Onthe Reducibility of a Class Nonlinear Almost Periodic Hamiltonian Systems

Abstract

Inthis paper, we consider the reducibility of a class of nonlinear almost periodic Hamiltonian systems. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by using KAM iteration, it is shown that the considered Hamiltonian system is reducible to an almost periodic Hamiltonian system with zero equilibrium points for most small enough parameters. As an example, we discuss the reducibility and stability of an almost periodic Hill’s equation.

1. Introduction

In this paper, we are concerned with the reducibility of the almost periodic Hamiltonian system

$$\dot{x}=(A+\epsilon Q(t))x+\epsilon g(t)+h(x,t),x\in {\mathbb{R}}^{2n},$$

(1)

where A has multiple possible eigenvalues, $Q(t),g(t),$ and $h(x,t)$ are all analytic almost periodic with respect to t, and $\epsilon >0$ is a sufficiently small parameter.

First, we review some relevant definitions for almost periodic systems. If $A(t)$ is an $d\times d$ almost periodic matrix, the equation

$$\frac{dx}{dt}=A(t)x,x\in {\mathbb{R}}^d$$

(2)

is reducible if there exists a regular almost periodic transformation

$$x=\kappa (t)z,$$

where $\kappa (t)$ and ${\kappa }^{-1}(t)$ are almost periodic and bounded, which transforms Equation (2) into

$$\frac{dz}{dt}=Dz,y\in {\mathbb{R}}^d$$

(3)

where D is constant.

In recent years, the reducibility for linear equations has attracted the attention and been studied by many researchers. The well known Floquet theorem states that every periodic differential Equation (2) can be reduced to a constant coefficient differential Equation (3) by means of a periodic change of variables with the same period as $A(t)$. But this result no longer holds true for the quasi-periodic and almost periodic linear equation; more details can be seen in [1]. If the coefficient matrix satisfies the “full spectrum” condition, Johnson and Sell [2] proved the reducibility of the quasi-periodic linear system (2).

Later, many authors [3,4,5,6,7,8,9,10,11,12] paid attention to the reducibility for the following quasi-periodic linear system:

$$\frac{dx}{dt}=(A+ϵQ(t))x,x\in {\mathbb{R}}^d.$$

(4)

where $ϵ$ is a sufficiently small parameter.

In 1996, Xu and You [13] studied the reducibility for the almost-periodic linear system

$$\frac{dx}{dt}=(A+\epsilon Q(t))x,x\in {\mathbb{R}}^n.$$

(5)

They proved that system (5) is reducible in the case that A has different eigenvalues, for most sufficiently small $\epsilon$ through KAM iteration and a “space structure”. Later, ref. [14] studied the case in which system (5) is Hamiltonian and A has possible multiple eigenvalues; they obtained reducibility results similar to those in [13].

In 2017, J. Li, C. Zhu, and S. Chen [15] studied the quasi-periodic case of (1). It was shown that for most sufficiently small parameters, under some assumptions of analyticity, non-resonant and non-degeneracy conditions, through a quasi-periodic symplectic change of variables, the considered system was changed into a quasi-periodic Hamiltonian system with zero equilibrium points.

Motivated by [13,14,15], we will extend the reducible results of [15] to the case of almost periodic Hamiltonian systems. Under some suitable assumptions, we will obtain a similar result.

Theorem 1.

Consider the almost periodic Hamiltonian system (1) in which A is a $2n\times 2n$ matrix that can be diagonalized with multiple possible eigenvalues ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{2n}$, $i=1,\cdots ,2n$, and $Q(t)={\sum }_{\Lambda \in \tau }{Q}_{\Lambda }(t)$, $g(t)={\sum }_{\Lambda \in \tau }{g}_{\Lambda }(t)$, and $h(x,t)={\sum }_{\Lambda \in \tau }{h}_{\Lambda }(x,t)$ are all analytic almost periodic functions on $D_{{\rho }_0}$; they have the same frequencies $\omega =({\omega }_1,{\omega }_2,\cdots )$ and spatial structure $(\tau ,[·])$. Moreover, $h(x,t)$ is analytic with respect to x on $B_{a_0}(0),h(0,t)=0,$ and $D_{x}h(0,t,\epsilon )=0.$ Here, $B_{a_0}(0)$ is a ball centered on 0 with radius $a_0$; $\epsilon \in (0,{\epsilon }_0)$ is a sufficiently small parameter. Suppose the following:

