# Invariant curves of quasi-periodic reversible mappings and its application

Yan Zhuang<EMAIL_ADDRESS>Daxiong Piao<EMAIL_ADDRESS>Yanmin Niu
<EMAIL_ADDRESS>School of Mathematical Sciences, Ocean University of
China, Qingdao 266100, P. R. China

###### Abstract

We consider the existence of invariant curves of real analytic reversible
mappings which are quasi-periodic in the angle variables. By the normal form
theorem, we prove that under some assumptions, the original mapping is changed
into its linear part via an analytic convergent transformation, so that
invariants curves are obtained. In the iterative process, by solving the
modified homological equations, we ensure that the transformed mapping is
still reversible. As an application, we investigate the invariant curves of a
class of nonlinear resonant oscillators, with the Birkhoff constants of the
corresponding Poincar$\acute{e}$ mapping all zeros or not.

###### keywords:

Invariant curves; quasi-periodic solutions; normal forms; Birkhoff constants;
reversible mappings.

###### MSC:

34C11 , 37J40.

††journal: Journal of Differential Equations

##  1 Introduction

In this paper, we consider the mappings of the form

$M:\theta_{1}=\theta+\gamma_{0}+f(\theta,r),\ \ r_{1}=r+g(\theta,r),$ (1.1)

where $f(\theta,r)$ and $g(\theta,r)$ are quasi-periodic in $\theta$ with
frequency $\omega=(\omega_{1},\omega_{2},...,\omega_{m})$, real analytic in a
neighborhood of $r=0$ and $f(\theta,0)=g(\theta,0)=0$. Suppose mapping $M$ is
reversible with respect to the involution $G:(x,y)\rightarrow(-x,y)$, that is,
$GMG=M^{-1}$.

Compared with the classical twist mappings

$\theta_{1}=\theta+\gamma_{0}+\sum_{k=0}^{+\infty}\gamma_{k}r^{k}+f(\theta,r),\
\ r_{1}=r+g(\theta,r),$ (1.2)

mapping $M$ in (1.1) can be seen as having no twist term with all Birkhoff
constants $\gamma_{1}=\gamma_{2}=\cdots=0$. If some of the Birkhoff constants
are not zero, $r^{-1}f,r^{-1}g$ are small and periodic in $\theta$, and
mappings (1.2) are area-preserving, Moser’s twist theorem[23] tells us that in
any neighborhood of the fixed point $r=0$, there are many analytic invariant
closed curves surrounding this point, which implies the stability of the fixed
point. When $f,g$ are quasi-periodic and the frequencies are sufficiently
incommensurable with $2\gamma_{0}^{-1}\pi$, Zharnitsky in [25] obtained the
same conclusion for exact symplectic map. In many applications, we may meet
non-Hamiltonian systems, or systems have some symmetrical characteristics.
Therefore, the theory for reversible system or for its corresponding Poincaré
mapping, that is, reversible mapping has received widespread attention. The
invariant curves theorem of reversible system was first obtained by Moser [16]
in 1965, and developed by himself [17, 18] and Sevryuk [22] for both
continuous and discrete systems. Based on the KAM technique, Liu [9] proved
the existence of invariant curves for quasi-periodic reversible mappings
(1.2), with $\gamma_{1}=1$ and the frequencies of $f,g$ and
$2\gamma_{0}^{-1}\pi$ satisfying the Diophantine condition. Furthermore,
within the framework of small twist theorem by Ortega[19], Liu extended the
results of mapping with intersection property to the reversible mapping under
periodic[10] and quasi-periodic [9] cases.

On the other hand, if all the Birkhoff constants are zero, there were several
papers concerning this equation. In 2000, Rüssmann [21] studied an area-
preserving mapping of the precise form

$x_{1}=x\cos\gamma_{0}-y\sin\gamma_{0}+\cdots,\ \
y_{1}=x\sin\gamma_{0}+y\cos\gamma_{0}+\cdots,$

where Birkhoff constants $\gamma_{1}=\gamma_{2}=\cdots=0$, and the dots denote
the higher order terms in $x$ and $y$. He proved that if there is a formal
change of variables transforming this mapping into the linearized normal form
and $\gamma_{0}$ satisfies the Bruno condition, then there is a convergent
change of variables taking the given mapping into its linear part. The
stability of the fixed point $(0,0)$ therefore obtained. Recently, Hu and Liu
et [2] considered the existence of invariant curves of real analytic area-
preserving mappings (1.2) by the normal form method. They applied the results
to the boundedness of all solutions for asymmetric oscillator. As far as we
know, there is few results about the existence of invariant curves for
reversible mapping (1.1).

By Rüssmann’s method and the assumption that a formal change takes (1.1) into
the linearized form $(\theta,r)\rightarrow(\theta+\gamma_{0},r)$, we aim to
construct a convergent transformation $U$ which can be commutated with $G$,
that is $GU=UG$, taking (1.1) into the linear part. In the follows, assume the
constant $\gamma_{0}$ in mapping $M$ and frequency $\omega$ of functions $f,g$
satisfy the so-called Diophantine condition

$\displaystyle\left|\frac{\langle
k,\omega\rangle\gamma_{0}}{2\pi}-j\right|\geq\frac{c_{0}}{|k|^{\sigma}},\quad
k\in\mathbb{Z}^{m}\setminus\\{0\\},j\in\mathbb{Z},$ (1.3)

where $c_{0}$ is a small positive constant, $\sigma>0$. It not difficult to
show that for $\sigma>m+1$, the Lebesgue measure for the set of $\gamma_{0}$
satisfying the above inequalities is positive for sufficient small $c_{0}$.

Now we can state our first main result.

###### Theorem 1.1

Assume that the mapping M in (1.1) is reversible with respect to the
involution $G:(\theta,r)\rightarrow(-\theta,r)$, $\gamma_{0}$ satisfies (1.3).
If there exists a formal change of variables

$U:\begin{cases}\theta=\xi+u(\xi,\eta),\\\ r=\eta+v(\xi,\eta),\end{cases}$
(1.4)

where $u,v$ are quasi-periodic in $\xi$, such that (1.1) is transformed into
the linearized form

$U^{-1}MU:\begin{cases}\xi_{1}=\xi+\gamma_{0},\\\ \eta_{1}=\eta,\end{cases}$
(1.5)

then there exists a real analytic convergent change of variables which
transforms (1.1) into (1.5) and commutate with $G$.

Based on the classical KAM approach, we will transform (1.1) closer to the
linear form (1.5) by a sequence of transformations. Meanwhile, the domain of
the transformed new map becomes narrower at each step in the iterative
process. Different from the elimination of mean values by the intersection
property, we use the unique property of the normal form. Moreover, in the
proof of the convergence of the compositions of infinitely transformations, we
take notice of the reversibility of every new mapping, which can be guaranteed
by setting the odd or even functions in the right hands of homological
equations.

At last, we give an application of the main theorem to show the boundedness of
all solutions for the following nonlinear resonant oscillator

$x^{\prime\prime}+\varphi(x)f(x^{\prime})+\omega^{2}x+g(x)=p(t),$ (1.6)

where the functions $f,\varphi,g,p$ satisfy assumptions $(H_{1})-(H_{5})$ in
Section 5. The boundedness conjecture of related problem
$x^{\prime\prime}+h(t,x)=0$ started from Littlewood [11], and in 1976 Morris
[14] gave a positive answer for $h(t,x)=2x^{3}+p(t)$. By using KAM theory, he
showed the existence of a family of invariant curves and proved the
boundedness of every solution of the superlinear case. Additional works for
general superlinear and sublinear nonlinearities $h(t,x)$ may be found in [1,
6, 7, 24, 3, 8]. The left class of problems is those with linear growth at
infinity, where the case $h(t,x)=n^{2}x+\varphi(x)-p(t)$ has been settled in
[5, 20]. These results are also based on KAM theory.

In this paper, we are concerned with the boundedness of solutions to (1.6),
where $h(t,x,x^{\prime})=\varphi(x)f(x^{\prime})+\omega^{2}x+g(x)-p(t)$ has
relation with $x^{\prime}$. Consequently, for $f\neq 0$, equation (1.6) will
no longer be a Hamiltonian system, which means that the classical twist
theorem of Moser is not applicable. By setting the parity of $f$ and $p$, we
regard (1.6) as a reversible system to solve this problem. In [4], Kunze,
Küpper and Liu investigated the reversible case
$h(t,x,x^{\prime})=f(x)x^{\prime}+n^{2}x+\varphi(x)-p(t)$ where $p(t)$ is
$2\pi$ periodic. By some smoothness hypothesis and a suitable form of small
twist theorem, they obtained the boundedness of solutions. A sharp result
concerning the unboundedness of solutions was also considered. Later, Li and
Ma proved the boundedness of solutions for
$h(t,x,x^{\prime})=f(x^{\prime})+\omega^{2}x+g(x)-p(t)$ with periodic $p(t)$
and $g(+\infty)-g(-\infty)\neq 0$ where
$g(\pm\infty)=\lim_{x\rightarrow\pm\infty}g(x)$ at $\omega\notin\emph{Q}$ and
$\omega\notin\emph{N}$ cases. Combining with the normal form case, we will
show similar twist condition
$(\varphi(+\infty)-\varphi(-\infty))f(+\infty)+(g(+\infty)-g(-\infty))\neq 0$
for (1.6) is no longer necessary.

Thus, our second main result is as following.

###### Theorem 1.2

Suppose that $(H_{1})-(H_{5})$ hold, and $\omega$ satisfies

$|k\omega^{-1}-l|\geq\frac{c_{0}}{|k|^{\sigma}},$

where $c_{0}>0,\sigma>0,k\in\mathbb{Z}\setminus 0,l\in\mathbb{Z}$. Then for
every solution $x(t)$ of (5.1), we have

$\sup_{t\in\mathbb{R}}(|x(t)|+|x^{\prime}(t)|)<+\infty.$

The paper is organized as follows. In Section 2 we recall some notations and
give some properties of the normal forms. The iteration lemma and the proof of
Theorem 1.1 are given in Sections 3 and 4. As an application, we discuss the
boundedness of all solutions for equation 5.1 in the last section.

##  2 Notations and normal form

In this section, we will provide some notations and give the definition of
normal form without considering the convergence of the power series.

Let $\mathbb{A}:=\mathbb{T}\times\mathbb{R}$ be an infinite cylinder, where
$\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$. The smooth diffeomorphisms on
$\mathbb{A}$ which homotopy to the identity maps are denoted by
$Diff_{0}^{\infty}(\mathbb{A})$. We regard as $f(\theta,r)\sim O(r^{s})$, if
the real analytic function $f(\theta,r)$ can be represented by

$f(\theta,r)=\sum_{k\geq s}f_{k}(\theta)r^{k},$

and denote $O(r^{s})$ as $O_{s}$ for short. We set the total space as

$\mathbb{P}=\\{\Phi:\mathbb{C}^{2}\rightarrow\mathbb{C}:\Phi(\theta,r)=\Phi_{1}(\theta,r)+\Phi_{2}(\theta,r)+\cdots,\Phi_{k}\in\mathbb{P}^{k},k=1,2,\cdots\\},$

where

$\mathbb{P}^{k}=\\{r^{k}\sum_{j\in\mathbb{Z}^{m}}\chi_{j}e^{i\langle
j,\omega\rangle\theta}:|\Im\theta|<t<1,|r|<\rho<1,\chi_{j}\in\mathbb{C},\overline{\chi}_{j}=\chi_{-j}\\},$

where $\overline{\chi}_{j}$ is conjugate of $\chi_{j}$.

Denote the mapping (1.1) in the form of

$M=I+\beta+F,$

where $I$ is the identity mapping, $\beta=(\gamma_{0},0)^{T}$ and
$F=(f(\theta,r),g(\theta,r))^{T}$. Then the linearized normal form (1.5) can
be expressed as $I+\beta$.

