)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{2}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(2,j)}_{2n_{2}}\mathcal{S}_{2}^{(d,j)}\left(\overline{\phi}_{n_{2}}(\theta),h_{2n_{2},0020}(\theta)\right),\end{array}\vspace{0.3cm}\\\
\begin{array}[]{lll}E_{32}^{d}&=&-\frac{1}{\sqrt{\ell\pi}}\frac{n_{2}^{2}}{\ell^{2}}\psi_{n_{2}}^{T}(0)\mathcal{S}_{2}^{(d,1)}\left(\phi_{n_{2}}(\theta),h_{0,1100}(\theta)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{2}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(2,j)}_{2n_{2}}\mathcal{S}_{2}^{(d,j)}\left(\phi_{n_{2}}(\theta),h_{2n_{2},1100}(\theta)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{2}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(1,j)}_{n_{2}+n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\phi_{n_{1}}(\theta),h_{(n_{2}+n_{1}),0110}(\theta)\right)\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}\psi_{n_{2}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(1,j)}_{n_{2}+n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\overline{\phi}_{n_{1}}(\theta),h_{(n_{2}+n_{1}),1010}(\theta)\right)\vspace{0.2cm}\\\
&&+\delta_{n_{1}n_{2}}\psi_{n_{2}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(1,j)}_{n_{2}-n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\phi_{n_{1}}(\theta),h_{(n_{2}-n_{1}),0110}(\theta)\right)\vspace{0.2cm}\\\
&&+\delta_{n_{1}n_{2}}\psi_{n_{2}}^{T}{(0)}\sum\limits_{j=1,2,3}b^{(1,j)}_{n_{2}-n_{1}}\mathcal{S}_{2}^{(d,j)}\left(\overline{\phi}_{n_{1}}(\theta),h_{(n_{2}-n_{1}),1010}(\theta)\right),\end{array}\end{array}\right.$
(2.54)

with

$\begin{array}[]{l}b^{(1,1)}_{k}=-\frac{n_{1}^{2}}{\ell^{2}},~{}~{}b^{(1,3)}_{k}=-\frac{k^{2}}{\ell^{2}},~{}~{}k=2n_{1},n_{2}+n_{1},n_{2}-n_{1},~{}~{}b^{(1,2)}_{k}=\left\\{\begin{array}[]{ll}\frac{n_{1}k}{\ell^{2}},&k=2n_{1},n_{2}+n_{1},\vspace{0.2cm}\\\
-\frac{n_{1}k}{\ell^{2}},&k=n_{2}-n_{1},\end{array}\right.\vspace{0.3cm}\\\
b^{(2,1)}_{k}=-\frac{n_{2}^{2}}{\ell^{2}},~{}~{}b^{(2,2)}_{k}=\frac{n_{2}k}{\ell^{2}},~{}~{}b^{(2,3)}_{k}=-\frac{k^{2}}{\ell^{2}},~{}~{}k=2n_{2},n_{2}+n_{1},n_{2}-n_{1}.\end{array}$

Clearly, we still need to compute $h_{n,1010}(\theta)$. We leave this tedious
calculation to Appendix C.

####  2.2.3 The normal form of double Hopf bifurcation truncated to third
terms

Let

$\begin{array}[]{l}B_{11}=C_{11}+\frac{3}{2}\left(D_{11}+E_{11}+E_{11}^{d}\right),~{}B_{12}=C_{12}+\frac{3}{2}\left(D_{12}+E_{12}+E_{12}^{d}\right),\vspace{0.3cm}\\\
B_{31}=C_{31}+\frac{3}{2}\left(D_{31}+E_{31}+E_{31}^{d}\right),~{}B_{32}=C_{32}+\frac{3}{2}\left(D_{32}+E_{32}+E_{32}^{d}\right).\end{array}$

From (2.25), (2.33), (2.40), (2.46), (2.50) and (2.52), we have the following
normal form of double Hopf bifurcation truncated to third terms:

$\dot{z}=Bz+\frac{1}{2!}\left(\begin{array}[]{c}\mathcal{H}\left(\left(B^{(1)}_{1}\mu_{1}+B^{(2)}_{1}\mu_{2}\right)z_{1}\right)\vspace{0.2cm}\\\
\mathcal{H}\left(\left(B^{(1)}_{3}\mu_{1}+B^{(2)}_{3}\mu_{2}\right)z_{3}\right)\end{array}\right)+\frac{1}{3!}\left(\begin{array}[]{c}\mathcal{H}\left(B_{11}z_{1}^{2}z_{2}+B_{12}z_{1}z_{3}z_{4}\right)\vspace{0.2cm}\\\
\mathcal{H}\left(B_{31}z_{3}^{2}z_{4}+B_{32}z_{1}z_{2}z_{3}\right)\end{array}\right).$
(2.55)

With the polar coordinates
$z_{1}=r_{1}e^{-i\Theta_{1}},z_{2}=r_{1}e^{i\Theta_{1}},z_{3}=r_{2}e^{-i\Theta_{2}},z_{4}=r_{2}e^{i\Theta_{2}}$,
we have the following amplitude equations for (2.55):

$\begin{cases}\frac{dr_{1}}{dt}=\left(\delta_{1}+p_{11}r_{1}^{2}+p_{12}r_{2}^{2}\right)r_{1},\vspace{0.1cm}\\\
\frac{dr_{2}}{dt}=\left(\delta_{2}+p_{21}r_{1}^{2}+p_{22}r_{2}^{2}\right)r_{2},\end{cases}$
(2.56)

where

$\begin{array}[]{l}\delta_{1}=\frac{1}{2}\mbox{Re}\left(B^{(1)}_{1}\mu_{1}+B^{(2)}_{1}\mu_{2}\right),~{}~{}\delta_{2}=\frac{1}{2}\mbox{Re}\left(B^{(1)}_{3}\mu_{1}+B^{(2)}_{3}\mu_{2}\right),\vspace{0.2cm}\\\
p_{11}=\frac{1}{3!}\mbox{Re}\left(B_{11}\right),~{}p_{12}=\frac{1}{3!}\mbox{Re}\left(B_{12}\right),~{}p_{21}=\frac{1}{3!}\mbox{Re}\left(B_{32}\right),~{}p_{22}=\frac{1}{3!}\mbox{Re}\left(B_{31}\right).\end{array}$

It follows from [21] that depending on whether $p_{11}$ and $p_{22}$ have the
same or opposite signs, there are two essential bifurcation cases: _simple_
case: $p_{11}p_{22}>0$, and _difficult_ case: $p_{11}p_{22}<0$.

For the _simple_ case or some subcases of the _difficult_ case, it is
sufficient to consider the normal form truncated to three-order terms.
However, for some subcases of the _difficult_ case, we have to calculate the
normal form up to fifth-order terms to determine the dynamics near the
bifurcation point.

##  3 Examples

In this section, taking

$f(u,v)=u\left(1-\frac{u}{a}\right)-\frac{buv}{1+u},~{}~{}g(u,v)=\frac{buv}{1+u}-cv,$

then (1.1) becomes the following predator-prey model with Holling type II
functional response:

$\left\\{\begin{array}[]{ll}\frac{\partial u(x,t)}{\partial
t}=d_{11}u_{xx}(x,t)+u\left(1-\frac{u}{a}\right)-\frac{buv}{1+u},&0<x<\ell\pi,t>0,\vspace{0.2cm}\\\
\frac{\partial v(x,t)}{\partial
t}=-d_{21}(v(x,t)u_{x}(x,t-\tau))_{x}+d_{22}v_{xx}(x,t)-cv+\frac{buv}{1+u},&0<x<\ell\pi,t>0,\vspace{0.2cm}\\\
u_{x}(0,t)=u_{x}(\ell\pi,t)=v_{x}(0,t)=v_{x}(\ell\pi,t)=0,&t\geq
0.\end{array}\right.$ (3.1)

System (3.1) has the positive constant steady state
$E_{*}(\gamma,v_{\gamma})$, where

$\gamma=\frac{c}{b-c},~{}~{}v_{\gamma}=\frac{(a-\gamma)(1+\gamma)}{ab},$

provided that $b>\frac{c(1+a)}{a}$ (or equivalently, $0<\gamma<a$) holds. For
$d_{21}=0$, $E_{*}(\gamma,v_{\gamma})$ is asymptotically stable for
$d_{11}\geq 0$ and $d_{22}\geq 0$ provided that $\frac{a-1}{2}<\gamma<a$. For
$d_{21}>0$, the stability and Hopf bifurcation for system (3.1) has been
detailedly investigated in [40, 39]. In what follows, we are interested in the
double Hopf bifurcation induced by the spatial memory diffusion coefficient
$d_{21}$ and the delay $\tau$. For this purpose, we need to investigate the
critical values of $d_{21}$ and $\tau$ at which (2.4) with (3.2) has two pairs
of purely imaginary roots. In the following, we use the same notations as in
[40] and simply introduce some results from [40, 39] for the analysis.