(1) 

There exists $z_0>0$ such that ${|\parallel Q\parallel |}_{z_0,{\rho }_0}<\infty ;$

(2) 

(Non-resonant conditions) $\lambda =({\lambda }_1,\cdots ,{\lambda }_{2n})$ and $\omega =({\omega }_1,{\omega }_2,\cdots )$ satisfy

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i|\ge \frac{{\alpha }_0}{{\Delta }^4(|k|){\Delta }^4(\left[k\right])},$$

(6)

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i+{\lambda }_j|\ge \frac{{\alpha }_0}{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}$$

(7)

for $1\le i,j\le 2n,i\ne j,$ and $k\in {\mathbb{Z}}^{\mathbb{N}}\setminus \{0\}$, where ${\alpha }_0>0$ is a small constant, and Δ is an approximation function.

(3) 

(Non-degeneracy conditions) Denote the solution of the equation $\dot{x}=Ax+\epsilon g(t)$ by $\underline{x}$. Let $\widehat{Q}(t)=Q(t)+{\epsilon }^{-1}{D}_{x}h(\underline{x}(t),t).$ Assume $A+\epsilon \overline{\widehat{Q}}:=A_{+}$ has eigenvalues ${\lambda }_1^{+},{\lambda }_2^{+},\cdots ,{\lambda }_{2n}^{+}$ that satisfy $|{\lambda }_i^{+}-{\lambda }_j^{+}|\ge 2\delta \epsilon >0$ and $|{\lambda }_i^{+}|\ge 2\delta \epsilon >0$, where $i\ne j$ and $i,j=1,2,\cdots ,2n.$

(4) 

$\parallel D_{xx}h(x,t,\epsilon )\parallel \le K_0,$$x\in B_{a_0}(0)$.

Then, there exists a positive Lebesgue measure, non-empty Cantor set $E^{\ast }\subset (0,{\epsilon }_0),$ such that for $\epsilon \in E^{\ast }$, there is an almost periodic symplectic transformation $x=\psi (t)y+\phi (t)$ that transforms (1) into

$$\dot{y}=By+h_{\infty }(y,t),$$

(8)

where $\psi (t)$ and $\phi (t)$ are almost periodic with the same frequencies and spatial structure as $Q(t)$, B is a real constant matrix, and $h_{\infty }(y,t)=O(y^2)$ as $y\to 0$. Moreover, $meas((0,{\epsilon }_0)\setminus E^{\ast })=o({\epsilon }_0)$ as ${\epsilon }_0\to 0$.

As an example, we will apply Theorem 1 in Section 4 to an almost-periodic Hill’s equation:

$$\ddot{x}+\epsilon a(t)x=0.$$

(9)

Under some appropriate assumptions, we have that, for most small $\epsilon$, Equation (9) is reducible. Furthermore, the zero equilibrium point of (9) is Lyapunov stable.

The basic framework of this paper is as follows. In Section 2, we recall some definitions and notations, present some results in the form of lemmas that will be useful in the proof of Theorem 1. The proof of Theorem 1 is presented in Section 3. In Section 4, we analyze the almost periodic Hill’s equation, Equation (9).

2. Some Preliminaries

Firstly, we present some definitions.

Definition 1.