Introduce an operator $R:\mathbb{P}\rightarrow\mathbb{P}$ as

$R\Phi=\Phi\circ(I+\beta)-\Phi=\Phi(\theta+\gamma_{0},r)-\Phi(\theta,r).$
(2.1)

It is obvious that $R$ is a linear mapping and satisfies

$R|_{\mathbb{P}^{k}}:\mathbb{P}^{k}\rightarrow\mathbb{P}^{k},k=1,2,\cdots.$

In the following, we will give the kernel and image of the image $R$.

###### Lemma 2.1

[12] Suppose (1.3) holds, then the kernel of $R$ is the set of all constant
functions in $\theta$, that is

$\mathbb{K}=\\{\Phi:\mathbb{C}^{2}\rightarrow\mathbb{C}:\Phi=\Phi(r)=\sum_{k=1}^{+\infty}\chi_{k}r^{k},\chi_{k}\in\mathbb{C}\\},$

and the image of $R$ is the set of all functions which has vanishing average
value in $\theta$, that is,

$\mathbb{M}=\\{\Phi:\mathbb{C}^{2}\rightarrow\mathbb{C}:\Phi=\Phi(\theta,r)=\sum_{k=1}^{+\infty}\sum_{j\in\mathbb{Z}^{m}\setminus\\{0\\}}\chi_{kj}e^{i\langle
j,\omega\rangle\theta}r^{k},\chi_{kj}\in\mathbb{C},\overline{\chi}_{kj}=\chi_{k(-j)}\\}.$

From the above discussions,

$\mathbb{P}=\mathbb{K}\oplus\mathbb{M},\quad{\mathbb{P}^{k}}={\mathbb{K}^{k}}\oplus{\mathbb{M}^{k}},\quad
k=1,2,\cdots,$ (2.2)

and

$\quad R|_{\mathbb{P}^{k}}\circ R^{-1}|_{\mathbb{P}^{k}}=I|_{\mathbb{M}^{k}}.$
(2.3)

Similarly, the mapping $R\times R$ on $\mathbb{P}\times\mathbb{P}$ is written
in the form

$(R\times
R)(\Phi,\Psi)^{T}=(R\Phi,R\Psi)^{T}=(\Phi\circ(I+\beta)-\Phi,\Psi\circ(I+\beta)-\Psi)^{T}.$

Moreover, it is easy to derive the properties in (2.2) and (2.3) for
$\mathbb{P}\times\mathbb{P}$ and $R\times R$.

Now we introduce the definition of normal forms and the uniqueness of the
normal form, and one can refer to [21, 2, 12] for details.

###### Definition 2.2

Assume (2.2)-(2.3) hold. $n$ is called a formal normal form of the mapping
$M$, if

: $n-(I+\beta)\in\mathbb{K}\times\mathbb{K}$,

: there is a change of variables $V=I+\eta$, $\eta\in(R\times
R)^{-1}(\mathbb{M}\times\mathbb{M})$, such that $V^{-1}\circ M\circ V=n$.

###### Lemma 2.3

[12] Assume (2.2)-(2.3) hold, if $I+\beta$ is a formal normal form of the
given mapping $M$, then the set of formal normal forms of $M$ is
$\\{I+\beta\\}$.

In what follows, we will meet the difference equation, the so-called
‘homological equation’:

$Ru(\theta,r)=h(\theta,r),$ (2.4)

where $R$ is defined in (2.1) and $h(\theta,r)$ is a quasi-periodic function
in $\mathbb{P}$.

###### Lemma 2.4

Suppose that $h(\theta,r)$ is real analytic and quasi-periodic in $\theta$
with frequency $\omega=(\omega_{1},\omega_{2},...,\omega_{m})$, the constant
$\gamma_{0}$ satisfies (1.3). Then for any $0<\tau<t$, difference equation
(2.4) has unique real analytic quasi-periodic solution $u(\theta,r)$ with
frequency $\omega$ and zero average value, that is,
$[u]=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}u(\theta,r)d\theta=0$, if
and only if $h(\theta,r)\in\mathbb{M}$. Moreover, we have the estimate

$\|u\|_{\tau}\leq\frac{c}{(t-\tau)^{m+\sigma}}\|h\|_{t},$

where $\|\cdot\|_{a}$ denotes the supremum norm in the domain
$\\{(\theta,r)\in\mathbb{C}^{2},|\Im\theta|<a,|r|<1\\}$. If
$h(-\theta-\gamma_{0},r)=h(\theta,r)$, then $u$ is odd in $\theta$; if
$h(-\theta-\gamma_{0},r)=-h(\theta,r)$, then $u$ is even in $\theta$.

The proof is similar with lemma 2 in [9], so we omit it here.

###### Lemma 2.5

Under the assumptions of diffeomorphism $M$ in (1.1), for every $N\geq 2$,
there exist a neighborhood of $r=0$ in $\mathbb{A}$, a smooth diffeomorphism
$V\in Diff_{0}^{\infty}(\mathbb{A})$, and N-1 numbers
$\gamma_{k}\in\mathbb{R},k=1,2,...,N-1$, such that $\hat{M}=V^{-1}\circ M\circ
V$ is in the form of

$\hat{M}(\theta,r)=\Big{(}\theta+\gamma_{0}+\sum_{k=1}^{N-1}\gamma_{k}r^{k}+\phi(\theta,r)r^{N},r+\varphi(\theta,r)r^{N}\Big{)},$

where $\phi,\varphi$ are smooth mappings. Moreover, $\hat{M}$ is reversible
with respect to the involution $G:(\theta,r)\mapsto(-\theta,r)$, and has the
invariant curve $r=0$.

Proof. Since $M$ in (1.1) has the invariant curve $r=0$, it can be written as

$M(\theta,r)=\Big{(}\theta+\gamma_{0}+\phi_{1}(\theta)r,\varphi_{1}(\theta)r\Big{)}+O(r^{2}),$
(2.5)

where $\phi_{1},\varphi_{1}\in C^{\infty}(\mathbb{T},\mathbb{R})$. $M\in
Diff_{0}^{\infty}(\mathbb{A})$ means that the determinant of Jacobian matrix
for $M$ is not vanishing at $r=0$, that is, $\varphi_{1}(\theta)\neq 0$.
Without loss of generality, we may assume $\varphi_{1}(\theta)>0$. In view of
(1.3) and Lemma 2.4, for every mapping $f\in
C^{\infty}(\mathbb{T}^{m},\mathbb{R})$, there is a unique function $g\in
C^{\infty}(\mathbb{T}^{m},\mathbb{R})$, such that

$f(\theta)=b+g(\theta)-g(\theta+\gamma_{0}),\quad\forall\theta\in\mathbb{T}^{m},$

where
$b=[f]=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}f(\theta)d\theta$.

Hence, for $\varphi_{1}(\theta)>0$, there exists a $g\in
C^{\infty}(\mathbb{T}^{m},\mathbb{R})$, such that,

$\ln{\varphi_{1}(\theta)}=[\ln{\varphi_{1}(\theta)}]+g(\theta)-g(\theta+\gamma_{0}),\quad\forall\theta\in\mathbb{T}^{m},$

that is,

$\varphi_{1}(\theta)=b_{1}\frac{g_{1}^{(2)}(\theta)}{g_{1}^{(2)}(\theta+\gamma_{0})},$
(2.6)

where $b_{1}=e^{[\ln{\varphi_{1}(\theta)}]}$, $g_{1}^{(2)}=e^{g(\theta)}\in
C^{\infty}(\mathbb{T}^{m},\mathbb{R})$.

Let

$V_{1}^{(2)}(\theta,r)=\Big{(}\theta,\frac{r}{\sqrt{g_{1}^{(2)}(\theta)g_{1}^{(2)}(-\theta)}}\Big{)}.$

Then under the transformation $V_{1}^{(2)}$, the original mapping $M$ is
changed into the form

$\displaystyle M_{1}^{(1)}(\theta,r)=$ $\displaystyle(V_{1}^{(2)})^{-1}\circ
M\circ V_{1}^{(2)}$ $\displaystyle=$
$\displaystyle(\theta+\gamma_{0}+\tilde{\phi_{1}}(\theta)r,\varphi_{1}(\theta)\frac{\sqrt{g_{1}^{(2)}(\theta+\gamma_{0})g_{1}^{(2)}(-\theta-\gamma_{0})}}{\sqrt{g_{1}^{(2)}(\theta)g_{1}^{(2)}(-\theta)}}\cdot
r)+O(r^{2}),$

where
$\tilde{\phi_{1}}=\phi_{1}(\theta)/\sqrt{g_{1}^{(2)}(\theta)g_{1}^{(2)}(-\theta)}$.

Since $M$ is reversible with respect to the involution $G$, that is,

$M\circ G\circ M(\theta,r)=G(\theta,r),$

it is easy to obtain that
$\varphi_{1}(\theta)\varphi_{1}(-\theta-\gamma_{0})r=r$, then we have
$\varphi_{1}^{-1}(\theta)=\varphi_{1}(-\theta-\gamma_{0})$.

By (2.6), one has that

$\varphi_{1}(-\theta-\gamma_{0})=b_{1}\frac{g_{1}^{(2)}(-\theta-\gamma_{0})}{g_{1}^{(2)}(-\theta)}.$
(2.7)

Hence, combining with (2.6) and (2.7), we obtain

$\varphi_{1}(\theta)\frac{\sqrt{g_{1}^{(2)}(\theta+\gamma_{0})g_{1}^{(2)}(-\theta-\gamma_{0})}}{\sqrt{g_{1}^{(2)}(\theta)g_{1}^{(2)}(-\theta)}}=1.$

It follows that

$M_{1}^{(1)}(\theta,r)=\Big{(}\theta+\gamma_{0}+\tilde{\phi_{1}}(\theta)r,r\Big{)}+O(r^{2}).$

Finally, the commutativity of $G$ and $V_{1}^{(2)}$, that is
$GV_{1}^{(2)}=V_{1}^{(2)}G$, yields that $M_{1}^{(1)}$ is also reversible with
respect to $G$.

From (1.3), there is a function $g_{1}^{(1)}\in
C^{\infty}(\mathbb{T}^{m},\mathbb{R})$ satisfying

$\tilde{\phi}_{1}(\theta)=\gamma_{1}+g_{1}^{(1)}(\theta)-g_{1}^{(1)}(\theta+\gamma_{0}),$
(2.8)

where $\gamma_{1}=[\tilde{\phi}_{1}(\theta)]$. We define

$V_{1}^{(1)}(\theta,r)=(\theta-\frac{g_{1}^{(1)}(\theta)-g_{1}^{(1)}(-\theta)}{2}\cdot
r,r),$

then $M_{1}^{(1)}$ is changed into

$\displaystyle M_{1}(\theta,r)=$ $\displaystyle(V_{1}^{(1)})^{-1}\circ
M_{1}^{(1)}\circ V_{1}^{(1)}$ (2.9) $\displaystyle=$
$\displaystyle\Big{(}\theta+\gamma_{0}+\big{(}\tilde{\phi_{1}}-\frac{g_{1}^{(1)}(\theta)-g_{1}^{(1)}(-\theta)}{2}+\frac{g_{1}^{(1)}(\theta+\gamma_{0})-g_{1}^{(1)}(-\theta-\gamma_{0})}{2}\big{)}\cdot
r,r\Big{)}+O(r^{2}),$

Since $M_{1}^{(1)}$ is reversible with respect to the involution $G$, that is,

$M_{1}^{(1)}\circ G\circ M_{1}^{(1)}(\theta,r)=G(\theta,r)=(-\theta,r),$

the left side of the above equality can be written as

$\displaystyle\Big{(}-\theta-\tilde{\phi_{1}}(\theta)r+\tilde{\phi_{1}}(-\theta-\gamma_{0})r,r\Big{)}+O(r^{2}),$

which leads to

$\tilde{\phi_{1}}(\theta)=\tilde{\phi_{1}}(-\theta-\gamma_{0}).$ (2.10)

A substitution of (2.8), (2.10) and

$\tilde{\phi_{1}}(-\theta-\gamma_{0})=\gamma_{1}+g_{1}^{(1)}(-\theta-\gamma_{0})-g_{1}^{(1)}(-\theta)$

into (2.9) yields that

$M_{1}(\theta,r)=V_{1}^{-1}\circ M\circ
V_{1}=(\theta+\gamma_{0}+\gamma_{1}r,r)+O(r^{2}),$ (2.11)

where $V_{1}=V_{1}^{(2)}\circ V_{1}^{(1)}$. Due to the commutativity of $G$
and $V_{1}^{(1)}$, $M_{1}$ is reversible with respect to $G$.