For $E_{*}(\gamma,v_{\gamma})$, we have

$\begin{array}[]{c}a_{11}=\frac{\gamma(a-1-2\gamma)}{a(1+\gamma)}\left\\{\begin{array}[]{ll}\leq
0,&\frac{a-1}{2}\leq\gamma<a,\vspace{0.2cm}\\\
>0,&0<\gamma<\frac{a-1}{2},\end{array}\right.\vspace{0.2cm}\\\
a_{12}=-c<0,~{}~{}a_{21}=\frac{a-\gamma}{a(1+\gamma)}>0,~{}~{}a_{22}=0.\end{array}$
(3.2)

Let

$\omega^{4}+P_{n}\omega^{2}+Q_{n}=0,$ (3.3)

where

$P_{n}=\left(d_{11}^{2}+d_{22}^{2}\right)(n/\ell)^{4}-2\left(d_{11}a_{11}+d_{22}a_{22}\right)(n/\ell)^{2}+a_{11}^{2}+a_{22}^{2}+2a_{12}a_{21},$
(3.4)

and

$Q_{n}=\widetilde{J}_{n}(0)\left(J_{n}+d_{21}v_{*}a_{12}(n/\ell)^{2}\right).$
(3.5)

It follows from [40] that if Eq.(3.3) has a positive roots $\omega_{n}^{+}$
(or $\omega_{n}^{-}$), then Eq.(2.4) has a pair of purely imaginary roots
$\pm\omega_{n}^{+}i$ (or $\pm\omega_{n}^{-}i$) at $\tau=\tau_{n,j}^{+}$ (or
$\tau=\tau_{n,j}^{-}$), where

$\omega_{n}^{\pm}=\sqrt{\frac{-P_{n}\pm\sqrt{\Delta_{n}}}{2}}.$ (3.6)

and

$\tau_{n,j}^{\pm}=\frac{1}{\omega_{n}^{\pm}}\left\\{\arccos\left\\{\frac{J_{n}-\left(\omega_{n}^{\pm}\right)^{2}}{d_{21}v_{*}a_{12}(n/\ell)^{2}}\right\\}+2j\pi\right\\},~{}~{}j\in\mathbb{N}_{0},~{}n\in\mathbb{N}.$
(3.7)

The number of the positive root of Eq.(3.3) depends on the signs of
$P_{n},Q_{n}$ and

$\Delta_{n}=P_{n}^{2}-4Q_{n}=T_{n}^{4}-4T_{n}^{2}J_{n}+4d_{21}^{2}v_{*}^{2}a_{12}^{2}(n/\ell)^{4}.$

Define

$d_{21}^{(n)}=\frac{1}{v_{*}|a_{12}|}\left(d_{11}d_{22}(n/\ell)^{2}+\frac{Det(A)}{(n/\ell)^{2}}-\left(d_{11}a_{22}+d_{22}a_{11}\right)\right),$
(3.8)
$d_{21}^{*(n)}=\frac{\sqrt{4T_{n}^{2}J_{n}-T_{n}^{4}}}{2v_{*}|a_{12}|(n/\ell)^{2}}.$
(3.9)

Then, for fixed $n$, it follows from [40] that $Q_{n}>0$ if and only if
$0<d_{21}<d_{21}^{(n)}$, and $\Delta_{n}>0$ for $4J_{n}\leq T_{n}^{2}$ if and
only if $d_{21}>d_{21}^{*(n)}$.

###  3.1 Dynamics near the double Hopf bifurcation point with the same spatial
profile

Taking the parameters as follows

$a=1,b=9,c=3,d_{11}=0.6,d_{22}=0.8,~{}\ell=2,$ (3.10)

we have $(u_{*},v_{*})=(1/2,1/12)$ and

$a_{11}=-\frac{1}{3},a_{12}=-3,a_{21}=\frac{1}{3},a_{22}=0.$

From (3.4), we have

$P_{n}=\frac{1}{16}n^{4}+\frac{1}{10}n^{2}-\frac{17}{9}\left\\{\begin{array}[]{ll}<0,&n=1,2,\vspace{0.2cm}\\\
>0,&n=3,4,\cdots.\end{array}\right.$ (3.11)

By (3.8), we have

$d_{21}^{(n)}=\frac{12n^{2}}{25}+\frac{16}{n^{2}}+\frac{16}{15},$

from which it is easy to verify that

$d_{21}^{(2)}=\frac{524}{75}<d_{21}^{(3)}=\frac{1612}{225}<d_{21}^{(4)}=\frac{731}{75}<d_{21}^{(1)}=\frac{1316}{75}<d_{21}^{(5)}=\frac{1028}{75}<d_{21}^{(6)}=\frac{4228}{225},$

and

$\min\limits_{n\in\mathbb{N}}\left\\{d_{21}^{(n)}\right\\}=d_{21}^{(2)}=\frac{524}{75}\doteq
6.9867,~{}d_{21}^{(n)}<d_{21}^{(n+1)}~{}\mbox{for any}~{}n\geq 5.$ (3.12)

When $0\leq d_{21}<d_{21}^{*(2)}$, the stability of the positive constant
steady state $(u_{*},v_{*})$ is independent of the delay and we have the
following stability result.

######  Proposition 3.1.

For system (3.1) with the parameters
$a=1,b=9,c=3,d_{11}=0.6,d_{22}=0.8,~{}\ell=2$, when $0\leq
d_{21}<d_{21}^{*(2)}=\frac{927}{134}$, the positive constant steady state
$(u_{*},v_{*})=(1/2,1/12)$ is locally asymptotically stable for any $\tau\geq
0$;

###### Proof.

From (3.5) and (3.12), it follows that when $d_{21}<d_{21}^{(2)}$, $Q_{n}>0$
for any $n\in\mathbb{N}$. This, together with (3.11), implies that when
$d_{21}<d_{21}^{(2)}$, Eq.(3.3) has no positive root for $n=3,4,\cdots$.

From (3.9), we have

$d_{21}^{*(2)}=\frac{927}{134}\doteq 6.9179<d_{21}^{*(1)}=\frac{6851}{633}.$

Notice that

$d_{21}^{*(2)}=\frac{927}{134}\doteq 6.9179<d_{21}^{(2)}=\frac{524}{75}\doteq
6.9867,$

which implies that for $0<d_{21}<d_{21}^{*(2)}\doteq 6.9179$, $\Delta_{1}<0$
and $\Delta_{2}<0$. Thus, for $d_{21}<d_{21}^{*(2)}$, Eq.(3.3) has no positive
root for $n=1,2$.

Combining the above discussion, we can conclude that for
$0<d_{21}<d_{21}^{*(2)}$, Eq.(3.3) has no positive root for any
$n\in\mathbb{N}$ and then the positive constant steady state
$(u_{*},v_{*})=(1/2,1/12)$ is asymptotically stable for any $\tau\geq 0$. ∎

When $d_{21}>d_{21}^{*(2)}\doteq 6.9179$, the stability of the positive
constant steady state $(u_{*},v_{*})$ is related to the delay. Fig.1(a)
illustrates the stability region and Hopf bifurcation curves in the
$d_{21}-\tau$ plane for $6.9\leq d_{21}\leq 7$ and $0\leq\tau\leq 20$.

When $d_{21}^{*(2)}\doteq 6.9179<d_{21}<d_{21}^{(2)}\doteq 6.9867$, Eq.(3.3)
has two positive roots for $n=2$ and no positive roots for $n\in\mathbb{N}$
and $n\neq 2$. Thus, the characteristic equation (2.4) has two sequences of
purely imaginary roots $\pm i\omega_{2}^{\pm}$ at $\tau=\tau_{2,j}^{\pm}$.
System (3.1) undergoes Hopf bifurcations at $\tau=\tau_{2,j}^{\pm}$, as shown
in Fig.1(b). Hopf bifurcation curves $\tau=\tau_{2,1}^{+}$ and
$\tau=\tau_{2,0}^{-}$ intersect at the point $P_{1}(6.9618,13.1290)$, which is
the double Hopf bifurcation point with $\omega_{2}^{-}\doteq 0.2222$ and
$\omega_{2}^{+}\doteq 0.6629$. This double Hopf bifurcation arises from the
interaction of two Hopf bifurcations with the same mode-2.

Figure 1:  Stability and bifurcation curves in $d_{21}$-$\tau$ plane. (a) for
the parameters in (3.10); (b) for the parameters in (3.15).

For this double Hopf bifurcation point $P_{1}(6.9618,13.1290)$, it follows
from the normal form theory derived in Section 2.2 with
$n_{1}=n_{2}=2,d_{21}^{c}=6.9618,\tau_{c}=13.1290$,
$\omega_{1}=\omega_{2}^{-}$ and $\omega_{2}=\omega_{2}^{+}$. The normal form
truncated to the third order terms is

$\begin{cases}\dot{r}_{1}=r_{1}\left(-4.1681\times
10^{-4}\mu_{1}+0.1332\mu_{2}-0.1428r_{1}^{2}+6.003r_{2}^{2}\right),\vspace{0.1cm}\\\
\dot{r}_{2}=r_{2}\left(0.0036\mu_{1}+0.1311\mu_{2}-5.4981r_{1}^{2}-2.2507r_{2}^{2}\right).\end{cases}$
(3.13)

Since $r_{1},r_{2}\geq 0$, system (3.13) has a zero equilibrium $E_{0}(0,0)$
for any $\mu_{1},\mu_{2}\in\mathbb{R}$, two boundary equilibria:

$E_{1}\left(\sqrt{0.9331\mu_{2}-0.0029\mu_{1}},~{}0\right),~{}~{}\mu_{1}<319.6009\mu_{2},$
$E_{2}\left(0,~{}\sqrt{0.0016\mu_{1}+0.0583\mu_{2}}\right),~{}\mu_{1}>-36.4988\mu_{2},$

and one interior positive equilibrium:

$E_{3}\left(\sqrt{6.1905\times
10^{-4}\mu_{1}+0.0326\mu_{2}},~{}\sqrt{0.8416\times
10^{-4}\mu_{1}-0.0214\mu_{2}}\right)$

for

$\mu_{1}>\max\left\\{-52.6919\mu_{2},254.4824\mu_{2}\right\\}.$

By analyzing the stability of these equilibria and noticing
$\mu_{1}=\tau-\tau_{c},\mu_{2}=d_{21}-d_{21}^{c}$, it is easy to obtain the
phase portrait and their dynamical topologies as shown in Fig.2(a), where the
curves $H_{1},H_{2},L_{1},L_{2}$ are defined by the following:

$\begin{array}[]{ll}H_{1}:~{}\tau-\tau_{c}=319.6\left(d_{21}-d_{21}^{c}\right),&H_{2}:~{}\tau-\tau_{c}=-36.4988\left(d_{21}-d_{21}^{c}\right),\vspace{0.2cm}\\\
L_{1}:~{}\tau-\tau_{c}=254.4824\left(d_{21}-d_{21}^{c}\right),d_{21}>d_{21}^{c},&L_{2}:~{}\tau-\tau_{c}=-52.6919\left(d_{21}-d_{21}^{c}\right),d_{21}<d_{21}^{c}.\end{array}$
(3.14)

The straight lines $H_{j},L_{j},j=1,2$, divide the vicinity of the double Hopf
bifurcation point $P_{1}$ into six regions. For each region, the corresponding
phase portrait is plotted in the right side of Fig.2.