We say a function f is quasi-periodic with the basic frequencies $\omega =({\omega }_1,{\omega }_2,\cdots ,{\omega }_s)$ if $f(t)=F({\theta }_1,{\theta }_2,\cdots ,{\theta }_s)$ where F is $2\pi$-periodic in ${\theta }_j={\omega }_{j}t$ for $j=1,2,\cdots ,s.$ Moreover, if $F(\theta )(\theta =({\theta }_1,{\theta }_2,\cdots ,{\theta }_s))$ is analytic on $D_{\rho }=\{\theta \in {\mathbb{C}}^s:|\Im {\theta }_j|\le \rho ,j=1,2,\cdots ,s\}$, then $f(t)$ is analytic quasi-periodic on $D_{\rho }$.

If $f(t)$ is analytic quasi-periodic, it can be expanded as a Fourier series

$$f(t)=\sum _{k\in {\mathbb{Z}}^s}{f}_k{e}^{\langle k,\omega \rangle \sqrt{-1}t}$$

with Fourier coefficients

$$f_k=\frac{1}{{(2\pi )}^s}{\int }_{{\mathbb{T}}^r}F(\theta )e^{-\langle k,\theta \rangle \sqrt{-1}}d\theta .$$

The norm is denoted as ${\left|\right|f\left|\right|}_{\rho }$:

$${\left|\right|f\left|\right|}_{\rho }=\sum _{k\in {\mathbb{Z}}^s}\left|f_k\right|e^{\left|k\right|\rho }.$$

Assume $R(t)={(r_{ij}(t))}_{1\le i,j\le m}$ is an $m\times m$ matrix. If all $r_{ij}(t)(i,j=1,2,\cdots ,m)$ are analytic quasi-periodic on $D_{\rho }$ with frequencies $\omega =({\omega }_1,{\omega }_2,\cdots ,{\omega }_s)$, then matrix $R(t)$ is said to be analytic quasi-periodic on $D_{\rho }$ with frequencies $\omega$.

The norm of $R(t)$ is defined as

$${\left|\right|R\left|\right|}_{\rho }=\underset{1\le i\le m}{max}\sum _{j=1}^m\left|\right|r_{ij}{\left|\right|}_{\rho }.$$

Obviously,

$$\left|\right|R_1{R}_2{\left|\right|}_{\rho }\le \left|\right|R_1{\left|\right|}_{\rho }\left|\right|R_2{\left|\right|}_{\rho }.$$

If R is a constant matrix, to simplify, we record ${\left|\right|R\left|\right|}_{\rho }$ as $\left|\right|R\left|\right|$. The average of $R(t)$ is $\overline{R}={({\overline{r}}_{ij})}_{1\le i,j\le m}$, where

$${\overline{r}}_{ij}=\underset{T\to \infty }{lim}\frac{1}{2T}{\int }_{-T}^T{r}_{ij}(t)dt,$$

The details can be found in [16].

Definition 2.

A function f is said to be an almost periodic function if $f(t)={\sum }_{m=1}^{\infty }{f}_m(t),$ where $f_m(t)$ are all quasi-periodic for $m=1,2,\cdots .$

In [13], we see that “spatial structure” and “approximation function” are very powerful tools to study almost periodic systems. We provide the definitions and notions from [17,18].

Definition 3

([17]). If τ is a set of some subsets of $\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, then $(\tau ,[·])$ is said to be a finite spatial structure if τ meets the following conditions:

1. 

$\varnothing \in \tau$;

2. 

If ${\Lambda }_r,{\Lambda }_s\in \tau$, then ${\Lambda }_r\cup {\Lambda }_s\in \tau$;

3. 

${\bigcup }_{\Lambda \in \tau }=\mathbb{N}$, where $[·]$ is a weight function defined on τ, satisfying $[\varnothing ]=0$ and $[{\Lambda }_r\cup {\Lambda }_s]\le \left[{\Lambda }_r\right]+\left[{\Lambda }_s\right]$.

Let $l\in {\mathbb{Z}}^{\mathbb{N}}$. Write the support set of l as

$$suppl=\{(s_1,s_2,\cdots ,s_n)|l_j\ne 0,j=s_1,s_2,\cdots ,s_n,l_j=0,asj=otherwise\}.$$

Write the weight value as $\left[l\right]=in{f}_{suppl\subset \Lambda ,\Lambda \in \tau }[\Lambda ].$ Denote

$$\left|l\right|=\sum _{s=1}^{\infty }\left|l_s\right|.$$

Definition 4

([18]). Δ is called an approximation function, if

1. 