Next, we expand mapping (2.11) to the $r^{2}$ terms with the form

$M_{1}(\theta,r)=(\theta+\gamma_{0}+\gamma_{1}r+\phi_{2}(\theta)r^{2},r+\varphi_{2}(\theta)r^{2})+O(r^{3}).$

Similar to the construction of $V_{1}^{(1)}$, we can find a constant $b_{2}$
and a function $g_{2}^{(2)}\in C^{\infty}(\mathbb{T}^{m},\mathbb{R})$, such
that

$\varphi_{2}(\theta)=b_{2}+g_{2}^{(2)}(\theta)-g_{2}^{(2)}(\theta+\gamma_{0}).$

Then under the transformation

$V_{2}^{(2)}(\theta,r)=\Big{(}\theta,r-\frac{g_{2}^{(2)}(\theta)+g_{2}^{(2)}(-\theta)}{2}\cdot
r^{2}\Big{)},$

we have

$\displaystyle M_{2}^{(1)}(\theta,r)=(V_{2}^{(2)})^{-1}\circ M_{1}\circ
V_{2}^{(2)}$
$\displaystyle=\Bigg{(}\theta+\gamma_{0}+\gamma_{1}r+\tilde{\phi_{2}}r^{2},r+\Big{(}\varphi_{2}(\theta)-\frac{g_{2}^{(2)}(\theta)+g_{2}^{(2)}(-\theta)}{2}+\frac{g_{2}^{(2)}(\theta+\gamma_{0})+g_{2}^{(2)}(-\theta-\gamma_{0})}{2}\Big{)}r^{2}\Bigg{)}+O(r^{3}),$

where
$\tilde{\phi_{2}}(\theta)=\phi_{2}(\theta)-\gamma_{1}\frac{g_{2}^{(2)}(\theta)+g_{2}^{(2)}(-\theta)}{2}$.
Since $M_{1}$ is reversible with respect to $G$, we obtain

$M_{2}^{(1)}(\theta,r)=\Big{(}\theta+\gamma_{0}+\gamma_{1}r+\tilde{\phi_{2}}(\theta)r^{2},r\Big{)}+O(r^{3}),$

and it is also reversible with respect to $G$.

Similarly, there exists a number $\gamma_{2}$ and a function $g_{2}^{(1)}\in
C^{\infty}(\mathbb{T}^{m},\mathbb{R})$, such that

$\tilde{\phi_{2}}(\theta)=\gamma_{2}+g_{2}^{(1)}(\theta)-g_{2}^{(1)}(\theta+\gamma_{0}).$

By the transformation

$V_{2}^{(1)}(\theta,r)=\Big{(}\theta-\frac{g_{2}^{(1)}(\theta)-g_{2}^{(1)}(-\theta)}{2}\cdot
r^{2},r\Big{)},$

we have

$M_{2}(\theta,r)=V_{2}^{-1}\circ M\circ
V_{2}=(\theta+\gamma_{0}+\gamma_{1}r+\gamma_{2}r^{2},r)+O(r^{3}),$ (2.12)

where $V_{2}=V_{1}\circ V_{2}^{(2)}\circ V_{2}^{(1)}$, and (2.12) is also
reversible with respect to $G$.

The rest may be deduced by analogy, therefore Lemma 2.5 is proved.
$\hfill\square$

We denote

$\hat{M}(\theta,r)=(\theta+\gamma_{0}+\sum_{k=1}^{N-1}\gamma_{k}r^{k},r)+O(r^{N})$

for short.

If some $\gamma_{k}\neq 0$, by Moser’s twist theorem [15], the mapping $M$ has
many invariant curves around the origin if the disturbance terms are
sufficiently small. Hence, we only restrict our attention to the existence of
invariant curves if $\gamma_{k}=0,k=1,2,\cdots$. By the Lemma 2.5, for
$N=s_{0}$ large, there exists a change of variables $V$, transforming (1.1)
into $M_{0}=V^{-1}\circ M\circ V$, that is,

$M_{0}:\begin{cases}\theta_{1}=\theta+\gamma_{0}+f_{s_{0}}(\theta,r),\\\
r_{1}=r+g_{s_{0}}(\theta,r).\end{cases}$ (2.13)

Moreover, $f_{s_{0}}(\theta,r)\sim O_{s_{0}},g_{s_{0}}(\theta,r)\sim
O_{s_{0}}$, and $M_{0}$ is reversible with respect to the involution
$G:(\theta,r)\mapsto(-\theta,r)$. In the rest part, we start with the
transformed map $M_{0}$.

##  3 The process of iteration

In this section, we will give the iteration theorem and its proof, which is of
vital importance in the approximation process of $M_{0}$ to its linear part.
In the classical iteration method, we use the twist condition and intersection
or area preserving properties to eliminate the mean values generated by
perturbations in the direction of angular and action variables. But for
reversible mapping $M_{0}$, there is no twist condition (all $\gamma_{k}=0$ ),
and no intersection or area preserving properties. How do we eliminate the
mean values? Fortunately, by Rüssmann’s method in [21], we use the uniqueness
of the formal normal forms to achieve this goal. Under the conditions of
Theorem 1.1, assume there is a formal change of variables transforming $M_{0}$
into the linearized form $I+\beta$.

###### Lemma 3.1

Assume the conditions of Theorem 1.1 hold, then we can derive a sequence of
changes of variables $\\{T_{n}\\}$, such that

$T_{n}^{-1}\circ M_{0}\circ T_{n}-(I+\beta)=O_{s_{n}},$ (3.1)

where $s_{n}=2^{\alpha+n}+1$, $\alpha$ is a sufficiently large positive
integer to be determined later and $T_{0}=I$. Moreover, for every $n$, the
transformed mapping $M_{n}:=T_{n}^{-1}\circ M_{0}\circ T_{n}$ is reversible
with respect to the involution $G:(\theta,r)\mapsto(-\theta,r)$.

Proof. For $n=0$, we choose $s_{0}=2^{\alpha}+1$. By (2.13), it is easy to see
(3.1) holds.

Suppose for all $n$, there exists $T_{n}$, such that

$T_{n}^{-1}\circ M_{0}\circ T_{n}-(I+\beta)\sim O_{s_{n}},$

then for $(n+1)-$th step, we need to find a change of variables $T_{n+1}$,
such that

$T_{n+1}^{-1}\circ M_{0}\circ T_{n+1}-(I+\beta)\sim O_{s_{n+1}}.$

Denote $T_{n}^{-1}\circ M_{0}\circ T_{n}=M_{n}$, and $T_{n+1}=T_{n}\circ\Delta
T_{n+1}$.

It is in a position to establish a transformation $\Delta T_{n+1}$, satisfying

$\Delta T_{n+1}^{-1}\circ M_{n}\circ\Delta T_{n+1}=M_{n+1}.$

In the sequel, we denote $s_{n}$ and $\Delta T_{n+1}$ by $s$ and $\Delta T$,
respectively.

Assume the change of variables $\Delta T$ has the form

$\Delta T:\begin{cases}\theta=\xi+u(\xi,\eta),\\\
r=\eta+v(\xi,\eta),\end{cases}$

and we denote $M_{n+1}$ as

$M_{n+1}:\begin{cases}\xi_{1}=\xi+\gamma_{0}+f_{\hat{s}}(\theta,r),\\\
\eta_{1}=\eta+g_{\hat{s}}(\theta,r).\end{cases}$

From $M_{n+1}=\Delta T^{-1}\circ M_{n}\circ\Delta T$, we have
$M_{n}\circ\Delta T=\Delta T\circ M_{n+1}$, it follows that

$\begin{cases}f_{\hat{s}}(\theta,r)=u(\xi,\eta)-u(\xi+\gamma_{0}+f_{\hat{s}},\eta+g_{\hat{s}})+f_{s}(\xi+u,\eta+v),\\\
g_{\hat{s}}(\theta,r)=v(\xi,\eta)-v(\xi+\gamma_{0}+f_{\hat{s}},\eta+g_{\hat{s}})+g_{s}(\xi+u,\eta+v),\end{cases}$
(3.2)

that is,

$\begin{cases}f_{\hat{s}}(\theta,r)=-(u(\xi+\gamma_{0},\eta)-u(\xi,\eta))+u(\xi+\gamma_{0},\eta)-u(\xi+\gamma_{0}+f_{\hat{s}},\eta+g_{\hat{s}})+f_{s}(\xi+u,\eta+v),\\\
g_{\hat{s}}(\theta,r)=-(v(\xi+\gamma_{0},\eta)-v(\xi,\eta))+v(\xi+\gamma_{0},\eta)-v(\xi+\gamma_{0}+f_{\hat{s}},\eta+g_{\hat{s}})+g_{s}(\xi+u,\eta+v).\end{cases}$
(3.3)

The important point to note here is that we need to ensure $M_{n+1}$ is also
reversible under transformation $\Delta T$. According to the concept of
reversible mappings, it implies the change of variable $\Delta T$ need to
commutes with the involution $G(\xi,\eta)=(-\xi,\eta)$. This requires
functions $u$ and $v$ satisfying

$u(-\xi,\eta)=-u(\xi,\eta),\quad\quad v(-\xi,\eta)=v(\xi,\eta).$ (3.4)

From Lemma 2.4, we try to decide the functions $u$ and $v$ from the following
modified difference equation:

$\begin{cases}u(\xi+\gamma_{0},\eta)-u(\xi,\eta)=\\{p_{s}\\}_{\mathbb{M}},\\\
v(\xi+\gamma_{0},\eta)-v(\xi,\eta)=\\{q_{s}\\}_{\mathbb{M}},\end{cases}$ (3.5)

where

$p_{s}(\xi,\eta)=\frac{1}{2}(f_{s}(\xi,\eta)+f_{s}(-\xi-\gamma_{0},\eta))\quad
q_{s}(\xi,\eta)=\frac{1}{2}(g_{s}(\xi,\eta)-g_{s}(-\xi-\gamma_{0},\eta)).$
(3.6)

It is easy to verify that
$p_{s}(-\xi-\gamma_{0},\eta)=p_{s}(\xi,\eta),q_{s}(-\xi-\gamma_{0},\eta)=-p_{s}(\xi,\eta)$.
$\\{p_{s}\\}_{\mathbb{M}}$ and $\\{q_{s}\\}_{\mathbb{M}}$ also possess this
property and
$[\\{p_{s}\\}_{\mathbb{M}}](\eta)=[\\{q_{s}\\}_{\mathbb{M}}](\eta)=0$. So the
functions $u$ and $v$ meet the condition (3.4).