Figure 2: The left column: partition of the region near the double Hopf
bifurcation point: (a) partition of the region near the point $P_{1}$ in
Fig.1(a) with $H_{j},L_{j},j=1,2$, being defined by (3.14) ; (b) partition of
the region near the point $P_{2}$ in Fig.1(b) with $H_{j},L_{j},j=1,2$, being
defined by (3.17). The right column: phase portraits for different parameter
regions in the left figures.

These phase portraits Fig.2 show that for the zero equilibrium $E_{0}$
of(3.13), it is stable in Region $①$ and unstable in other regions. Notice
that the zero equilibrium $E_{0}$ of(3.13) corresponds to the positive
equilibrium $E_{*}$ of the original system (1.1). For
$(d_{21},\tau)=(6.96,12.5)$ in Region $①$, numerical simulations in Fig.3 show
the stability of the positive equilibrium $E_{*}$ of the original system(1.1).

Figure 3: For $(d_{21},\tau)=(6.95,12.5)$ in Region $①$, the positive
equilibrium $E_{*}$ of (1.1) is asymptotically stable.

The boundary equilibria $E_{1}$ and $E_{2}$ of correspond to the periodic
solution of the original system (1.1). From the phase portraits ②, ③ and ⑥ in
Fig.2, it is shown that near the neighbourhood of the point $P_{1}$, there
exist two types of stable periodic solutions, respectively, bifurcating from
the Hopf bifurcation at $\tau=\tau_{2,0}^{-}$ and $\tau=\tau_{2,1}^{+}$.
Figs.4 and 5 numerically illustrate these two types of periodic solutions
(1.1) bifurcating from $\tau_{2,0}^{-}$ and $\tau_{2,1}^{+}$ for the
parameters $(d_{21},\tau)$ in Regions $②$ and $⑥$, respectively.

Figure 4: For $(d_{21},\tau)=(6.96,12.5)$ in Region $②$, the positive
equilibrium $E_{*}$ is unstable and the periodic solution bifurcating from the
Hopf bifurcating at $\tau_{2,0}^{-}$ appears. The initial value is chosen as
$u(x,0)=u_{*}+0.005\cos(x),v(x,0)=v_{*}-0.005\cos(x)$.

Figure 5: For $(d_{21},\tau)=(6.945,13.9)$ in Region $⑥$, the positive
equilibrium $E_{*}$ is unstable and the periodic solution bifurcating from the
Hopf bifurcating at $\tau_{2,1}^{+}$ appears. The initial value is chosen as
$u(x,0)=u_{*}+0.01\cos(x),v(x,0)=v_{*}-0.01\cos(x)$.

The positive equilibrium $E_{3}$ correspond to the quasi-periodic solution of
the original system(1.1), which exist for $(d_{21},\tau)$ in Regions ④ and ⑤
of Fig.2(a). Figs.6(a) and 6(d) numerically illustrate this quasi-periodic
solution for $(d_{21},\tau)$ in Region $⑤$. Figs.6(b) and 6(e) are the
truncated curves of Figs.6(a) and 6(d), respectively, for fixed space
$x=\pi/5$, and Figs.6(c) and 6(f) are the truncated curves of Figs.6(a) and
6(d), respectively, for fixed time $t=3000$. Fig.7 illustrates the phase
portrait of $u(x,t)$ and $v(x,t)$ in the $u$-$v$ plane for fixed space
$x=\pi/5$, which looks like a “bird”.

Figure 6: For $(d_{21},\tau)=(6.95,14)$ in Region $⑤$, the positive
equilibrium $E_{*}$ is unstable and there exists the stable quasi-periodic
solution with the spatial profile like $\cos(x)$. $(a)$ and $(d)$: the
spatiotemporal dynamics of the prey $u$ and predator $v$; $(b)$ and $(e)$: the
truncated curves of $(a)$ and $(d)$ for fixed space $x=\pi/5$ showing the
evolution of $u$ and $v$ in time; $(c)$ and $(f)$: the truncated curves of
$(a)$ and $(d)$ for fixed time $t=3000$ showing the spatial profiles of $u$
and $v$. The initial value is chosen as
$u(x,0)=u_{*}+0.02\cos(x),v(x,0)=v_{*}+0.01\cos(x)$.  Figure 7:  For
$(d_{21},\tau)=(6.95,14)$ in Region $⑤$ and fixed space $x=\pi/5$, the
evolution of the dynamics of the prey $u$ and predator $v$ in the $u$-$v$
plane.

###  3.2 Dynamics near the double Hopf bifurcation point with different
spatial modes

Taking the same parameters as used in [39]:

$a=1,~{}b=\frac{3}{10},~{}c=\frac{1}{10},~{}d_{11}=\frac{6}{10},~{}d_{22}=\frac{8}{10},~{}\ell=2,$
(3.15)

we have $(u_{*},v_{*})=(1/5,5/2)$. For the dynamical classification near the
double Hopf bifurcation point, we introduce the results of linear analysis
from [39], as shown in Fig.1(b). In Fig.1(b), Hopf bifurcation curves
$\tau=\tau_{1,0}^{+}$ and $\tau=\tau_{2,0}^{+}$ intersect at the point
$P_{2}(4.1350,4.0276)$, which is the double Hopf bifurcation point. This
double Hopf bifurcation arises from the interaction of spatially inhomogeneous
Hopf bifurcations with mode-$1$ and mode-$2$. For this point $P_{2}$, we can
obtain $\omega_{1}^{+}\doteq 0.2671,\omega_{2}^{+}\doteq 0.3666$.

To investigate the dynamics near this point $P_{2}$, we need to calculate the
corresponding normal form. Setting $(d_{21}^{c},\tau_{c})=(13.4531,0.6811)$,
$\omega_{1}=\omega_{1}^{+}$ and $\omega_{2}=\omega_{2}^{+}$, and then
employing the algorithm developed in Section 2, we have the following normal
form for this double Hopf bifurcation point $P_{2}$

$\begin{cases}\dot{r}_{1}=r_{1}\left(0.0781\mu_{1}+0.1224\mu_{2}-1.4203r_{1}^{2}-4.2174r_{2}^{2}\right),\vspace{0.1cm}\\\
\dot{r}_{2}=r_{2}\left(0.0595\mu_{1}+0.1653\mu_{2}-1.8176r_{1}^{2}-2.3315r_{2}^{2}\right).\end{cases}$
(3.16)

System (3.16) has the similar dynamics to (3.13). The dynamical classification
of (3.16) is plotted in Fig.1(b), where

$\begin{array}[]{ll}H_{1}:~{}\tau-\tau_{c}=-2.7794\left(d_{21}-d_{21}^{c}\right),&H_{2}:~{}\tau-\tau_{c}=-1.5672\left(d_{21}-d_{21}^{c}\right),\vspace{0.2cm}\\\
L_{1}:~{}\tau-\tau_{c}=0.2140\left(d_{21}-d_{21}^{c}\right),d_{21}>d_{21}^{c},&L_{2}:~{}\tau-\tau_{c}=-5.991\left(d_{21}-d_{21}^{c}\right),d_{21}<d_{21}^{c}.\end{array}$
(3.17)

It follows from Fig.2(b) and the corresponding phase portraits that there
exist stable periodic solutions with spatial profile like $\cos(x)$ and
$\cos(x/2)$, respectively, for $\left(d_{21},\tau\right)$ in Region ② and
Region ⑥. For $\left(d_{21},\tau\right)$ in Region ③, there exist a connection
orbit from the periodic solutions with spatial profile like $\cos(x)$ to the
one spatial profile like $\cos(x/2)$, but in the reverse direction for
$\left(d_{21},\tau\right)$ in Region ⑧.

However, for $\left(d_{21},\tau\right)$ in Region ⑦ of Fig.2(b), there exists
a bistaiblity phenomenon, i.e., the coexistence of two types of stable
periodic solutions with spatial profile like $\cos(x)$ and $\cos(x/2)$. Taking
$(d_{21},\tau)=(4.4,4.3)$ in Region $④$, Figs.8 and 9 numerically illustrate
this bistaiblity phenomenon. With the same parameter $(d_{21},\tau)=(4.4,4.3)$
and different initial values, Figs.8(a) and 8(b) show that the solution
$(u,v)$ finally converges to the periodic solutions with spatial profile like
$\cos(x/2)$ for the initial values
$u(x,0)=u_{*}+0.1\cos(x/2),v(x,0)=v_{*}+0.1\cos(x/2)$ , but Figs.8(c) and 8(d)
show that the solution $(u,v)$ finally converges to the periodic solutions
with spatial profile like $\cos(x)$ for the initial values
$u(x,0)=u_{*}+0.1\cos(x),v(x,0)=v_{*}+0.1\cos(x)$. For fixed space $x=\pi/5$,
Fig.9 numerically illustrate the orbit of $(u,v)$ in the $u$-$v$ plane.

(a)

(b)

(c)

(d)

Figure 8:  For $(d_{21},\tau)=(4.4,4.3)$ in Region $⑦$ of Fig.2(b), two types
of stable spatially inhomogeneous periodic solutions with different spatial
profiles coexist. (a)-(b) The initial value is chosen as
$u(x,0)=u_{*}+0.1\cos(x/2),v(x,0)=v_{*}+0.1\cos(x/2)$.; (c)-(d) The initial
value is chosen as $u(x,0)=u_{*}+0.1\cos(x),v(x,0)=v_{*}+0.1\cos(x)$.  Figure
9:  For $(d_{21},\tau)=(4.4,4.3)$ in Region $⑦$ of Fig.2(b) and fixed space
$x=\pi/5$, two stable periodic orbits in $u$-$v$ plane coexist.

##  4 Discussion

In this paper, we have paid our attention on the development of an algorithm
for computing the normal form of the double Hopf bifurcation induced by the
memory-based diffusion coefficient and memory delay for the memory-based
diffusion system. The calculating formulae of the second and third terms in
the normal form are explicitly derived from those in the original system. This
algorithm, which is for the memory-based diffusion system, where the delay
appears in the directional diffusion terms and the diffusion terms are
nonlinear, is the counterpart of the existing algorithm for the related theory
of the classical reaction-diffusion system. By analysing the corresponding
normal form, we can determine the dynamical classification near the double
Hopf bifurcation point.