$\Delta :[0,\infty )\to [1,\infty )$ is increasing, and $\Delta (0)=1$ is satisfied;

2. 

$\frac{log\Delta (t)}{t}$ is decreasing on $[0,\infty )$;

3. 

${\int }_0^{\infty }\frac{log\Delta (t)}{t^2}dt<\infty .$

Remark 1.

If Δ is an approximation function, from Definition 4, it follows that ${\Delta }^4$ is also an approximation function.

Definition 5.

Let $R(t)={\sum }_{\Lambda \in \tau }{R}_{\Lambda }(t)$. If $R_{\Lambda }(t)$ are quasi-periodic matrix functions with basic frequencies $\omega =\{{\omega }_s|s\in \Lambda \}$, then $R(t)$ is said to be an almost periodic matrix function with spatial structure $(\tau ,[·])$ and basic frequencies ω.

We also write the average of R(t) as $\overline{R}$, where

$$\overline{R}=\underset{T\to \infty }{lim}\frac{1}{2T}{\int }_{-T}^{T}R(t)dt.$$

Let $R(t)={\sum }_{\Lambda \in \tau }{R}_{\Lambda }(t)$. For $\mu >0$ and $\varrho >0$,

$${|\parallel R\parallel |}_{\mu ,\varrho }=\sum _{\Lambda \in \tau }{e}^{\mu [\Lambda ]}{\parallel R_{\Lambda }\parallel }_{\varrho }$$

is the weighted norms with finite spatial structure $(\tau ,[·])$. From [13], we can select the weighted function

$$[\Lambda ]=1+\sum _{s\in \Lambda }{log}^q(1+|s|),q>2.$$

Also, we will present some lemmas in this section, which are useful for the proof of our main result.

Lemma 1

([8]). Let $g:B_{\sigma }(0)\subset {\mathbb{R}}^n\to {\mathbb{R}}^n$ be a $C^2$ function with $g(0)=0,D_{x}g(0)=0,$ $\left|\right|D_{xx}g(x)\left|\right|\le M,$ and $x\in B_{\sigma }(0).$ Then, $\left|\right|g(x)\left|\right|\le \frac{M}{2}{\left|\right|x\left|\right|}^2$ and $\left|\right|D_{x}g(x)\left|\right|\le M\left|\right|x\left|\right|$.

Lemma 2

([8]). Suppose that $B_0$ is an $m\times m$ matrix with the eigenvalues ${\mu }_1^0,\cdots ,{\mu }_m^0$, which satisfy $|{\mu }_i^0|>\nu ,|{\mu }_i^0-{\mu }_j^0|>\nu ,i\ne j,1\le i,$ and $j\le m.$ Let $S_0$ be a nonsingular matrix with $S_0^{-1}{B}_0{S}_0=diag({\mu }_1^0,\cdots ,{\mu }_m^0),$ ${\beta }_0=max\{||S_0\left|\right|,\left|\right|S_0^{-1}\left|\right|\}$, and ϱ is a value such that

$$0<\varrho <\frac{\nu }{(3m-1){\beta }_0^2}.$$

If $B_1$ verifies $\left|\right|B_1-B_0\left|\right|\le \varrho ,$ then the following results hold:

(1) 

$B_1$ has m different nonzero eigenvalues ${\mu }_1^1,\cdots ,{\mu }_m^1$.

(2) 

There is a nonsingular matrix $S_1$ such that $S_1^{-1}{B}_1{S}_1=diag({\mu }_1^1,\cdots ,{\mu }_m^1)$, which satisfies $\left|\right|S_1\left|\right|,\left|\right|S_1^{-1}\left|\right|\le {\beta }_1,$ where ${\beta }_1=2{\beta }_0$.

Lemma 3.