Thus, (3.3) can be rewritten as

$\begin{cases}f_{\hat{s}}(\theta,r)=-\\{p_{s}\\}_{\mathbb{M}}+u(\xi+\gamma_{0},\eta)-u(\xi+\gamma_{0}+f_{\hat{s}},\eta+g_{\hat{s}})+f_{s}(\xi+u,\eta+v)+p_{s}(\xi,\eta)-p_{s}(\xi,\eta),\\\
g_{\hat{s}}(\theta,r)=-\\{q_{s}\\}_{\mathbb{M}}+v(\xi+\gamma_{0},\eta)-v(\xi+\gamma_{0}+f_{\hat{s}},\eta+g_{\hat{s}})+g_{s}(\xi+u,\eta+v)+q_{s}(\xi,\eta)-q_{s}(\xi,\eta).\end{cases}$
(3.7)

Since $f_{s}\sim O_{s},g_{s}\sim O_{s}$, it follows that $p_{s}\sim
O_{s},q_{s}\sim O_{s}$, and by (3.2) and (3.5), there are

$u\sim O_{s},v\sim O_{s},f_{\hat{s}}\sim O_{s},g_{\hat{s}}\sim O_{s},$

and

$\displaystyle
u(\xi+\gamma_{0}+f_{\hat{s}},\eta+g_{\hat{s}})-u(\xi+\gamma_{0},\eta)$
$\displaystyle=$ $\displaystyle D_{\xi}u(\xi+\gamma_{0},\eta)\cdot
f_{\hat{s}}+D_{\eta}u(\xi+\gamma_{0},\eta)\cdot g_{\hat{s}}$
$\displaystyle\sim$ $\displaystyle O_{2s-1}.$

In the same way, we have

$v(\xi+\gamma_{0}+f_{\hat{s}},\eta+g_{\hat{s}})-v(\xi+\gamma_{0},\eta)\sim
O_{2s-1},$ $f_{s}(\xi+u,\eta+v)-f_{s}(\xi,\eta)\sim O_{2s-1},$
$g_{s}(\xi+u,\eta+v)-g_{s}(\xi,\eta)\sim O_{2s-1},$
$f_{s}(-\xi-\gamma_{0}-f_{s},\eta+g_{s})-f_{s}(-\xi-\gamma_{0},\eta)\sim
O_{2s-1}.$

In the following, we will prove

$f_{s}(\xi+u,\eta+v)-p_{s}(\xi,\eta)\sim O_{2s-1},\quad
g_{s}(\xi+u,\eta+v)-q_{s}(\xi,\eta)\sim O_{2s-1}.$ (3.8)

From the definitions of $p_{s}$ and $q_{s}$, we have

$f_{s}(\xi+u,\eta+v)-p_{s}(\xi,\eta)=\frac{1}{2}(f_{s}(\xi+u,\eta+v)-f_{s}(\xi,\eta))+\frac{1}{2}(f_{s}(\xi+u,\eta+v)-f_{s}(-\xi-\gamma_{0},\eta)),$
$g_{s}(\xi+u,\eta+v)-q_{s}(\xi,\eta)\\\
=\frac{1}{2}(g_{s}(\xi+u,\eta+v)-g_{s}(\xi,\eta))+\frac{1}{2}(g_{s}(\xi+u,\eta+v)+g_{s}(-\xi-\gamma_{0},\eta)).$

Since the mapping $M_{n}$ is reversible, i.e., $M_{n}GM_{n}=G$, it follows
that

$f_{s}(-\xi-\gamma_{0}-f_{s},\eta+g_{s})-f_{s}(\xi,\eta)=0,\ \ \
g_{s}(-\xi-\gamma_{0}-f_{s},\eta+g_{s})+g_{s}(\xi,\eta)=0.$ (3.9)

Hence, by (3.9), we have

$\displaystyle f_{s}(\xi+u,\eta+v)-f_{s}(-\xi-\gamma_{0},\eta)$
$\displaystyle=f_{s}(\xi+u,\eta+v)-f_{s}(-\xi-\gamma_{0}-f_{s},\eta+g_{s})+f_{s}(-\xi-\gamma_{0}-f_{s},\eta+g_{s})-f_{s}(-\xi-\gamma_{0},\eta)$
$\displaystyle=f_{s}(\xi+u,\eta+v)-f_{s}(\xi,\eta)+f_{s}(-\xi-\gamma_{0}-f_{s},\eta+g_{s})-f_{s}(-\xi-\gamma_{0},\eta)$
$\displaystyle\sim O_{2s-1}.$

By the same reason, $g_{s}(\xi+u,\eta+v)-g_{s}(-\xi-\gamma_{0},\eta)\sim
O_{2s-1}$. As a consequence, (3.8) holds.

By (3.7), we have

$\begin{cases}f_{\hat{s}}(\xi,\eta)=\\{p_{s}\\}_{\mathbb{K}}+O_{2s-1},\\\
g_{\hat{s}}(\xi,\eta)=\\{q_{s}\\}_{\mathbb{K}}+O_{2s-1}.\end{cases}$

Then it follows that

$M_{n+1}=\Delta T^{-1}\circ M_{n}\circ\Delta
T=I+\beta+\\{H\\}_{\mathbb{K}\times\mathbb{K}}+O_{2s-1},$ (3.10)

where $H=(p_{s},q_{s}),\hat{s}=2s-1$.

For (3.5) and (3.10), nothing changes if we replace $H$ by

$H^{*}=(p_{s}^{*},q_{s}^{*})=\sum_{k=s}^{2s-2}H_{k},$

that is,

$(u(\xi,\eta),v(\xi,\eta))=R^{-1}\\{H^{*}\\}_{\mathbb{M}\times\mathbb{M}},$

where $Ru=u(\xi+\gamma_{0},\eta)-u(\xi,\eta)$, instead of (3.5), and

$M_{2s-1}=\Delta T^{-1}\circ M_{s}\circ\Delta
T=I+\beta+\\{H^{*}\\}_{\mathbb{K}\times\mathbb{K}}+O_{2s-1},$ (3.11)

instead of (3.10).

In the following, we show that $\\{H^{*}\\}_{\mathbb{K}\times\mathbb{K}}=0$.
Otherwise, if $\\{H^{*}\\}_{\mathbb{K}\times\mathbb{K}}=K_{m}+O_{m+1}$,
$K_{m}\in K^{m}\times K^{m},K_{m}\neq 0,s\leq m\leq 2s-2$. Then from (3.11),
we have

$\Delta T^{-1}\circ M_{s}\circ\Delta T=I+\beta+K_{m}+O_{m+1}+O_{2s-1},$

which contradicts our assumption for $M_{0}$ and Lemma 2.3. Hence,
$\\{H^{*}\\}_{\mathbb{K}\times\mathbb{K}}=0$. As a consequence,

$f_{2s-1}(\xi,\eta)\sim O_{2s-1},\quad g_{2s-1}(\xi,\eta)\sim O_{2s-1},$

and there exists a $T_{n+1}=T_{n}\circ\Delta T_{n+1}$ such that (3.1) holds.
Therefore the Lemma is proved.$\hfill\square$

Now, we introduce the iteration lemma, which is used infinitely times to
transforming $M_{0}$ close to $I+\beta$. Since the iteration lemma is one step
in the iteration process, we write $s$ instead of $s_{n}$ and $\widetilde{M}$
instead of mapping $M_{n}$ in (3.1).

Set several complex domains

$D=\\{(\theta,r):|\Im\theta|<t,|r|<\rho\\},$
$B=\\{(\theta,r):|\Im\theta|<\tau,|r|<\varrho\\},$
$B^{(k)}=\\{(\theta,r):|\Im\theta|<t-\frac{k(t-\tau)}{4},|r|<\rho-\frac{k(\rho-\varrho)}{4}\\},k=1,2,3,$

with $\tau<t<1,\varrho<\rho<1$. It is easy to get $B\subset B^{(3)}\subset
B^{(2)}\subset B^{(1)}\subset D$. In what follows, we denote the norm
$|f|_{D}=\sup_{(\theta,r)\in D}|f(\theta,r)|$.

###### Lemma 3.2

(Iteration Lemma) Suppose that the real analytic mapping

$\widetilde{M}:\begin{cases}\theta_{1}=\theta+\gamma_{0}+f_{s}(\theta,r),\\\
r_{1}=r+g_{s}(\theta,r),\end{cases}$ (3.12)

where

$f_{s}(\theta,r)=\sum_{k\geq
s}^{\infty}\sum_{j\in\mathbb{Z}^{l}}f_{kj}e^{i\langle
j,\omega\rangle\theta}r^{k},\quad g_{s}(\theta,r)=\sum_{k\geq
s}^{\infty}\sum_{j\in\mathbb{Z}^{l}}g_{kj}e^{i\langle
j,\omega\rangle\theta}r^{k},$

satisfy

: $\widetilde{M}$ is reversible with respect to
$G:(\theta,r)\rightarrow(-\theta,r)$, and the constant $\gamma_{0}$ satisfies
(1.3);

:
$|f_{s}|_{D}+|g_{s}|_{D}<d,d<\min\\{\frac{t-\tau}{4},\frac{\rho-\varrho}{4}\\}$;

:
$\nu=c_{7}d(t-\tau)^{-m-\sigma}(\rho-\varrho)^{-1}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho})<\min\\{\frac{t-\tau}{4},\frac{\rho-\varrho}{4},\frac{1}{5}\\}$,
where $c_{7}$ and $c_{i}(i=1,2,3,4,5,6,8,9)$ are positive constants, which
will determined later, and depend only on $c_{0},\omega,\sigma$.

Then there exists a change of variables $\Delta T$ such that $\widetilde{M}$
is transformed into $\overline{M}=\Delta T^{-1}\circ\widetilde{M}\circ\Delta
T$ with the form

$\overline{M}:\begin{cases}\xi_{1}=\xi+\gamma_{0}+f_{2s-1}(\xi,\eta),\\\
\eta_{1}=\eta+g_{2s-1}(\xi,\eta),\end{cases}$ (3.13)

and $\overline{M}$ is reversible with respect to $G$. Moreover, the following
estimates hold:

$|u|_{B^{(1)}}+|v|_{B^{(1)}}<c_{2}d(t-\tau)^{-m-\sigma}(\rho-\varrho)^{-1},$
(3.14)
$|u_{\xi}|_{B^{(1)}}+|u_{\eta}|_{B^{(1)}}+|v_{\xi}|_{B^{(1)}}+|v_{\eta}|_{B^{(1)}}<c_{3}d(t-\tau)^{-m-\sigma}(\rho-\varrho)^{-1}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho}),$
(3.15)
$|f_{2s-1}(\xi,\eta)|_{B}+|g_{2s-1}(\xi,\eta)|_{B}\leq\frac{c_{6}(\rho-\varrho)^{-1}(de^{\frac{1}{2}(-2s-1)(\rho-\varrho)}+d^{2}(t-\tau)^{-m-\sigma}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho}))}{1-\nu}.$
(3.16)

Proof. Firstly, the existence of such a change

$\Delta T:\begin{cases}\theta=\xi+u(\xi,\eta),\\\
r=\eta+v(\xi,\eta)\end{cases}$ (3.17)

is guaranteed by Lemma 3.1. Secondly, we give some estimates of $\Delta T$ and
$\overline{M}$.