Employing the obtained theoretical results to the predator-prey system with
Holling-II functional response, we investigate the spatio-temporal dynamics
due to the interaction of double Hopf bifurcation for two cases: (i) double
Hopf bifurcation with the same spatial mode; (ii) double Hopf bifurcation with
the different spatial modes. For the former, we find two types of stable
spatially inhomogeneous periodic solutions with the same spatial mode and
similar oscillatory frequency, and the quasi-periodic solution in some region
near the double Hopf bifurcation point. For latter, we find two types of
stable spatially inhomogeneous periodic solutions with different spatial mode
and oscillatory frequency, and the coexistence of two stable periodic
solutions with different spatial mode in some region near the double Hopf
bifurcation point. In both cases, the pattern transitions from a unstable
periodic solution to a stable periodic solution are found.

We would also like to iterate that the theoretical results in Section 2 are
derived for the non-resonant and weakly resonant double Hopf bifurcations.
Thus it is not applicable for the strongly resonant double Hopf bifurcation.
The algorithm for computing the normal form for the case of strongly double
Hopf bifurcation for the memory-based diffusion system (1.1) still remains
open and we leave this for the further investigation.

##  Appendix A Expressions of $A_{i_{1}i_{2}i_{3}i_{4}}^{(d,j)},j=1,2,3.$

$A_{1010}^{(d,1)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi_{n_{1}}^{(1)}(-1)\phi_{n_{2}}^{(2)}(0)+\phi_{n_{2}}^{(1)}(-1)\phi_{n_{1}}^{(2)}(0)\end{array}\right),$
$A_{1001}^{(d,1)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi_{n_{1}}^{(1)}(-1)\overline{\phi}_{n_{2}}^{(2)}(0)+\overline{\phi}_{n_{2}}^{(1)}(-1)\phi_{n_{1}}^{(2)}(0)\end{array}\right),$

$A_{0110}^{(d,1)}=\overline{A^{(d,1)}_{1001}},\quad\quad
A_{0101}^{(d,1)}=\overline{A^{(d,1)}_{1010}},$

$A_{2000}^{(d,1)}=A_{2000}^{(d,2)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi_{n_{1}}^{(1)}(-1)\phi_{n_{1}}^{(2)}(0)\end{array}\right),$

$A_{0020}^{(d,1)}=A_{0020}^{(d,2)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi_{n_{2}}^{(1)}(-1)\phi_{n_{2}}^{(2)}(0)\end{array}\right),$

$A_{1100}^{(d,1)}=A_{1100}^{(d,2)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
2\mbox{Re}\left\\{\phi_{n_{1}}^{(1)}(-1)\overline{\phi}_{n_{1}}^{(2)}(0)\right\\}\end{array}\right),$

$A_{0011}^{(d,1)}=A_{0011}^{(d,2)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
2\mbox{Re}\left\\{\phi_{n_{2}}^{(1)}(-1)\overline{\phi}_{n_{2}}^{(2)}(0)\right\\}\end{array}\right),$

$A_{1010}^{(d,2)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi_{n_{1}}^{(1)}(-1)\phi_{n_{2}}^{(2)}(0)\end{array}\right),~{}~{}A_{1010}^{(d,3)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi_{n_{2}}^{(1)}(-1)\phi_{n_{1}}^{(2)}(0)\end{array}\right),$

$A_{1001}^{(d,2)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi_{n_{1}}^{(1)}(-1)\overline{\phi}_{n_{2}}^{(2)}(0)\end{array}\right),~{}~{}A_{1001}^{(d,3)}=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\overline{\phi}_{n_{2}}^{(1)}(-1)\phi_{n_{1}}^{(2)}(0)\end{array}\right),$

$A_{0200}^{(d,1)}=A_{0200}^{(d,2)}=\overline{A^{(d,1)}_{2000}},\quad\quad
A_{0002}^{(d,1)}=A_{0002}^{(d,2)}=\overline{A^{(d,1)}_{0020}},$

$A_{0110}^{(d,2)}=\overline{A^{(d,2)}_{1001}},\quad
A_{0110}^{(d,3)}=\overline{A^{(d,3)}_{1001}},\quad
A_{0101}^{(d,2)}=\overline{A^{(d,2)}_{1010}},\quad
A_{0101}^{(d,3)}=\overline{A^{(d,3)}_{1010}}.$

##  Appendix B  The calculation of
$\bf{\mbox{Proj}_{S}\left(\left(D_{w,w_{x},w_{xx}}f^{(1,2)}_{2}(z,0,0)\right)U_{2}^{(2,d)}(z,0)(\theta)\right)}$

Denote $\varphi(\theta)=\Phi(\theta)z_{x}$ and

$\begin{array}[]{lll}&&F_{2}^{d}(\varphi(\theta),w,w_{x},w_{xx})=F_{2}^{d}(\Phi(\theta)z_{x}+w,0)\vspace{0.3cm}\\\
&=&-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\left(\varphi^{(1)}_{xx}(-1)+w_{xx}^{(1)}(-1)\right)\left(\varphi^{(2)}(0)+w^{(2)}(0)\right)\end{array}\right)\vspace{0.3cm}\\\
&&-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\left(\varphi^{(1)}_{x}(-1)+w_{x}^{(1)}(-1)\right)\left(\varphi^{(2)}_{x}(0)+w_{x}^{(2)}(0)\right)\end{array}\right),\end{array}$
$\begin{array}[]{l}\widetilde{S}_{2}^{(d,1)}(\varphi(\theta),w)=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(1)}_{xx}(-1)w^{(2)}(0)\end{array}\right),\vspace{0.3cm}\\\
\widetilde{S}_{2}^{(d,2)}(\varphi(\theta),w_{x})=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(1)}_{x}(-1)w_{x}^{(2)}(0)\end{array}\right)-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(2)}_{x}(0)w_{x}^{(1)}(-1)\end{array}\right),\vspace{0.3cm}\par\\\
\widetilde{S}_{2}^{(d,3)}(\varphi(\theta),w_{xx})=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\varphi^{(2)}(0)w_{xx}^{(1)}(-1)\end{array}\right).\end{array}$

Then, we have

$\begin{array}[]{lll}&&\left.D_{w,w_{x},w_{xx}}F_{2}^{d}(\varphi(\theta),w,w_{x},w_{xx})\right|_{w,w_{x},w_{xx}=0}U_{2}^{(2,d)}(z,0)(\theta)\vspace{0.2cm}\\\
&=&\widetilde{S}_{2}^{(d,1)}\left(\varphi(\theta),h(\theta,z)\right)+\widetilde{S}_{2}^{(d,2)}\left(\varphi(\theta),h_{x}(\theta,z)\right)\vspace{0.2cm}\\\
&&+\widetilde{S}_{2}^{(d,3)}\left(\varphi(\theta),h_{xx}(\theta,z)\right)\par\par\par\end{array}$

and

$\begin{array}[]{lll}&&\left(\begin{array}[]{c}\left[\widetilde{S}_{2}^{(d,1)}\left(\varphi(\theta),h(\theta,z)\right),\beta_{\nu}^{(1)}\right]\vspace{0.2cm}\\\
\left[\widetilde{S}_{2}^{(d,1)}\left(\varphi(\theta),h(\theta,z)\right),\beta_{\nu}^{(2)}\right]\end{array}\right)\vspace{0.2cm}\\\
&=&-(n_{1}/\ell)^{2}\sum\limits_{n\in\mathbb{N}_{0}}b_{n_{1},n,\nu}\left(\mathcal{S}_{2}^{(d,1)}\left(\phi_{n_{1}}(\theta)z_{1}(\theta),h_{n}(\theta,z)\right)+\mathcal{S}_{2}^{(d,1)}\left(\overline{\phi}_{n_{1}}(\theta)z_{2},h_{n}(\theta,z)\right)\right)\vspace{0.2cm}\\\
&&-(n_{2}/\ell)^{2}\sum\limits_{n\in\mathbb{N}_{0}}b_{n_{2},n,\nu}\left(\mathcal{S}_{2}^{(d,1)}\left(\phi_{n_{2}}(\theta)z_{3},h_{n}(\theta,z)\right)+\mathcal{S}_{2}^{(d,1)}\left(\overline{\phi}_{n_{2}}(\theta)z_{4},h_{n}(\theta,z)\right)\right),\end{array}$

$\begin{array}[]{lll}&&\left(\begin{array}[]{c}\left[\widetilde{S}_{2}^{(d,2)}\left(\varphi(\theta),h_{x}(\theta,z)\right),\beta_{\nu}^{(1)}\right]\vspace{0.2cm}\\\
\left[\widetilde{S}_{2}^{(d,2)}\left(\varphi(\theta),h_{x}(\theta,z)\right),\beta_{\nu}^{(2)}\right]\end{array}\right)\vspace{0.2cm}\\\
&=&(n_{1}/\ell)\sum\limits_{n\in\mathbb{N}_{0}}(n/\ell)b^{s}_{n_{1},n,\nu}\left(\mathcal{S}_{2}^{(d,2)}\left(\phi_{n_{1}}(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}^{(d,2)}\left(\overline{\phi}_{n_{1}}(\theta)z_{2},h_{n}(\theta,z)\right)\right)\vspace{0.2cm}\\\
&&+(n_{2}/\ell)\sum\limits_{n\in\mathbb{N}_{0}}(n/\ell)b^{s}_{n_{2},n,\nu}\left(\mathcal{S}_{2}^{(d,2)}\left(\phi_{n_{2}}(\theta)z_{3},h_{n}(\theta,z)\right)+\mathcal{S}_{2}^{(d,2)}\left(\overline{\phi}_{n_{2}}(\theta)z_{4},h_{n}(\theta,z)\right)\right),\end{array}$