Suppose $u,v$ in (3.17) and $p_{s}$, $q_{s}$ in (3.6) have expansions of the
type:

$u(\xi,\eta)=\sum_{k=s}^{2s-2}\sum_{j\in\mathbb{Z}^{m}}u_{kj}e^{i\langle
j,\omega\rangle\xi}\eta^{k},\quad
v(\xi,\eta)=\sum_{k=s}^{2s-2}\sum_{j\in\mathbb{Z}^{m}}v_{kj}e^{i\langle
j,\omega\rangle\xi}\eta^{k},$ (3.18)
$p_{s}(\xi,\eta)=\sum_{k=s}^{+\infty}\sum_{j\in\mathbb{Z}^{m}}p_{kj}e^{i\langle
j,\omega\rangle\xi}\eta^{k},\quad
q_{s}(\xi,\eta)=\sum_{k=s}^{+\infty}\sum_{j\in\mathbb{Z}^{m}}q_{kj}e^{i\langle
j,\omega\rangle\xi}\eta^{k}.$

Then,

$\\{p_{s}^{*}\\}_{\mathbb{M}}=\sum_{k=s}^{2s-2}\sum_{j\in\mathbb{Z}^{m}\setminus\\{0\\}}p_{kj}e^{i\langle
j,\omega\rangle\xi}\eta^{k},\quad\\{q_{s}^{*}\\}_{\mathbb{M}}=\sum_{k=s}^{2s-2}\sum_{j\in\mathbb{Z}^{m}\setminus\\{0\\}}q_{kj}e^{i\langle
j,\omega\rangle\xi}\eta^{k}.$ (3.19)

Substituting (3.18) and (3.19) into (3.5), we have

$u_{kj}(e^{i\langle j,\omega\rangle\gamma_{0}}-1)=p_{kj},\quad
v_{kj}(e^{i\langle j,\omega\rangle\gamma_{0}}-1)=q_{kj},\quad s\leq k\leq
2s-1,\ j\in\mathbb{Z}^{m}\setminus\\{0\\},$

hence,

$u_{kj}=\frac{p_{kj}}{e^{i\langle j,\omega\rangle\gamma_{0}}-1},\quad
v_{kj}=\frac{q_{kj}}{e^{i\langle j,\omega\rangle\gamma_{0}}-1},\quad s\leq
k\leq 2s-1,\ j\in\mathbb{Z}^{m}\setminus\\{0\\},$

and $u_{k0}=0,v_{k0}=0$.

Since the constant $\gamma_{0}$ and $\omega$ satisfy (1.3), it means that

$|e^{i\langle j,\omega\rangle\gamma_{0}}-1|\geq\frac{4c_{0}}{|j|^{\sigma}}.$

Denote $p_{s}=\sum_{k=s}^{+\infty}p_{k}(\xi)\eta^{k}$,
$p_{k}(\xi)=\sum_{j\in\mathbb{Z}^{m}}p_{kj}e^{i\langle j,\omega\rangle\xi}$,
then

$p_{s}^{*}=\sum_{k=s}^{2s-2}p_{k}(\xi)\eta^{k},\quad
p_{k}(\xi)=\sum_{j\in\mathbb{Z}^{m}}p_{kj}e^{i\langle j,\omega\rangle\xi}.$

By Cauchy’s estimate and the analyticity of $p_{s}$ and $q_{s}$, we have

$|p_{k}|_{D}\leq d\rho^{-k},\ \ |p_{kj}|_{D}\leq d\rho^{-k}e^{-|j||\omega|t}.$

Denote a narrower strip

$D^{*}=\\{(\xi,\eta):|\Im\xi|<t-\delta_{1},,|\eta|<\rho-\delta_{2}\\},$

where $\delta_{1}=\frac{t-\tau}{5},\delta_{2}=\frac{\rho-\varrho}{5}$, and it
is easy to prove $B\subset B^{(3)}\subset B^{(2)}\subset B^{(1)}\subset
D^{*}\subset D$.

Then we have the estimate

$\displaystyle|u|_{D^{*}}=$
$\displaystyle\left|\sum_{k=s}^{2s-2}\sum_{j\in\mathbb{Z}^{m}\setminus\\{0\\}}\frac{p_{kj}}{e^{i\langle
j,\omega\rangle\gamma_{0}}-1}e^{i\langle j,\omega\rangle\xi}\eta^{k}\right|$
$\displaystyle\leq$
$\displaystyle\sum_{k=s}^{2s-2}\sum_{j\in\mathbb{Z}^{m}\setminus\\{0\\}}\frac{|j|^{\sigma}}{4c_{0}}d\rho^{-k}e^{-|j||\omega|t}e^{|j||\omega|(t-\delta_{1})}(\rho-\delta_{2})^{k}$
$\displaystyle=$
$\displaystyle\frac{d}{4c_{0}}\sum_{k=s}^{2s-2}(1-\frac{\delta_{2}}{\rho})^{k}\sum_{j\in\mathbb{Z}^{m}\setminus\\{0\\}}|j|^{\sigma}e^{-|j||\omega|\delta_{1}}$
$\displaystyle<$ $\displaystyle
c_{1}d\delta_{1}^{-m-\sigma}\frac{\rho}{\delta_{2}}<\frac{1}{2}c_{2}d(t-\tau)^{-m-\sigma}(\rho-\varrho)^{-1},$

where $c_{2}>10c_{1}$, and $c_{1},c_{2}$ are positive constants depending on
$c_{0},\sigma,\omega$. In a similar way,

$|v|_{D^{*}}<\frac{1}{2}c_{2}d(t-\tau)^{-m-\sigma}(\rho-\varrho)^{-1}.$

From the above discussions, we have (3.14), and by Cauchy’s estimate, we get
(3.15):

$|u_{\xi}|_{D^{*}}+|u_{\eta}|_{D^{*}}+|v_{\xi}|_{D^{*}}+|v_{\eta}|_{D^{*}}<c_{3}d(t-\tau)^{-m-\sigma}(\rho-\varrho)^{-1}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho}),$

where $c_{3}>c_{2}$ is a positive constant.

The last step is to estimate $f_{2s-1}$ and $g_{2s-1}$, which satisfy the
equation (3.3), that is

$\begin{cases}f_{2s-1}(\xi,\eta)=-\\{p_{s}^{*}\\}_{\mathbb{M}}+u(\xi+\gamma_{0},\eta)-u(\xi+\gamma_{0}+f_{2s-1},\eta+g_{2s-1})+f_{s}(\xi+u,\eta+v),\\\
g_{2s-1}(\xi,\eta)=-\\{q_{s}^{*}\\}_{\mathbb{M}}+v(\xi+\gamma_{0},\eta)-v(\xi+\gamma_{0}+f_{2s-1},\eta+g_{2s-1})+g_{s}(\xi+u,\eta+v).\end{cases}$

Since

$\displaystyle f_{s}(\xi+u,\eta+v)-\\{p_{s}^{*}\\}_{\mathbb{M}}$
$\displaystyle=\big{(}f_{s}(\xi+u,\eta+v)-f_{s}(\xi,\eta)\big{)}+\big{(}f_{s}(\xi,\eta)-p_{s}(\xi,\eta)\big{)}+\big{(}p_{s}(\xi,\eta)-p_{s}^{*}(\xi,\eta)\big{)}+\big{(}p_{s}^{*}(\xi,\eta)-\\{p_{s}^{*}\\}_{\mathbb{M}}\big{)},$

we divide the estimate of
$|f_{s}(\xi+u,\eta+v)-\\{p_{s}^{*}\\}_{\mathbb{M}}|_{B}$ into four parts.

By (3.14), it follows that

$\displaystyle|f_{s}(\xi+u,\eta+v)-f_{s}(\xi,\eta)|_{B}$
$\displaystyle\leq|D_{\xi}f_{s}|\cdot|u|+|D_{\eta}f_{s}|\cdot|v|$
$\displaystyle\leq
c_{4}d^{2}(t-\tau)^{-m-\sigma}(\rho-\varrho)^{-1}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho}),$

where $c_{4}>c_{3}$ is a positive constant. Combining with (3.6) and (3.9), we
have

$\displaystyle|f_{s}(\xi,\eta)-p_{s}(\xi,\eta)|_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}|f_{s}(\xi,\eta)-f_{s}(-\xi-\gamma_{0},\eta)|$
$\displaystyle=$
$\displaystyle\frac{1}{2}|f_{s}(-\xi-\gamma_{0}-f_{s},\eta+g_{s})-f_{s}(-\xi-\gamma_{0},\eta)|$
$\displaystyle\leq$ $\displaystyle
c_{4}d^{2}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho}).$

In view of $p_{s}=\sum_{k=s}^{+\infty}p_{k}(\xi)\eta^{k}$ and

$|p_{k}|_{B^{(2)}}\leq|p_{s}|_{D}\left(\frac{\rho+\varrho}{2}\right)^{-k}<d(\frac{\rho+\varrho}{2})^{-k},$

it yields

$\displaystyle|p_{s}(\xi,\eta)-p_{s}^{*}(\xi,\eta)|_{B}$ $\displaystyle=$
$\displaystyle\left|\sum_{k=2s-1}^{+\infty}p_{k}(\xi)\eta^{k}\right|\leq\sum_{k=2s-1}^{+\infty}|p_{k}|_{B^{(2)}}|\varrho|^{k}$
$\displaystyle\leq$
$\displaystyle\sum_{k=2s-1}^{+\infty}d\left(\frac{\rho+\varrho}{2}\right)^{-k}|\varrho|^{k}$
$\displaystyle=$ $\displaystyle
d\left(1-\frac{\rho-\varrho}{\rho+\varrho}\right)^{2s-1}\sum_{k=0}^{+\infty}(1-\frac{\rho-\varrho}{\rho+\varrho})^{k}$
$\displaystyle\leq$ $\displaystyle
c_{5}de^{-\frac{1}{2}(2s-1)(\rho-\varrho)}(\rho-\varrho)^{-1},$

where $c_{5}>c_{4}$ is a positive constant.

From these estimates, we have

$|f_{s}(\xi+u,\eta+v)-\\{p_{s}^{*}\\}_{\mathbb{M}}|_{B}<\frac{1}{2}c_{6}\Big{(}\rho-\varrho)^{-1}(de^{\frac{1}{2}(-2s-1)(\rho-\varrho)}+d^{2}(t-\tau)^{-m-\sigma}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho})\Big{)},$

where $c_{6}>2c_{5}$ is a positive constant.

As a consequence,

$\displaystyle|f_{2s-1}(\xi,\eta)|_{B}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}c_{6}(\rho-\varrho)^{-1}(de^{\frac{1}{2}(-2s-1)(\rho-\varrho)}+d^{2}(t-\tau)^{-m-\sigma}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho}))$
$\displaystyle+|u(\xi+\gamma_{0},\eta)-u(\xi+\gamma_{0}+f_{2s-1},\eta+g_{2s-1})|$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}c_{6}(\rho-\varrho)^{-1}(de^{\frac{1}{2}(-2s-1)(\rho-\varrho)}+d^{2}(t-\tau)^{-m-\sigma}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho}))$
$\displaystyle+\frac{1}{2}c_{7}d(t-\tau)^{-m-\sigma}(\rho-\varrho)^{-1}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho})(|f_{2s-1}|_{B}+|g_{2s-1}|_{B}),$

where $c_{7}$ is a positive constant. Similarly, there is same estimate for
$|g_{2s-1}(\xi,\eta)|_{B}$.

From the above discussion, we obtain

$|f_{2s-1}(\xi,\eta)|_{B}+|g_{2s-1}(\xi,\eta)|_{B}\leq\frac{c_{6}(\rho-\varrho)^{-1}(de^{\frac{1}{2}(-2s-1)(\rho-\varrho)}+d^{2}(t-\tau)^{-m-\sigma}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho}))}{1-\nu},$

by denoting
$\nu=c_{7}d(t-\tau)^{-m-\sigma}(\rho-\varrho)^{-1}(\frac{1}{t-\tau}+\frac{1}{\rho-\varrho})$.
The proof is completed.$\hfill\square$

Obviously, applying the iteration lemma to $M_{n}$ in (3.1), we obtain
$M_{n+1}$ with estimates (3.14)-(3.16), so that iteration process can
continue. The specific steps and the convergence of composed mappings are
shown in the next section.