$\begin{array}[]{lll}&&\left(\begin{array}[]{c}\left[\widetilde{S}_{2}^{(d,3)}\left(\varphi(\theta),h_{xx}(\theta,z)\right),\beta_{\nu}^{(1)}\right]\vspace{0.2cm}\\\
\left[\widetilde{S}_{2}^{(d,3)}\left(\varphi(\theta),h_{xx}(\theta,z)\right),\beta_{\nu}^{(2)}\right]\end{array}\right)\vspace{0.2cm}\\\
&=&-\sum\limits_{n\in\mathbb{N}_{0}}(n/\ell)^{2}b_{n_{1},n,\nu}\left(\mathcal{S}_{2}^{(d,3)}\left(\phi_{n_{1}}(\theta)z_{1},h_{n}(\theta,z)\right)+\mathcal{S}_{2}^{(d,3)}\left(\overline{\phi}_{n_{1}}(\theta)z_{2},h_{n}(\theta,z)\right)\right)\vspace{0.2cm}\\\
&&-\sum\limits_{n\in\mathbb{N}_{0}}(n/\ell)^{2}b_{n_{2},n,\nu}\left(\mathcal{S}_{2}^{(d,3)}\left(\phi_{n_{2}}(\theta)z_{3},h_{n}(\theta,z)\right)+\mathcal{S}_{2}^{(d,3)}\left(\overline{\phi}_{n_{2}}(\theta)z_{4},h_{n}(\theta,z)\right)\right),\end{array}$

where, for $\nu=n_{1},n_{2},$ $b_{n_{j},n,\nu}$ is defined as in (2.49) and

$b^{s}_{n_{j},n,\nu}=\int_{0}^{\ell\pi}\xi_{n_{j}}(x)\xi_{n}(x)\gamma_{\nu}(x)dx=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{2\ell\pi}},&n=2n_{j},\nu=n_{j},\vspace{0.2cm}\\\
\frac{1}{\sqrt{2\ell\pi}},&n=n_{1}+n_{2},\nu=n_{j+(-1)^{j+1}},\vspace{0.2cm}\\\
\frac{1}{\sqrt{2\ell\pi}},&n=n_{2}-n_{1},n_{j}=n_{2},\nu=n_{1},\vspace{0.2cm}\\\
-\frac{1}{\sqrt{2\ell\pi}},&n=n_{2}-n_{1},n_{j}=n_{1},\nu=n_{2},\vspace{0.2cm}\\\
0,&\mbox{otherwise},\end{array}\right.$

and for
$\phi(\theta)=\left(\phi^{(1)}(\theta)\quad\phi^{(2)}(\theta)\right)^{T},y(\theta)=\left(y^{(1)}(\theta)\quad
y^{(2)}(\theta)\right)^{T}\in C\left([-1,0],\mathbb{R}^{2}\right)$,

$\left\\{\begin{array}[]{l}\mathcal{S}_{2}^{(d,1)}(\phi(\theta),y(\theta))=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi^{(1)}(-1)y^{(2)}(0)\end{array}\right),\vspace{0.3cm}\\\
\mathcal{S}_{2}^{(d,2)}(\phi(\theta),y(\theta))=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi^{(1)}(-1)y^{(2)}(0)\end{array}\right)-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi^{(2)}(0)y^{(1)}(-1)\end{array}\right),\vspace{0.3cm}\par\\\
\mathcal{S}_{2}^{(d,3)}(\phi(\theta),y(\theta))=-2d_{21}^{c}\tau_{c}\left(\begin{array}[]{c}0\vspace{0.2cm}\\\
\phi^{(2)}(0)y^{(1)}(-1)\end{array}\right).\end{array}\right.$ (B.1)

Then, from (2.35), (2.37), (2.47) and (2.48), we have

$\begin{array}[]{lll}&&\left(D_{w,w_{x},w_{xx}}f^{(1,2)}_{2}(z,0,0)\right)U_{2}^{(2,d)}(z,0)(\theta)\vspace{0.3cm}\\\
&=&\Psi(0)\left(\begin{array}[]{c}\left[\left.D_{w,w_{x},w_{xx}}F_{2}^{d}(\varphi(\theta),w,w_{x},w_{xx})\right|_{w,w_{x},w_{xx}=0}U_{2}^{(2,d)}(z,0)(\theta),\beta_{\nu}^{(1)}\right]\vspace{0.2cm}\\\
\left[\left.D_{w,w_{x},w_{xx}}F_{2}^{d}(\varphi(\theta),w,w_{x},w_{xx})\right|_{w,w_{x},w_{xx}=0}U_{2}^{(2,d)}(z,0)(\theta),\beta_{\nu}^{(2)}\right]\end{array}\right)_{\nu=n_{1}}^{\nu=n_{2}}\end{array}$

and then we obtian (2.52).

##  Appendix C Calculation of $h_{n,q_{1}q_{2}q_{3}q_{4}}(\theta)$.

From [12] , we have

$\begin{array}[]{lll}&&M_{2}^{2}\left(h_{n}(\theta,z)\gamma_{n}(x)\right)\vspace{0.2cm}\\\
&=&D_{z}\left(h_{n}(\theta,z)\gamma_{n}(x)\right)Bz-
A_{\mathcal{Q}^{1}}\left(h_{n}(\theta,z)\gamma_{n}(x)\right),\end{array}$

which leads to

$\begin{array}[]{lll}&&\left(\begin{array}[]{c}\left[M_{2}^{2}\left(h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{n}^{(1)}\right]\vspace{0.2cm}\\\
\left[M_{2}^{2}\left(h_{n}(\theta,z)\gamma_{n}(x)\right),\beta_{n}^{(2)}\right]\end{array}\right)\vspace{0.2cm}\\\
&=&2i\omega_{1c}\left(h_{n,2000}(\theta)z_{1}^{2}-h_{n,0200}(\theta)z_{2}^{2}\right)+2i\omega_{2c}\left(h_{n,0020}(\theta)z_{3}^{2}-h_{n,0002}(\theta)z_{4}^{2}\right)\\\
&&+i\left(\omega_{1c}+\omega_{2c}\right)h_{n,1010}(\theta)z_{1}z_{3}+i\left(\omega_{1c}-\omega_{2c}\right)h_{n,1001}(\theta)z_{1}z_{4}\\\
&&-i\left(\omega_{1c}-\omega_{2c}\right)h_{n,0110}(\theta)z_{2}z_{3}-i\left(\omega_{1c}+\omega_{2c}\right)h_{n,0101}(\theta)z_{2}z_{4}\vspace{0.2cm}\\\
&&-\left(\dot{h}_{n}(\theta,z)+X_{0}(\theta)\left(\mathscr{L}_{0}\left(h_{n}(\theta,z)\right)-\dot{h}_{n}(0,z)\right)\right),\end{array}$
(C.1)

where

$\mathscr{L}_{0}\left(h_{n}(\theta,z)\right)=-\tau_{c}(n/\ell)^{2}\left(D_{1}h_{n}(0,z)+D_{2}h_{n}(-1,z)\right)+\tau_{c}Ah_{n}(\theta,z).$

By (2.23), we get

$f_{2}^{2}(z,0,0)=X_{0}(\theta)\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right)-\pi\left(X_{0}(\theta)\widetilde{F}_{2}\left(\Phi(\theta)z_{x},0\right)\right).$

###  C.1 Case 1: $n_{1}\not=n_{2}$.

By (2.15), (2.41), (2.42) and (2.43), we have , for $n_{2}\not=n_{1}$,

$\begin{array}[]{lll}&&\left(\begin{array}[]{c}\left[f_{2}^{2}(z,0,0),\beta_{n}^{(1)}\right]\vspace{0.1cm}\\\
\left[f_{2}^{2}(z,0,0),\beta_{n}^{(2)}\right]\end{array}\right)\vspace{0.2cm}\\\
&=&\begin{cases}\left\\{\begin{array}[]{l}\frac{1}{\sqrt{\ell\pi}}X_{0}(\theta)\left(A_{2000}z_{1}^{2}+A_{0200}z_{2}^{2}+A_{0020}z_{3}^{2}\right.\vspace{0.2cm}\\\
\left.+A_{0002}z_{4}^{2}+A_{1100}z_{1}z_{2}+A_{0011}z_{3}z_{4}\right),\end{array}\right.&n=0,\vspace{0.3cm}\\\
\frac{1}{\sqrt{2\ell\pi}}X_{0}(\theta)\left(\widetilde{A}_{2000}z_{1}^{2}+\widetilde{A}_{0200}z_{2}^{2}+\widetilde{A}_{1100}z_{1}z_{2}\right),&n=2n_{1},n_{2}\not=2n_{1},n_{2}\not=3n_{1},\vspace{0.3cm}\\\
\left\\{\begin{array}[]{l}\frac{1}{\sqrt{2\ell\pi}}\left(X_{0}(\theta)I_{2}-\Phi_{n_{2}}(\theta)\Psi_{n_{2}}(0)\right)\left(\widetilde{A}_{2000}z_{1}^{2}\right.\vspace{0.2cm}\\\
\left.+\widetilde{A}_{0200}z_{2}^{2}+\widetilde{A}_{1100}z_{1}z_{2}\right),\end{array}\right.&n=2n_{1},n_{2}=2n_{1},\vspace{0.3cm}\\\
\left\\{\begin{array}[]{l}\frac{1}{\sqrt{2\ell\pi}}X_{0}(\theta)\left(\widetilde{A}_{2000}z_{1}^{2}+\widetilde{A}_{0200}z_{2}^{2}+\widetilde{A}_{1100}z_{1}z_{2}\right.\vspace{0.2cm}\\\
\left.+\widetilde{A}_{1010}z_{1}z_{3}+\widetilde{A}_{1001}z_{1}z_{4}+\widetilde{A}_{0110}z_{2}z_{3}+\widetilde{A}_{0101}z_{2}z_{4}\right),\end{array}\right.&n=2n_{1},n_{2}=3n_{1},\vspace{0.3cm}\\\
\frac{1}{\sqrt{2\ell\pi}}X_{0}(\theta)\left(\widetilde{A}_{0020}z_{3}^{2}+\widetilde{A}_{0002}z_{4}^{2}+\widetilde{A}_{0011}z_{3}z_{4}\right),&n=2n_{2},\vspace{0.3cm}\\\
\left\\{\begin{array}[]{l}\frac{1}{\sqrt{2\ell\pi}}X_{0}(\theta)\left(\widehat{A}_{1010}z_{1}z_{3}+\widehat{A}_{1001}z_{1}z_{4}+\widehat{A}_{0110}z_{2}z_{3}\right.\vspace{0.2cm}\\\
\left.+\widehat{A}_{0101}z_{2}z_{4}\right),\end{array}\right.&n=n_{1}+n_{2},\vspace{0.3cm}\\\
\left\\{\begin{array}[]{l}\frac{1}{\sqrt{2\ell\pi}}X_{0}(\theta)\left(\widetilde{A}_{1010}z_{1}z_{3}+\widetilde{A}_{1001}z_{1}z_{4}\right.\vspace{0.2cm}\\\
\left.+\widetilde{A}_{0110}z_{2}z_{3}+\widetilde{A}_{0101}z_{2}z_{4}\right),\end{array}\right.&n=n_{2}-n_{1},n_{2}\not=2n_{1},n_{2}\not=3n_{1},\vspace{0.2cm}\\\
\left\\{\begin{array}[]{l}\frac{1}{\sqrt{2\ell\pi}}\left(X_{0}(\theta)I_{2}-\Phi_{n_{1}}(\theta)\Psi_{n_{1}}(0)\right)\left(\widetilde{A}_{1010}z_{1}z_{3}\right.\\\
\left.+\widetilde{A}_{1001}z_{1}z_{4}+\widetilde{A}_{0110}z_{2}z_{3}+\widetilde{A}_{0101}z_{2}z_{4}\right),\end{array}\right.&n=n_{2}-n_{1},n_{2}=2n_{1},\vspace{0.2cm}\\\
\left\\{\begin{array}[]{l}\frac{1}{\sqrt{2\ell\pi}}X_{0}(\theta)\left(\widetilde{A}_{1010}z_{1}z_{3}+\widetilde{A}_{1001}z_{1}z_{4}+\widetilde{A}_{0110}z_{2}z_{3}\right.\vspace{0.2cm}\\\
\left.+\widetilde{A}_{0101}z_{2}z_{4}+\widetilde{A}_{2000}z_{1}^{2}+\widetilde{A}_{0200}z_{2}^{2}+\widetilde{A}_{1100}z_{1}z_{2}\right),\end{array}\right.&n=n_{2}-n_{1},n_{2}=3n_{1},\end{cases}\end{array}$
(C.2)

where $\widetilde{A}_{j_{1}j_{2}j_{3}j_{4}}$ is defined by (2.44), (2.45) and
the following (C.3)

$\left\\{\begin{array}[]{l}\widetilde{A}_{j_{1}j_{2}j_{3}j_{4}}=A_{j_{1}j_{2}j_{3}j_{4}}-\frac{n_{2}^{2}}{\ell^{2}}\left(A_{j_{1}j_{2}j_{3}j_{4}}^{(d,1)}+A_{j_{1}j_{2}j_{3}j_{4}}^{(d,2)}\right),\vspace{0.2cm}\\\
j_{3},j_{4}=0,1,2,\quad j_{3}+j_{4}=2,\quad j_{1}=j_{2}=0,\end{array}\right.$
(C.3)

and

$\left\\{\begin{array}[]{l}\widehat{A}_{j_{1}j_{2}j_{3}j_{4}}=A_{j_{1}j_{2}j_{3}j_{4}}-\frac{n_{1}n_{2}}{\ell^{2}}A_{j_{1}j_{2}j_{3}j_{4}}^{(d,1)}-\frac{n_{1}^{2}}{\ell^{2}}A_{j_{1}j_{2}j_{3}j_{4}}^{(d,2)}-\frac{n_{2}^{2}}{\ell^{2}}A_{j_{1}j_{2}j_{3}j_{4}}^{(d,3)},\vspace{0.2cm}\\\
j_{1},j_{2},j_{3},j_{4}=0,1,\quad j_{1}+j_{2}=1,\quad
j_{3}+j_{4}=1.\end{array}\right.$ (C.4)

Hence, from (C.1), (C.2) and matching the coefficients of
$z_{1}^{2},z_{1}z_{2},z_{1}z_{3},z_{2}z_{3},z_{3}^{2}$, we have

$n=0,~{}\left\\{\begin{array}[]{ll}z_{1}^{2}:&\begin{cases}\dot{h}_{0,2000}(\theta)-2i\omega_{1c}h_{0,2000}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{0,2000}(0)-L_{0}(h_{0,2000}(\theta))=\frac{1}{\sqrt{\ell\pi}}A_{2000},\end{cases}\vspace{0.2cm}\\\
z_{3}^{2}:&\begin{cases}\dot{h}_{0,0020}(\theta)-2i\omega_{2c}h_{0,0020}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{0,0020}(0)-L_{0}\left(h_{0,0020}(\theta)\right)=\frac{1}{\sqrt{\ell\pi}}A_{0020},\end{cases}\vspace{0.2cm}\\\
z_{1}z_{2}:&\begin{cases}\dot{h}_{0,1100}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{0,1100}(0)-L_{0}(h_{0,1100}(\theta))=\frac{1}{\sqrt{\ell\pi}}A_{1100},\end{cases}\vspace{0.2cm}\\\
z_{3}z_{4}:&\begin{cases}\dot{h}_{0,0011}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{0,0011}(0)-L_{0}(h_{0,0011}(\theta))=\frac{1}{\sqrt{\ell\pi}}A_{0011},\end{cases}\end{array}\right.$
(C.5) $\begin{array}[]{l}n=2n_{1},\vspace{0.2cm}\\\
n_{2}\not=2n_{1},\end{array}~{}\left\\{\begin{array}[]{ll}z_{1}^{2}:&\begin{cases}\dot{h}_{2n_{1},2000}(\theta)-2i\omega_{1c}h_{2n_{1},2000}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{2n_{1},2000}(0)-\mathscr{L}_{0}(h_{2n_{1},2000}(\theta))=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{2000},\end{cases}\vspace{0.2cm}\\\
z_{1}z_{2}:&\begin{cases}\dot{h}_{2n_{1},1100}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{2n_{1},1100}(0)-\mathscr{L}_{0}(h_{2n_{1},1100}(\theta))=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{1100},\end{cases}\vspace{0.2cm}\\\
z_{3}z_{4}:&\begin{cases}\dot{h}_{2n_{1},0011}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{2n_{1},0011}(0)-\mathscr{L}_{0}(h_{2n_{1},0011}(\theta))=(0\quad
0)^{T},\end{cases}\end{array}\right.$ (C.6)
$\begin{array}[]{l}n=2n_{1},\vspace{0.2cm}\\\
n_{2}=2n_{1},\end{array}~{}\left\\{\begin{array}[]{ll}z_{1}^{2}:&\begin{cases}\dot{h}_{2n_{1},2000}(\theta)-2i\omega_{1c}h_{2n_{1},2000}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\Phi_{n_{2}}(\theta)\Psi_{n_{2}}(0)\widetilde{A}_{2000},\vspace{0.1cm}\\\
\dot{h}_{2n_{1},2000}(0)-\mathscr{L}_{0}(h_{2n_{1},2000}(\theta))=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{2000},\end{cases}\vspace{0.2cm}\\\
z_{1}z_{2}:&\begin{cases}\dot{h}_{2n_{1},1100}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\Phi_{n_{2}}(\theta)\Psi_{n_{2}}(0)\widetilde{A}_{1100},\vspace{0.1cm}\\\
\dot{h}_{2n_{1},1100}(0)-\mathscr{L}_{0}(h_{2n_{1},1100}(\theta))=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{1100},\end{cases}\vspace{0.2cm}\\\
z_{3}z_{4}:&\begin{cases}\dot{h}_{2n_{1},0011}(\theta)=(0,0)^{T},\vspace{0.1cm}\\\
\dot{h}_{2n_{1},0011}(0)-\mathscr{L}_{0}(h_{2n_{1},0011}(\theta))=(0\quad
0)^{T},\end{cases}\end{array}\right.$ (C.7)

$n=2n_{2},~{}\left\\{\begin{array}[]{ll}z_{3}^{2}:&\begin{cases}\dot{h}_{2n_{2},0020}(\theta)-2i\omega_{2c}h_{2n_{2},0020}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{2n_{2},0020}(0)-\mathscr{L}_{0}(h_{2n_{2},0020}(\theta))=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{0020},\end{cases}\vspace{0.2cm}\\\
z_{1}z_{2}:&\begin{cases}\dot{h}_{2n_{2},1100}(\theta)=(0,0)^{T},\vspace{0.1cm}\\\
\dot{h}_{2n_{2},1100}(0)-\mathscr{L}_{0}(h_{2n_{2},1100}(\theta))=(0\quad
0)^{T},\end{cases}\vspace{0.2cm}\\\
z_{3}z_{4}:&\begin{cases}\dot{h}_{2n_{2},0011}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{2n_{2},0011}(0)-\mathscr{L}_{0}(h_{2n_{2},0011}(\theta))=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{0011},\end{cases}\end{array}\right.$
(C.8) ${\small
n=n_{1}+n_{2},\left\\{\begin{array}[]{ll}z_{1}z_{3}:&\begin{cases}\dot{h}_{n_{1}+n_{2},1010}(\theta)-i\left(\omega_{1c}+\omega_{2c}\right)h_{n_{1}+n_{2},1010}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{n_{1}+n_{2},1010}(0)-\mathscr{L}_{0}\left(h_{n_{1}+n_{2},1010}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widehat{A}_{1010},\end{cases}\vspace{0.2cm}\\\
z_{1}z_{4}:&\begin{cases}\dot{h}_{n_{1}+n_{2},1001}(\theta)-i\left(\omega_{1c}-\omega_{2c}\right)h_{n_{1}+n_{2},1001}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{n_{1}+n_{2},1001}(0)-\mathscr{L}_{0}\left(h_{n_{1}+n_{2},1001}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widehat{A}_{1001},\end{cases}\vspace{0.2cm}\\\
z_{2}z_{3}:&\begin{cases}\dot{h}_{n_{1}+n_{2},0110}(\theta)-i\left(\omega_{2c}-\omega_{1c}\right)h_{n_{1}+n_{2},0110}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{n_{1}+n_{2},0110}(0)-\mathscr{L}_{0}\left(h_{n_{1}+n_{2},0110}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widehat{A}_{0110},\end{cases}\end{array}\right.}$
(C.9) ${\small\begin{array}[]{l}n=n_{2}-n_{1},\vspace{0.2cm}\\\
n_{2}\not=2n_{1},\end{array}\left\\{\begin{array}[]{ll}z_{1}z_{3}:&\begin{cases}\dot{h}_{n_{2}-n_{1},1010}(\theta)-i\left(\omega_{1c}+\omega_{2c}\right)h_{n_{2}-n_{1},1010}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{n_{2}-n_{1},1010}(0)-\mathscr{L}_{0}\left(h_{n_{2}-n_{1},1010}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{1010},\end{cases}\vspace{0.2cm}\\\
z_{1}z_{4}:&\begin{cases}\dot{h}_{n_{2}-n_{1},1001}(\theta)-i\left(\omega_{1c}-\omega_{2c}\right)h_{n_{2}-n_{1},1001}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{n_{2}-n_{1},1001}(0)-\mathscr{L}_{0}\left(h_{n_{2}-n_{1},1001}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{1001},\end{cases}\vspace{0.2cm}\\\
z_{2}z_{3}:&\begin{cases}\dot{h}_{n_{2}-n_{1},0110}(\theta)-i\left(\omega_{2c}-\omega_{1c}\right)h_{n_{2}-n_{1},0110}(\theta)=(0\quad
0)^{T},\vspace{0.1cm}\\\
\dot{h}_{n_{2}-n_{1},0110}(0)-\mathscr{L}_{0}\left(h_{n_{2}-n_{1},0110}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{0110},\end{cases}\end{array}\right.}$
(C.10)