##  4 Proof of Theorem 1.1

Set some sequences of variables and domains:

$t_{n}=\frac{t_{0}}{2}\big{(}1+(\frac{2}{3})^{n}\big{)},t_{0}<1,t=t_{n},\tau=t_{n+1},$
$\rho_{n}=\frac{t_{0}}{2}\big{(}1+(\frac{2}{3})^{n}\big{)},\rho_{0}<1,\rho=\rho_{n},\varrho=\rho_{n+1},$
$d_{n+1}=(\frac{3}{2})^{n}d_{n}^{\frac{4}{3}},d=d_{0}<1,$
$e_{n}=(\frac{3}{2})^{3n+9}d_{n},e_{n+1}=e_{n}^{\frac{4}{3}},$
$D_{n}=\\{(\xi,\eta):|\Im\xi|<t_{n},|\eta|<\rho_{n}\\}.$

In this section, we will verify that there exists a convergent change of
variables, transforming (1.1) into (1.4). For this purpose, we need to prove
for every $n$, there is a transformation $T_{n}$ such that $M_{0}$ is
transformed into $M_{n}$,

$M_{n}:\begin{cases}\xi_{1}=\xi+\gamma_{0}+f_{s_{n}}(\xi,\eta),\\\
\eta_{1}=\eta+g_{s_{n}}(\xi,\eta),\end{cases}$

and

$|f_{s_{n}}|_{D_{n}}+|g_{s_{n}}|_{D_{n}}<d_{n}.$ (4.1)

By Lemma 3.2, the existence of transformation $T_{n}$ is obtained. Thus we
just check (4.1) for all $n$.

When $n=0$, $|f_{s_{0}}|_{D_{0}}+|g_{s_{0}}|_{D_{0}}<d=d_{0}$. Supposing for
all $n$, the nonlinear part of $M_{n}$ satisfies (4.1), it is in a position to
prove that

$|f_{s_{n+1}}|_{D_{n+1}}+|g_{s_{n+1}}|_{D_{n+1}}<d_{n+1}.$

Firstly, we have to guarantee the conditions of Lemma 3.2, i.e.,

$d_{n}<\min\\{\frac{t_{n}-t_{n+1}}{4},\frac{\rho_{n}-\rho_{n+1}}{4}\\},$
$\nu=c_{7}d_{n}(t_{n}-t_{n+1})^{-m-\sigma}(\rho_{n}-\rho_{n+1})^{-1}(\frac{1}{t_{n}-t_{n+1}}+\frac{1}{\rho_{n}-\rho_{n+1}})<\min\\{\frac{t_{n}-t_{n+1}}{4},\frac{\rho_{n}-\rho_{n+1}}{4},\frac{1}{5}\\},$

which means that

$(\frac{3}{2})^{n}d_{n}<\min\\{\frac{t_{0}}{24},\frac{\rho_{0}}{24}\\},$

and

$c_{7}d_{n}(\frac{t_{0}}{6})^{-m-\sigma}(\frac{\rho_{0}}{6})^{-1}\Big{(}(\frac{t_{0}}{6})^{-1}+(\frac{\rho_{0}}{6})^{-1}\Big{)}(\frac{3}{2})^{n(m+\sigma+2)}<\min\\{\frac{t_{0}}{24},\frac{\rho_{0}}{24},\frac{1}{5}\\}.$

From the definition of $d_{n}$, we have
$d_{n}=(\frac{2}{3})^{3n+9}\Big{(}(\frac{3}{2})^{9}d_{0}\Big{)}^{(\frac{4}{3})^{n}}$.
Obviously, $d_{0}$ depends on $t_{0},\rho_{0},m,\sigma$ and $c_{0}$, thus we
can choose $\alpha$ large enough such that $d_{0}$ is sufficiently small,
finally all above inequalities hold. By Lemma 3.2, we derive

$\displaystyle|f_{s_{n+1}}(\xi,\eta)|_{D_{n+1}}+|g_{s_{n+1}}(\xi,\eta)|_{D_{n+1}}$
$\displaystyle\leq\frac{c_{6}(\rho_{n}-\rho_{n+1})^{-1}(d_{n}e^{\frac{1}{2}(2s_{n}-1)(\rho_{n}-\rho_{n+1})}+(d_{n})^{2}(t_{n}-t_{n+1})^{-m-\sigma}(\frac{1}{t_{n}-t_{n+1}}+\frac{1}{\rho_{n}-\rho_{n+1}}))}{1-\nu}.$

On the one hand, we will prove the inequality

$\frac{c_{6}(\rho_{n}-\rho_{n+1})^{-1}d_{n}e^{-\frac{1}{2}(2s_{n}-1)(\rho_{n}-\rho_{n+1})}}{1-\nu}<\frac{1}{2}d_{n+1}.$

Since

$\displaystyle\frac{c_{6}(\rho_{n}-\rho_{n+1})^{-1}d_{n}e^{-\frac{1}{2}(2s_{n}-1)(\rho_{n}-\rho_{n+1})}}{1-\nu}$
$\displaystyle<\frac{5}{4}c_{6}(\frac{\rho_{0}}{6})^{-1}(\frac{3}{2})^{n}e^{-\frac{\rho_{0}}{12}(\frac{2}{3})^{n}(2s_{n}-1)}d_{n}$
$\displaystyle=\frac{5}{4}c_{6}(\frac{\rho_{0}}{6})^{-1}e^{-\frac{\rho_{0}}{12}(\frac{2}{3})^{n}(2s_{n}-1)}d_{n}^{-\frac{1}{3}}(\frac{3}{2})^{n}d_{n}^{\frac{4}{3}}$
$\displaystyle<\frac{1}{2}\Big{(}c_{8}(\frac{\rho_{0}}{6})^{-1}e^{-\frac{\rho_{0}}{12}(\frac{2}{3})^{n}(2s_{n}-1)}d_{n}^{-\frac{1}{3}}\Big{)}d_{n+1},$

where $c_{8}>\frac{5}{2}c_{6}$ is a positive constant, we will only verify the
term
$c_{8}(\frac{\rho_{0}}{6})^{-1}e^{-\frac{\rho_{0}}{12}(\frac{2}{3})^{n}(2s_{n}-1)}d_{n}^{-\frac{1}{3}}<1$.
In fact

$\displaystyle
c_{8}(\frac{\rho_{0}}{6})^{-1}e^{-\frac{\rho_{0}}{12}(\frac{2}{3})^{n}(2s_{n}-1)}d_{n}^{-\frac{1}{3}}$
$\displaystyle<\frac{6c_{8}}{\rho_{0}}(e^{-\frac{\rho_{0}}{12}2^{\alpha}})^{(\frac{4}{3})^{n}}\Big{(}\big{(}(\frac{3}{2})^{9}d_{0}\big{)}^{(\frac{4}{3})^{n}}(\frac{2}{3})^{3n+9}\Big{)}^{-\frac{1}{3}}$
$\displaystyle<\Big{(}(\frac{6c_{8}}{\rho_{0}})^{3}e^{-\frac{\rho_{0}}{12}2^{\alpha}}\Big{)}^{\frac{1}{3}(\frac{4}{3})^{n}}\Big{(}(\frac{3}{2})^{9}d_{0}\Big{)}^{-\frac{1}{3}(\frac{4}{3})^{n}}\big{(}(\frac{3}{2})^{3n+9}\big{)}^{\frac{1}{3}}$
$\displaystyle<\Big{(}(\frac{6c_{8}}{\rho_{0}})^{3}e^{-\frac{\rho_{0}}{12}2^{\alpha}}\Big{)}^{\frac{1}{3}(\frac{4}{3})^{n}}\Big{(}(\frac{3}{2})^{9}d_{0}\Big{)}^{-\frac{1}{3}(\frac{4}{3})^{n}}\big{(}(\frac{3}{2})^{9}\big{)}^{\frac{1}{3}(\frac{4}{3})^{n}}$
$\displaystyle=\Big{(}(\frac{6c_{8}}{\rho_{0}})^{3}(\frac{3}{2})^{9}\frac{1}{e^{\frac{\rho_{0}}{12}2^{\alpha}}(\frac{3}{2})^{9}d_{0}}\Big{)}^{\frac{1}{3}(\frac{4}{3})^{n}}$
$\displaystyle=\Big{(}(\frac{6c_{8}}{\rho_{0}})^{3}\frac{1}{e^{\frac{\rho_{0}}{12}2^{\alpha}}d_{0}}\Big{)}^{\frac{1}{3}(\frac{4}{3})^{n}}.$

Therefore when
$(\frac{6c_{8}}{\rho_{0}})^{3}\frac{1}{e^{\frac{\rho_{0}}{12}2^{\alpha}}d_{0}}<1$,
i.e.,

$\alpha>\log_{2}^{\frac{12}{\rho_{0}}\ln{\frac{216c_{9}}{\rho_{0}^{3}d_{0}}}},$
(4.2)

we have

$c_{8}(\frac{\rho_{0}}{6})^{-1}e^{-\frac{\rho_{0}}{12}(\frac{2}{3})^{n}(2s_{n}-1)}d_{n}^{-\frac{1}{3}}<1,$

and

$\frac{c_{6}(\rho_{n}-\rho_{n+1})^{-1}d_{n}e^{-\frac{1}{2}(2s_{n}-1)(\rho_{n}-\rho_{n+1})}}{1-\nu}<\frac{1}{2}d_{n+1},$

where $c_{9}=c_{8}^{3}$.

On the other hand, we will prove the inequality

$\frac{c_{6}(\rho_{n}-\rho_{n+1})^{-1}d_{n}^{2}(t_{n}-t_{n+1})^{-m-\sigma}(\frac{1}{t_{n}-t_{n+1}}+\frac{1}{\rho_{n}-\rho_{n+1}})}{1-\nu}<\frac{1}{2}d_{n+1}.$

Due to

$\displaystyle\frac{c_{6}(\rho_{n}-\rho_{n+1})^{-1}d_{n}^{2}(t_{n}-t_{n+1})^{-m-\sigma}(\frac{1}{t_{n}-t_{n+1}}+\frac{1}{\rho_{n}-\rho_{n+1}})}{1-\nu}$
$\displaystyle<\frac{5}{4}c_{6}\frac{36}{\rho_{0}}(\frac{6}{t_{0}})^{m+\sigma}(\frac{1}{\rho_{0}}+\frac{1}{t_{0}})(\frac{3}{2})^{n(m+\sigma+1)}d_{n}^{\frac{2}{3}}(\frac{3}{2})^{n}d_{n}^{\frac{4}{3}}$
$\displaystyle<\frac{1}{2}\Big{(}\frac{36c_{8}}{\rho_{0}}(\frac{6}{t_{0}})^{m+\sigma}(\frac{1}{\rho_{0}}+\frac{1}{t_{0}})(\frac{3}{2})^{n(m+\sigma+1)}d_{n}^{\frac{2}{3}}\Big{)}d_{n+1},$

we turn to prove

$\frac{36c_{8}}{\rho_{0}}(\frac{6}{t_{0}})^{m+\sigma}(\frac{1}{\rho_{0}}+\frac{1}{t_{0}})(\frac{3}{2})^{n(m+\sigma+1)}d_{n}^{\frac{2}{3}}<1.$
(4.3)

The estimate of the left hand in (4.3) follows

$\displaystyle\frac{36c_{8}}{\rho_{0}}(\frac{6}{t_{0}})^{m+\sigma}(\frac{1}{\rho_{0}}+\frac{1}{t_{0}})(\frac{3}{2})^{n(m+\sigma+1)}d_{n}^{\frac{2}{3}}$
$\displaystyle=\frac{36c_{8}}{\rho_{0}}(\frac{6}{t_{0}})^{m+\sigma}(\frac{1}{\rho_{0}}+\frac{1}{t_{0}})(\frac{3}{2})^{n(m+\sigma+1)}\Big{(}\big{(}(\frac{3}{2})^{9}d_{0}\big{)}^{(\frac{4}{3})^{n}}(\frac{2}{3})^{3n+9}\Big{)}^{\frac{2}{3}}$
$\displaystyle<\frac{36c_{8}}{\rho_{0}}(\frac{6}{t_{0}})^{m+\sigma}(\frac{1}{\rho_{0}}+\frac{1}{t_{0}})(\frac{3}{2})^{n(m+\sigma-1)-6}\big{(}(\frac{3}{2})^{6}d_{0}^{\frac{2}{3}}\big{)}^{(\frac{4}{3})^{n}}$
$\displaystyle<\Big{(}\frac{36c_{8}}{\rho_{0}}(\frac{6}{t_{0}})^{m+\sigma}(\frac{1}{\rho_{0}}+\frac{1}{t_{0}})(\frac{3}{2})^{(m+\sigma-1)}(\frac{3}{2})^{6}d_{0}^{\frac{2}{3}}\Big{)}^{(\frac{4}{3})^{n}}$
$\displaystyle=\Big{(}\frac{36c_{8}}{\rho_{0}}(\frac{6}{t_{0}})^{m+\sigma}(\frac{1}{\rho_{0}}+\frac{1}{t_{0}})(\frac{3}{2})^{(m+\sigma+5)}d_{0}^{\frac{2}{3}}\Big{)}^{(\frac{4}{3})^{n}}.$

Thus, by the choice of

$d_{0}<[\frac{\rho_{0}}{36c_{8}}(\frac{t_{0}}{6})^{m+\sigma}(\frac{1}{\rho_{0}}+\frac{1}{t_{0}})^{-1}(\frac{2}{3})^{(m+\sigma+5)}]^{\frac{3}{2}},$
(4.4)

(4.3) is established.