${\small\begin{array}[]{l}n=n_{2}-n_{1},\vspace{0.2cm}\\\
n_{2}=2n_{1}\end{array}\left\\{\begin{array}[]{ll}z_{1}z_{3}:&\begin{cases}\dot{h}_{n_{2}-n_{1},1010}(\theta)-i\left(\omega_{1c}+\omega_{2c}\right)h_{n_{2}-n_{1},1010}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\Phi_{n_{1}}(\theta)\Psi_{n_{1}}(0)\widetilde{A}_{1010},\vspace{0.1cm}\\\
\dot{h}_{n_{2}-n_{1},1010}(0)-\mathscr{L}_{0}\left(h_{n_{2}-n_{1},1010}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{1010},\end{cases}\vspace{0.2cm}\\\
z_{1}z_{4}:&\begin{cases}\dot{h}_{n_{2}-n_{1},1001}(\theta)-i\left(\omega_{1c}-\omega_{2c}\right)h_{n_{2}-n_{1},1001}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\Phi_{n_{1}}(\theta)\Psi_{n_{1}}(0)\widetilde{A}_{1001},\vspace{0.1cm}\\\
\dot{h}_{n_{2}-n_{1},1001}(0)-\mathscr{L}_{0}\left(h_{n_{2}-n_{1},1001}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{1001},\end{cases}\vspace{0.2cm}\\\
z_{2}z_{3}:&\begin{cases}\dot{h}_{n_{2}-n_{1},0110}(\theta)-i\left(\omega_{2c}-\omega_{1c}\right)h_{n_{2}-n_{1},0110}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\Phi_{n_{1}}(\theta)\Psi_{n_{1}}(0)\widetilde{A}_{0110},\vspace{0.1cm}\\\
\dot{h}_{n_{2}-n_{1},0110}(0)-\mathscr{L}_{0}\left(h_{n_{2}-n_{1},0110}(\theta)\right)=\frac{1}{\sqrt{2\ell\pi}}\widetilde{A}_{0110}.\end{cases}\end{array}\right.}$
(C.11)

Solving (C.5), (C.6), (C.7),(C.8),(C.9),(C.10) and (C.11), we obtain

$\left\\{\begin{array}[]{l}h_{0,2000}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}(2i\omega_{1c})\right)^{-1}A_{2000}e^{2i\omega_{1c}\theta},\vspace{0.2cm}\\\
h_{0,0020}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}(2i\omega_{2c})\right)^{-1}A_{0020}e^{2i\omega_{2c}\theta},\vspace{0.2cm}\\\
h_{0,1100}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}(0)\right)^{-1}A_{1100},\vspace{0.2cm}\\\
h_{0,0011}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}(0)\right)^{-1}A_{0011},\end{array}\right.$
(C.12)
$\left\\{\begin{array}[]{lll}h_{2n_{1},2000}(\theta)&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}(2i\omega_{1c})\right)^{-1}\widetilde{A}_{2000}e^{2i\omega_{1c}\theta}\vspace{0.2cm}\\\
h_{2n_{1},1100}(\theta)&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}(0)\right)^{-1}\widetilde{A}_{1100},\vspace{0.2cm}\\\
h_{2n_{1},0011}(\theta)&=&(0\quad
0)^{T},\end{array}~{}~{}\mbox{for}~{}~{}n_{2}\not=2n_{1},\right.$ (C.13)
$\left\\{\begin{array}[]{l}h_{2n_{2},0020}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{2}}(2i\omega_{2c})\right)^{-1}\widetilde{A}_{0020}e^{2i\omega_{2c}\theta}\vspace{0.2cm}\\\
h_{2n_{2},1100}(\theta)=(0,0)^{T},\vspace{0.2cm}\\\
h_{2n_{2},0011}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{2}}(0)\right)^{-1}\widetilde{A}_{0011},\end{array}\right.$
(C.14)
$\left\\{\begin{array}[]{l}h_{n_{1}+n_{2},1010}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{n_{1}+n_{2}}\left(i\left(\omega_{1c}+\omega_{2c}\right)\right)\right)^{-1}\widehat{A}_{1010}e^{i\left(\omega_{1c}+\omega_{2c}\right)\theta}\vspace{0.2cm}\\\
h_{n_{1}+n_{2},1001}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{n_{1}+n_{2}}\left(i\left(\omega_{1c}-\omega_{2c}\right)\right)\right)^{-1}\widehat{A}_{1001}e^{i\left(\omega_{1c}-\omega_{2c}\right)\theta}\vspace{0.2cm}\\\
h_{n_{1}+n_{2},0110}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{n_{1}+n_{2}}\left(i\left(\omega_{2c}-\omega_{1c}\right)\right)\right)^{-1}\widehat{A}_{0110}e^{i\left(\omega_{2c}-\omega_{1c}\right)\theta},\end{array}\right.$
(C.15) $\left\\{\begin{array}[]{lll}&&h_{2n_{1},2000}(\theta)\vspace{0.2cm}\\\
&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}\left(2i\omega_{1c}\right)\right)^{-1}\left(\widetilde{A}_{2000}-C_{1}\widetilde{\mathcal{M}}_{2n_{1}}\left(i\omega_{2c}\right)\phi_{n_{2}}(0)\right.\vspace{0.2cm}\\\
&&\left.-C_{2}\widetilde{\mathcal{M}}_{2n_{1}}\left(-i\omega_{2c}\right)\overline{\phi}_{n_{2}}(0)\right)e^{2i\omega_{1c}\theta}\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}C_{1}\phi_{n_{2}}(\theta)+\frac{1}{\sqrt{2\ell\pi}}C_{2}\overline{\phi}_{n_{2}}(\theta),\vspace{0.2cm}\\\
&&h_{2n_{1},1100}(\theta)\vspace{0.2cm}\\\
&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}\left(0\right)\right)^{-1}\left(\widetilde{A}_{1100}-C_{3}\widetilde{\mathcal{M}}_{2n_{1}}\left(i\omega_{2c}\right)\phi_{n_{2}}(0)\right.\vspace{0.2cm}\\\
&&\left.-C_{4}\widetilde{\mathcal{M}}_{2n_{1}}\left(-i\omega_{2c}\right)\overline{\phi}_{n_{2}}(0)\right)+\frac{1}{\sqrt{2\ell\pi}}C_{3}\phi_{n_{2}}(\theta)+\frac{1}{\sqrt{2\ell\pi}}C_{4}\overline{\phi}_{n_{2}}(\theta),\vspace{0.2cm}\\\
&&h_{2n_{1},0011}(\theta)=(0\quad
0)^{T},\end{array}~{}~{}\mbox{for}~{}~{}n_{2}=2n_{1},\right.$

where

$C_{1}=\frac{1}{i(\omega_{2c}-2\omega_{1c})}\psi_{n_{2}}^{T}(0)\widetilde{A}_{2000},\quad
C_{2}=-\frac{1}{i(\omega_{2c}+2\omega_{1c})}\overline{\psi}_{n_{2}}^{T}(0)\widetilde{A}_{2000},$
$C_{3}=\frac{1}{i\omega_{2c}}\psi_{n_{2}}^{T}(0)\widetilde{A}_{1100},\quad
C_{4}=-\frac{1}{i\omega_{2c}}\overline{\psi}_{n_{2}}^{T}(0)\widetilde{A}_{1100},$
$\left\\{\begin{array}[]{lll}h_{n_{2}-n_{1},1010}(\theta)&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(i(\omega_{1c}+\omega_{2c})\right)\right)^{-1}\widetilde{A}_{1010}e^{i\left(\omega_{1c}+\omega_{2c}\right)\theta},\vspace{0.2cm}\\\
h_{n_{2}-n_{1},1001}(\theta)&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(i(\omega_{1c}-\omega_{2c})\right)\right)^{-1}\widetilde{A}_{1001}e^{i\left(\omega_{1c}-\omega_{2c}\right)\theta},\vspace{0.2cm}\\\
h_{n_{2}-n_{1},0110}(\theta)&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(i(\omega_{2c}-\omega_{1c})\right)\right)^{-1}\widetilde{A}_{0110}e^{i\left(\omega_{2c}-\omega_{1c}\right)\theta},\end{array}~{}~{}\mbox{for}~{}~{}n_{2}\not=2n_{1},\right.$