From the above discussion, we conclude

$|f_{s_{n+1}}(\xi,\eta)|_{D_{n+1}}+|g_{s_{n+1}}(\xi,\eta)|_{D_{n+1}}<d_{n+1},$

if (4.2) and (4.4) hold. Hence we get the inequality (4.1), which completes
the induction.

For one thing, the change of variables $T_{n}$ can be written as
$T_{n}=T_{0}\circ\Delta T_{1}\circ\Delta T_{2}\circ\cdots\Delta T_{n}$, then
$T_{n+1}=T_{n}\circ\Delta T_{n+1}$, and the transformation $T_{n+1}$ can be
expressed as

$T_{n+1}:\begin{cases}\theta=\xi+u_{n+1}(\xi,\eta)\\\
r=\eta+v_{n+1}(\xi,\eta),\end{cases}$

where

$u_{n+1}=u_{0}+u_{1}+...+u_{n},\quad\quad v_{n+1}=v_{0}+v_{1}+...+v_{n}.$
(4.5)

The convergence of the transformation sequence $\\{T_{n+1}\\}$ is decided by
their nonlinear parts (4.5).

By Lemma 3.2, we have

$\displaystyle|u_{n}|_{D_{n}}+|v_{n}|_{D_{n}}<c_{2}d_{n}(t_{n}-t_{n+1})^{-m-\sigma}(\rho_{n}-\rho_{n+1})^{-1}$
$\displaystyle=c_{2}(\frac{6}{t_{0}})^{m+\sigma}\frac{6}{\rho_{0}}(\frac{3}{2})^{n(m+\sigma+1)}(\frac{2}{3})^{3n+9}\big{(}(\frac{3}{2})^{9}d_{0}\big{)}^{(\frac{4}{3})^{n}}.$

When $d_{0}<(\frac{2}{3})^{m+\sigma+7}$, there is
$|u_{n}|_{D_{n}}+|v_{n}|_{D_{n}}\rightarrow 0$ as $n\rightarrow\infty$. It
follows that the sequence $\\{u_{n+1}\\}$ and $\\{v_{n+1}\\}$ are uniformly
bounded in
$D_{\infty}=\\{(\xi,\eta):|\Im\xi|<\frac{t_{0}}{2},|\eta|<\frac{\rho_{0}}{2}\\}$.
Hence, one can choose a subsequence of $\\{T_{n}\\}$, which converges to a
transformation $T$ in $D_{\infty}$.

For another, since the nonlinear parts of $M_{n}$ satisfies
$|f_{n}|_{D_{n}}+|g_{n}|_{D_{n}}<d_{n}\rightarrow 0$ as $n\rightarrow\infty$,
it implies that the mapping $M_{n}={T_{n}}^{-1}\circ M_{0}\circ T_{n}$ tends
to a linearized normal form (1.5) in $D_{\infty}$.

In conclusion, there is a convergent transformation $V\circ T$, where $V$ is
defined in Lemma 2.5, such that the mapping (1.1) is reduced to the formal
normal form (1.5).

##  5 Application

In this section, we will apply Theorem 1.1 to the equation

$x^{\prime\prime}+\varphi(x)f(x^{\prime})+\omega^{2}x+g(x)=p(t).$ (5.1)

Suppose that

$(H_{1})$: functions $\varphi,f,g$ and $p$ are real analytic in $x,x^{\prime}$
and $t$;

$(H_{2})$: $f$ and $p$ are even functions, $p(t+2\pi)=p(t)$;

$(H_{3})$:
$\lim_{x\rightarrow\pm\infty}\varphi(x)=:\varphi(\pm\infty)\in\mathbb{R},\quad\lim_{|x|\rightarrow+\infty}x^{4}\varphi^{(4)}(x)=0;$

$(H_{4})$:
$\lim_{x\rightarrow\pm\infty}f(x)=:f(+\infty)\in\mathbb{R},\quad\lim_{|x|\rightarrow+\infty}x^{4}f^{(4)}(x)=0;$

$(H_{5})$:
$\lim_{x\rightarrow\pm\infty}g(x)=:g(\pm\infty)\in\mathbb{R},\quad\lim_{|x|\rightarrow+\infty}x^{4}g^{(4)}(x)=0.$

###### Theorem 5.1

Suppose that $(H_{1})-(H_{5})$ hold, and $\omega$ satisfies

$|k\omega^{-1}-l|\geq\frac{c_{0}}{|k|^{\sigma}},$

where $c_{0}>0,\sigma>0,k\in\mathbb{Z}\setminus 0,l\in\mathbb{Z}$. Then for
every solution $x(t)$ of (5.1), we have

$\sup_{t\in\mathbb{R}}(|x(t)|+|x^{\prime}(t)|)<+\infty.$

In order to obtain the boundedness of all solutions of (5.1), it is sufficient
to prove that its Poincaré mapping can be written as a twist mapping with
small enough perturbations. Under some transformations, if some Birkhoff
constants of the Poincaré mapping are not zero, we use classical twist theorem
or small twist theorem for reversible mappings to derive the boundedness.
Otherwise, if all Birkhoff constants of the Poincaré mapping vanish, we apply
Theorem 1.1 to achieve the goal. In the following, we will give the proof of
Theorem 5.1, which is similar to the proof in [4] and [13]. Thus we give the
sketch of the proof.

We first rewrite (5.1) as

$\begin{cases}x^{\prime}=-\omega y\\\ y^{\prime}=\omega
x+\omega^{-1}\varphi(x)f(\omega
y)+\omega^{-1}g(x)-\omega^{-1}p(t).\end{cases}$ (5.2)

From $(H_{2})$, it follows that (5.2) is reversible with respect to the
involution $G(x,y)=(x,-y)$.

By polar coordinates change $x=r\cos\theta,\ y=r\sin\theta,$ the system (5.3)
is transformed into

$\begin{cases}r^{\prime}=\omega^{-1}\big{(}\varphi(r\cos\theta)f(\omega
r\sin\theta)+g(r\cos\theta)\big{)}\sin\theta-\omega^{-1}p(t)\sin\theta\\\
\theta^{\prime}=\omega+\omega^{-1}r^{-1}\big{(}\varphi(r\cos\theta)f(\omega
r\sin\theta)+g(r\cos\theta)\big{)}\cos\theta-\omega^{-1}r^{-1}p(t)\cos\theta.\end{cases}$
(5.3)

Observing that

$|\omega^{-1}r^{-1}\big{(}\varphi(r\cos\theta)f(\omega
r\sin\theta)+g(r\cos\theta)\big{)}\cos\theta-\omega^{-1}r^{-1}p(t)\cos\theta|\leq
Cr^{-1},$

for some $C>0$, we may consider (5.3) assuming that $r(t)>2C\omega^{-1}$ for
all $t\in\mathbb{R}$ along a solution $t\mapsto(r(t),\theta(t))$. Therefore,

$\theta^{\prime}\geq\frac{1}{2}\omega>0,\quad t\in\mathbb{R},$

which means that $t\mapsto\theta(t)$ ia globally invertible. Denoting by
$\theta\mapsto t(\theta)$ the inverse function, we have that
$\theta\mapsto(r(t(\theta)),t(\theta))$ solves the system

$\begin{cases}\frac{dr}{d\theta}=\Phi(r,t,\theta)\\\
\frac{dt}{d\theta}=\Psi(r,t,\theta),\end{cases}$ (5.4)

where

$\Phi(r,t,\theta)=\frac{\omega^{-1}\big{(}\varphi(r\cos\theta)f(\omega
r\sin\theta)+g(r\cos\theta)\big{)}\sin\theta-\omega^{-1}p(t)\sin\theta}{\omega+\omega^{-1}r^{-1}\big{(}\varphi(r\cos\theta)f(\omega
r\sin\theta)+g(r\cos\theta)\big{)}\cos\theta-\omega^{-1}r^{-1}p(t)\cos\theta},$
$\Psi(r,t,\theta)=\frac{1}{\omega+\omega^{-1}r^{-1}\big{(}\varphi(r\cos\theta)f(\omega
r\sin\theta)+g(r\cos\theta)\big{)}\cos\theta-\omega^{-1}r^{-1}p(t)\cos\theta}.$

Now noting that the action, angle and time variables are $r,t$ and $\theta$,
respectively. Since $\Psi(r,-t,-\theta)=\Psi(r,t,\theta)$ and
$\Phi(r,-t,-\theta)=-\Phi(r,t,\theta)$, we see that system (5.4) is reversible
under the transformation $(r,t)\mapsto(r,-t)$.

To estimate error terms, we introduce some notations.

###### Definition 5.2

: Assume function $f(\theta,r,t)$ is $O_{n}(r^{-j})$, if $f$ is smooth in
$(r,t)$, continue in $\theta$, periodic of period $2\pi$ in $\theta$ and $t$,
moreover

$|r^{k+j}\frac{\partial^{k+l}f}{\partial r^{k}\partial t^{l}}|\leq C,\quad
0\leq k+l\leq n,$

where $C$ is a positive constant.

: Suppose function $f(\theta,r,t)$ is $o_{n}(r^{-j})$, if $f$ is smooth in
$(r,t)$, continue in $\theta$, periodic of period $2\pi$ in $\theta$ and $t$,
moreover

$\lim_{r\rightarrow\infty}r^{k+j}\frac{\partial^{k+l}f}{\partial r^{k}\partial
t^{l}}=0,\quad 0\leq k+l\leq n,$

uniformly in $(\theta,t)$.

It is obvious that

$\Phi(r,t,\theta)\in O_{4}(1),\quad\Psi(r,t,\theta)\in O_{4}(1),$

and (5.4) can be rewritten as

$\begin{cases}\frac{dr}{d\theta}=\omega^{-2}\big{(}\varphi(r\cos\theta)f(\omega
r\sin\theta)+g(r\cos\theta)\big{)}\sin\theta-\omega^{-2}p(t)\sin\theta+O_{4}(r^{-1})\\\
\frac{dt}{d\theta}=\omega^{-1}-\omega^{-3}r^{-1}\big{(}\varphi(r\cos\theta)f(\omega
r\sin\theta)+g(r\cos\theta)\big{)}\cos\theta+\omega^{-3}r^{-1}p(t)\cos\theta+O_{4}(r^{-2}).\end{cases}$
(5.5)

Since the Poincaré mapping of (5.5) is not sufficiently close to a twist map,
we need to transform (5.5) further.