and for $n_{2}=2n_{1},$ we have

$\left\\{\begin{array}[]{lll}h_{n_{2}-n_{1},1010}(\theta)&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(i(\omega_{1c}+\omega_{2c})\right)\right)^{-1}\left(\widetilde{A}_{1010}-C_{5}\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(i\omega_{1c}\right)\phi_{n_{1}}(0)\right.\vspace{0.2cm}\\\
&&\left.-C_{6}\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(-i\omega_{1c}\right)\overline{\phi}_{n_{1}}(0)\right)e^{i\left(\omega_{1c}+\omega_{2c}\right)\theta}\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}C_{5}\phi_{n_{1}}(\theta)+\frac{1}{\sqrt{2\ell\pi}}C_{6}\overline{\phi}_{n_{1}}(\theta),\vspace{0.2cm}\\\
h_{n_{2}-n_{1},1001}(\theta)&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(i(\omega_{1c}-\omega_{2c})\right)\right)^{-1}\left(\widetilde{A}_{1001}-C_{7}\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(i\omega_{1c}\right)\phi_{n_{1}}(0)\right.\vspace{0.2cm}\\\
&&\left.-C_{8}\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(-i\omega_{1c}\right)\overline{\phi}_{n_{1}}(0)\right)e^{i\left(\omega_{1c}-\omega_{2c}\right)\theta}\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}C_{7}\phi_{n_{1}}(\theta)+\frac{1}{\sqrt{2\ell\pi}}C_{8}\overline{\phi}_{n_{1}}(\theta),\vspace{0.2cm}\\\
h_{n_{2}-n_{1},0110}(\theta)&=&\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(i(\omega_{2c}-\omega_{1c})\right)\right)^{-1}\left(\widetilde{A}_{0110}-C_{9}\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(i\omega_{1c}\right)\phi_{n_{1}}(0)\right.\vspace{0.2cm}\\\
&&\left.-C_{10}\widetilde{\mathcal{M}}_{n_{2}-n_{1}}\left(-i\omega_{1c}\right)\overline{\phi}_{n_{1}}(0)\right)e^{i\left(\omega_{2c}-\omega_{1c}\right)\theta}\vspace{0.2cm}\\\
&&+\frac{1}{\sqrt{2\ell\pi}}C_{9}\phi_{n_{1}}(\theta)+\frac{1}{\sqrt{2\ell\pi}}C_{10}\overline{\phi}_{n_{1}}(\theta),\vspace{0.2cm}\end{array}\right.$

where

$\begin{array}[]{ll}C_{5}=-\frac{1}{i\omega_{2c}}\psi_{n_{1}}^{T}(0)\widetilde{A}_{1010},&C_{6}=-\frac{1}{i(2\omega_{1c}+\omega_{2c})}\overline{\psi}_{n_{1}}^{T}(0)\widetilde{A}_{1010},\vspace{0.2cm}\\\
C_{7}=\frac{1}{i\omega_{2c}}\psi_{n_{1}}^{T}(0)\widetilde{A}_{1001},&C_{8}=-\frac{1}{i(2\omega_{1c}-\omega_{2c})}\overline{\psi}_{n_{1}}^{T}(0)\widetilde{A}_{1001},\vspace{0.2cm}\\\
C_{9}=\frac{1}{i(2\omega_{1c}-\omega_{2c})}\psi_{n_{1}}^{T}(0)\widetilde{A}_{0110},&C_{10}=-\frac{1}{i\omega_{2c}}\overline{\psi}_{n_{1}}^{T}(0)\widetilde{A}_{0110}.\end{array}$

###  C.2 Case 2: $n_{1}=n_{2}$.

By (2.15), (2.41), (2.42) and (2.43), for $n_{2}=n_{1}$, we have

$\begin{array}[]{lll}&&\left(\begin{array}[]{c}\left[f_{2}^{2}(z,0,0),\beta_{n}^{(1)}\right]\vspace{0.1cm}\\\
\left[f_{2}^{2}(z,0,0),\beta_{n}^{(2)}\right]\end{array}\right)\vspace{0.2cm}\\\
&=&\begin{cases}\left\\{\begin{array}[]{l}\frac{1}{\sqrt{\ell\pi}}X_{0}(\theta)\left(A_{2000}z_{1}^{2}+A_{0200}z_{2}^{2}+A_{0020}z_{3}^{2}+A_{0002}z_{4}^{2}\right.\vspace{0.2cm}\\\
+A_{1100}z_{1}z_{2}+A_{0011}z_{3}z_{4}+\breve{A}_{1010}z_{1}z_{3}+\breve{A}_{1001}z_{1}z_{4}\vspace{0.2cm}\\\
\left.+\breve{A}_{0110}z_{2}z_{3}+\breve{A}_{0101}z_{2}z_{4}\right),\end{array}\right.&n=0,\vspace{0.3cm}\\\
\left\\{\begin{array}[]{l}\frac{1}{\sqrt{2\ell\pi}}X_{0}(\theta)\left(\widetilde{A}_{2000}z_{1}^{2}+\widetilde{A}_{0200}z_{2}^{2}+\widetilde{A}_{1100}z_{1}z_{2}\right.\vspace{0.2cm}\\\
+\widetilde{A}_{0020}z_{3}^{2}+\widetilde{A}_{0002}z_{4}^{2}+\widetilde{A}_{0011}z_{3}z_{4}+\widehat{A}_{1010}z_{1}z_{3}\vspace{0.2cm}\\\
\left.+\widehat{A}_{1001}z_{1}z_{4}+\widehat{A}_{0110}z_{2}z_{3}+\widehat{A}_{0101}z_{2}z_{4}\right),\end{array}\right.&n=2n_{1},\vspace{0.3cm}\\\
\end{cases}\end{array}$ (C.16)

where $\widetilde{A}_{j_{1}j_{2}j_{3}j_{4}}$ and
$\widehat{A}_{j_{1}j_{2}j_{3}j_{4}}$ are defined by (2.44), (2.45), (C.3) and
(C.4), and $\breve{A}_{j_{1}j_{2}j_{3}j_{4}}$ is defined by the following

$\left\\{\begin{array}[]{l}\breve{A}_{j_{1}j_{2}j_{3}j_{4}}=A_{j_{1}j_{2}j_{3}j_{4}}+\frac{n_{1}^{2}}{\ell^{2}}\left(A_{j_{1}j_{2}j_{3}j_{4}}^{(d,1)}-A_{j_{1}j_{2}j_{3}j_{4}}^{(d,2)}-A_{j_{1}j_{2}j_{3}j_{4}}^{(d,3)}\right),\vspace{0.2cm}\\\
j_{1},j_{2},j_{3},j_{4}=0,1,\quad j_{1}+j_{2}=1,\quad
j_{3}+j_{4}=1.\end{array}\right.$

From (C.1) and (C.16), we have

$\left\\{\begin{array}[]{l}h_{0,2000}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}(2i\omega_{1c})\right)^{-1}A_{2000}e^{2i\omega_{1c}\theta},\vspace{0.2cm}\\\
h_{0,0020}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}(2i\omega_{2c})\right)^{-1}A_{0020}e^{2i\omega_{2c}\theta},\vspace{0.2cm}\\\
h_{0,1100}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}(0)\right)^{-1}A_{1100},\vspace{0.2cm}\\\
h_{0,0011}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}(0)\right)^{-1}A_{0011},\vspace{0.2cm}\\\
h_{0,1010}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}\left(i\left(\omega_{1c}+\omega_{2c}\right)\right)\right)^{-1}\breve{A}_{1010}e^{i\left(\omega_{1c}+\omega_{2c}\right)\theta},\vspace{0.2cm}\\\
h_{0,1001}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}\left(i\left(\omega_{1c}-\omega_{2c}\right)\right)\right)^{-1}\breve{A}_{1001}e^{i\left(\omega_{1c}-\omega_{2c}\right)\theta},\vspace{0.2cm}\\\
h_{0,0110}(\theta)=\frac{1}{\sqrt{\ell\pi}}\left(\widetilde{\mathcal{M}}_{0}\left(i\left(\omega_{2c}-\omega_{1c}\right)\right)\right)^{-1}\breve{A}_{0110}e^{i\left(\omega_{2c}-\omega_{1c}\right)\theta},\end{array}\right.$
(C.17)

and

$\left\\{\begin{array}[]{l}h_{2n_{1},2000}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}(2i\omega_{1c})\right)^{-1}\widetilde{A}_{2000}e^{2i\omega_{1c}\theta},\vspace{0.2cm}\\\
h_{2n_{1},0020}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}(2i\omega_{2c})\right)^{-1}\widetilde{A}_{0020}e^{2i\omega_{2c}\theta}\vspace{0.2cm}\\\
h_{2n_{1},1100}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}(0)\right)^{-1}\widetilde{A}_{1100},\vspace{0.2cm}\\\
h_{2n_{1},0011}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}(0)\right)^{-1}\widetilde{A}_{0011},\vspace{0.2cm}\\\
h_{2n_{1},1010}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}\left(i\left(\omega_{1c}+\omega_{2c}\right)\right)\right)^{-1}\widehat{A}_{1010}e^{i\left(\omega_{1c}+\omega_{2c}\right)\theta}\vspace{0.2cm}\\\
h_{2n_{1},1001}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}\left(i\left(\omega_{1c}-\omega_{2c}\right)\right)\right)^{-1}\widehat{A}_{1001}e^{i\left(\omega_{1c}-\omega_{2c}\right)\theta}\vspace{0.2cm}\\\
h_{2n_{1},0110}(\theta)=\frac{1}{\sqrt{2\ell\pi}}\left(\widetilde{\mathcal{M}}_{2n_{1}}\left(i\left(\omega_{2c}-\omega_{1c}\right)\right)\right)^{-1}\widehat{A}_{0110}e^{i\left(\omega_{2c}-\omega_{1c}\right)\theta}.\end{array}\right.$
(C.18)

Notice that for $n_{1}=n_{2}$, $2n_{1}=2n_{2}=n_{1}+n_{2}$ and
$n_{2}-n_{1}=0$. Thus, substituting (C.17) and (C.18) into (2.51), (2.53) and
(2.54), we can calculate $E_{ij}$ and $E_{ij}^{d}$.

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