Let

$\lambda=r+S_{1}(\theta,r),\quad t=t,$

where

$S_{1}(\theta,r)=-\omega^{-2}\int_{0}^{\theta}\big{(}\varphi(r\cos\phi)f(\omega
r\sin\phi)+g(r\cos\phi)\big{)}\sin\phi d\phi.$

Under this transformation, (5.5) is transformed into

$\begin{cases}\frac{d\lambda}{d\theta}=-\omega^{-2}p(t)\sin\theta+O_{4}(\lambda^{-1}),\\\
\frac{dt}{d\theta}=\omega^{-1}-\omega^{-3}\lambda^{-1}\big{(}\varphi(\lambda\cos\theta)f(\omega\lambda\sin\theta)+g(\lambda\cos\theta)\big{)}\cos\theta+\omega^{-3}\lambda^{-1}p(t)\cos\theta+O_{4}(\lambda^{-2}).\end{cases}$
(5.6)

Introduce a transformation

$\lambda=\lambda,\quad\tau=t+S_{2}(\theta,\lambda),$

where

$S_{2}(\theta,\lambda)=\omega^{-3}\lambda^{-1}\int_{0}^{\theta}\big{(}\varphi(\lambda\cos\phi)f(\omega\lambda\sin\phi)\cos\phi-\lambda
J_{1}(\lambda)\big{)}+\big{(}g(\lambda\cos\phi)\cos\phi-\lambda
J_{2}(\lambda)\big{)}d\phi,$

with

$J_{1}(\lambda)=\frac{1}{2\pi\lambda}\int_{0}^{2\pi}\varphi(\lambda\cos\phi)f(\omega\lambda\sin\phi)\cos\phi
d\phi,$
$J_{2}(\lambda)=\frac{1}{2\pi\lambda}\int_{0}^{2\pi}g(\lambda\cos\phi)\cos\phi
d\phi.$

By this transformation, (5.6) is transformed into

$\begin{cases}\frac{d\lambda}{d\theta}=-\omega^{-2}p(\tau)\sin\theta+O_{4}(\lambda^{-1})\\\
\frac{d\tau}{d\theta}=\omega^{-1}-\omega^{-3}\big{(}J_{1}(\lambda)+J_{2}(\lambda)\big{)}+\omega^{-3}\lambda^{-1}p(\tau)\cos\theta+O_{4}(\lambda^{-2}).\end{cases}$
(5.7)

Furthermore, we can find a transformation
$(\lambda,\tau)\rightarrow(\lambda,\varsigma)$, where

$\varsigma=\tau+\lambda^{-1}S_{3}(\theta,\tau),$ (5.8)

and $S_{3}(\theta,\tau)$ is determined by solving equation

$\omega^{-3}p(\tau)\cos\theta+\frac{\partial
S_{3}}{\partial\theta}+\omega^{-1}\frac{\partial S_{3}}{\partial\tau}=0.$

By this transformation, we eliminate
$\omega^{-3}\lambda^{-1}p(\tau)\cos\theta$ item in the second equation of
(5.7).

Since

$\lim_{\lambda\rightarrow+\infty}\lambda^{k+1}J_{1}^{(k)}(\lambda)=(-1)^{k}k!\frac{1}{\pi}\big{(}\varphi(+\infty)-\varphi(-\infty)\big{)}f(+\infty),$

and

$\lim_{\lambda\rightarrow+\infty}\lambda^{k+1}J_{2}^{(k)}(\lambda)=(-1)^{k}k!\frac{1}{\pi}(g(+\infty)-g(-\infty)),\quad
0\leq k\leq 4,$

(5.7) can be rewritten as

$\begin{cases}\frac{d\lambda}{d\theta}=-\omega^{-2}p(\varsigma)\sin\theta+O_{4}(\lambda^{-1})\\\
\frac{d\varsigma}{d\theta}=\omega^{-1}-\frac{\omega^{-3}}{\pi}\lambda^{-1}\Big{(}\big{(}\varphi(+\infty)-\varphi(-\infty)\big{)}f(+\infty)+\big{(}g(+\infty)-g(-\infty)\big{)}\Big{)}+o_{4}(\lambda^{-1}).\end{cases}$
(5.9)

We also recalled (5.9) is reversible with respect to
$(\lambda,\varsigma)\mapsto(\lambda,-\varsigma)$.

Denote $\lambda=\rho^{-1},$ then (5.9) can be rewritten as

$\begin{cases}\frac{d\rho}{d\theta}=\omega^{-2}\rho^{2}p(\varsigma)\sin\theta+o_{4}(\rho^{2}),\\\
\frac{d\varsigma}{d\theta}=\omega^{-1}-\frac{\omega^{-3}}{\pi}\rho\Big{(}\big{(}\varphi(+\infty)-\varphi(-\infty)\big{)}f(+\infty)+\big{(}g(+\infty)-g(-\infty)\big{)}\Big{)}+o_{4}(\rho).\end{cases}$

Therefore, we derive the corresponding Poincaré mapping with the form

$\begin{cases}\rho_{1}=\rho_{0}+\rho_{0}^{2}l(\varsigma_{0})+o_{4}(\rho_{0}^{2}),\\\
\varsigma_{1}=\varsigma_{0}+\gamma_{0}+\gamma_{1}\rho_{0}+o_{4}(\rho_{0}),\end{cases}$
(5.10)

where $\gamma_{0}=2\pi\omega^{-1}$,
$l(\varsigma_{0})=\omega^{-2}\int_{0}^{2\pi}p(\varsigma_{0}+\omega^{-1}\theta)\sin\theta
d\theta$ and

$\gamma_{1}=-2\omega^{-3}\big{(}(\varphi(+\infty)-\varphi(-\infty))f(+\infty)+(g(+\infty)-g(-\infty))\big{)}.$
(5.11)

From the normal form theory, we see that one of the following two cases
occurs.

Case 1. If
$(\varphi(+\infty)-\varphi(-\infty))f(+\infty)+(g(+\infty)-g(-\infty))\neq 0$,
that is, $\gamma_{1}\neq 0$, then by twist theorem for reversible mappings
(see [9]), there are many invariant curves for $\rho\ll 1$. If $\gamma_{1}=0$,
we need to continue looking for changes, such that (5.10) is of the following
form

$\begin{cases}\rho_{1}=\rho_{0}+c_{1}\rho_{0}^{k}+o(\rho_{0}^{k+1}),\\\
\varsigma_{1}=\varsigma_{0}+\gamma_{0}+\gamma_{l}\rho_{0}^{l}+o(\rho_{0}^{l+1}),\end{cases}$

where $c_{1},\gamma_{1}$ are constants and $l>k$. Similarly, as long as the
coefficient of the twist term $\gamma_{l}$ is not zero, by twist theorem in
[9], we obtain many invariant curves if $\rho\ll 1$, i.e., $r\gg 1$, which
implies the existence of quasi-periodic solutions of (5.1).

Case 2. There is a change of variables such that (5.10) can be transformed
into a linear normal form

$\begin{cases}\rho_{1}=\rho_{0}\\\
\varsigma_{1}=\varsigma_{0}+\gamma_{0}.\end{cases}$

By Theorem 1.1, there are sequence of invariant curves tending to
$\rho_{0}=0$.

In conclusion, the mapping (5.10) has many invariant curves tending to
$\rho_{0}=0$, which means the invariant curves of the Poincaré map of (5.1)
tend to infinity. Thus for equation (5.1), the existence of quasi-periodic
solutions is got. Moreover for the initial value lying between two invariant
curves, the solution is globally bounded. As the invariant curves tend to
infinity, all solutions of (5.1) are bounded, therefore we finish the proof of
Theorem 5.1.

## Acknowledgment

Y. Z was partially supported by the NSFC (grant no. 11971059) and the
Fundamental Research Funds for the Central Universities (grant no. 202261096).
D. P was supported in part by the NSFC (grant no. 11971059). Y. N was
supported by the NSFC (grant no. 12201587) , the China Postdoctoral Science
Foundation (grant no. 2020M682236) and the Shandong Provincial Natural Science
Foundation, China (grant no. A010704).

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$\displaystyle=-V\rho^{2}\sqrt{\frac{\pi}{g}}\exp\left\\{-\frac{\Delta_{0\bar{1}}^{2}}{4gK^{2}}\right\\}$
$\displaystyle\approx\frac{V\rho^{2}\Delta_{0\bar{1}}^{2}}{4gK^{2}}\sqrt{\frac{\pi}{g}}+const.$
(89)

We note that this leading quadratic term in $\Delta_{0\bar{1}}$ is expected
for arbitrary $V_{0}(q)$, which can be shown by expanding Eq. 74.

Figure 16: Schematic Fock phase diagram, the numbers label $(C\text{(mod
3)},m)$ in Eq. 92.

Using $C_{3}$ and combining the contribution from all edges of MBZ, similar to
the discussion in Eq. 80, we identify the following

$\displaystyle C$ $\displaystyle=\frac{3}{2\pi}(\theta_{0}-\theta_{\bar{1}})$
$\displaystyle\phi_{BZ}$
$\displaystyle=-3\left(\int_{K_{0}}^{\bar{K}_{1}}+\int_{\bar{K}_{0}}^{K_{1}}\right)d\mathbf{k}^{\prime}\cdot\mathbf{A}(\mathbf{k}^{\prime})$
(90)

Therefore,

$\displaystyle\Delta_{0\bar{1}}=-\frac{2\pi C-\Phi_{BZ}}{3}+2m\pi$ (91)

The $C_{3}$ symmetry demands the contribution from three edges to be exactly
the same. Then, we find the Fock energy as a function of the Chern number

$H_{Fock}=H^{0\bar{1}}_{Fock}+H^{1\bar{2}}_{Fock}+H^{2\bar{0}}_{Fock}=3H^{0\bar{1}}_{Fock}=\frac{V\rho^{2}\pi^{5/2}}{3g^{3/2}K^{2}}\left(C-3m-\frac{\Phi_{BZ}}{2\pi}\right)^{2}+const.$
(92)

The phase diagram is plotted in Fig. 16. We not only derive the rounding but
also predict the Little-Parks effect: The Hall crystal/Wigner crystal becomes
less stable when parent Berry flux approaches $(2n+1)\pi$, which may become
useful when discussing physics at lower densities. At reduced fillings and
without a moiré potential, as we discussed in Sec. VIII, the correlated fermi
liquid is almost frozen, i.e. it behaves like a crystal at short distances and
short times. It is natural to assume that the WC/AHC short-range order has a
lattice constant set by electron density. However, the picture in Fig. 16
suggests a route for the Fock energy to shrink the WC/AHC lattice so that the
new BZ (with an area larger than the electron density) encloses a Berry flux
quantized to an integer multiple of $2\pi$. The outcome of this competition
between two crystalline orders is unclear to us, since the energy of the
shrunk crystal also depends on the fate of the doped vacancies. We leave this
to future study.

##  Appendix G 7-band Fock-only and Hartree-Fock phase diagrams

We present in Fig. 17 the Fock-only and Hartree-Fock phase diagrams computed
with a 7-conduction band projection. This provides a consistency (and
convergence) check with our 4-band projection results presented in the main
text. As seen, the Hartree correction to the Fock-only phase diagram expands
the region of $|C|=1$ in parameter space, in agreement with the results in the
main text.

Figure 17: Chern number of mean-field band obtained with Fock only [Left] and
Hartree-Fock [right], respectively. These results are obtained from the
10-orbital model and the same parameters as Fig. 3, with the exception of
using 7-bands in the mean-field projection.

##  Appendix H Comparison of Fermi liquid and Hall crystal: Hartree Fock
energies

We present in Fig. 18 the Hartree-Fock energy difference of the translation
symmetry preserving Fermi liquid and translation symmetry breaking Hall
crystal for wavevector $\mathbf{G}_{M}$ associated with moiré twist angle of
0.77∘ for $u_{d}=-36$ meV. As seen, for a range of displacement field
energies, the Fermi liquid state is higher in energy than the Hall crystal. As
alluded to in the main text, this energy difference is not altogether
surprising, as the Hartree-Fock framework is biased towards finding
translation symmetry broken (and gapped) phases of matter.

Figure 18: Energy difference of translation symmetry preserving Fermi liquid
and translation symmetry breaking Hall crystal for wavevector $\vec{G}_{M}$
associated with moiré twist angle of 0.77∘ for $u_{d}=-36$ meV.