Consider the differential equation

$$\dot{X}=AX+\epsilon g(t),X\in {\mathbb{R}}^{2m},$$

(10)

where A is a $2m\times 2m$ constant matrix, which can be diagonalized and where the eigenvalues ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{2m}$ of A satisfy $|{\lambda }_i|\ge \xi$, and ξ is a positive constant. Also, $g(t)={\sum }_{\Lambda \in \tau }{g}_{\Lambda }(t)$ is an analytic almost periodic function on $D_{\rho }$, of which its frequencies are $\omega =({\omega }_1,{\omega }_2,\cdots )$ and spatial structure is $(\tau ,[·])$. If

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i|\ge \frac{\alpha }{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}$$

for all $k\in {\mathbb{Z}}^{\mathbb{N}}\setminus \{0\}$, $\alpha >0$. Set $0<{\rho }_1<\rho$ and $0<z_1<z.$ Then, for Equation (10), there is a unique analytic almost periodic solution $X(t)$ that has the same frequencies and spatial structure as $g(t)$ and satisfies

$${|\parallel X\parallel |}_{z-z_1,\rho -{\rho }_1}\le \epsilon \frac{c\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }{|\parallel g\parallel |}_{z,\rho },$$

where $\mathrm{\Gamma }(\rho )={sup}_{t\ge 0}\{{\Delta }^4(t)e^{-\rho t}\}.$

Proof. 

Make the change of variable $X=BY$, and let $h(t)=B^{-1}g.$ Equation (10) becomes

$$\dot{Y}=DY+\epsilon h(t),Y\in {\mathbb{R}}^{2m},$$

(11)

where $D=B^{-1}AB=diag({\lambda }_1,\cdots ,{\lambda }_{2m})$.

Let $Y={\sum }_{\Lambda \in \tau }{y}_{\Lambda },h={\sum }_{\Lambda \in \tau }{h}_{\Lambda }$, and

$$y_{\Lambda }=({\tilde{y}}_{\Lambda }^{ij}),{\tilde{y}}_{\Lambda }^{ij}=\sum _{suppk\subset \Lambda }{\tilde{y}}_{\Lambda k}^{ij}{e}^{\sqrt{-1}\langle k,\omega \rangle t},$$

$$h_{\Lambda }=({\tilde{h}}_{\Lambda }^{ij}),{\tilde{h}}_{\Lambda }^{ij}=\sum _{suppk\subset \Lambda }{\tilde{h}}_{\Lambda k}^{ij}{e}^{\sqrt{-1}\langle k,\omega \rangle t}.$$

By (11), we have

$${\tilde{y}}_{\Lambda k}^{ij}=\epsilon \frac{{\tilde{h}}_{\Lambda k}^{ij}}{\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i}.$$

So,

$$\begin{array}{ccc}\hfill \left|\right|{\tilde{y}}_{\Lambda }^{ij}{\left|\right|}_{\rho -{\rho }_1}& \le & \epsilon \sum _{suppk\subset \Lambda }\frac{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}{\alpha }\left|{\tilde{h}}_{\Lambda k}^{ij}\right|e^{(\rho -{\rho }_1)\left|k\right|}\hfill \\ & \le & \epsilon \frac{\mathrm{\Gamma }({\rho }_1){\Delta }^4([\Lambda ])}{\alpha }\left|\right|{\tilde{h}}_{\Lambda }^{ij}{\left|\right|}_{\rho }.\hfill \end{array}$$

Thus,

$$\left|\right|y_{\Lambda }{\left|\right|}_{\rho -{\rho }_1}\le \epsilon \frac{\mathrm{\Gamma }({\rho }_1){\Delta }^4([\Lambda ])}{\alpha }\left|\right|h_{\Lambda }{\left|\right|}_{\rho }.$$

From Definition 5, we have

$$\begin{array}{ccc}\hfill {|\parallel Y\parallel |}_{z-z_1,\rho -{\rho }_1}& =& \sum _{\Lambda \in \tau }\left|\right|{\tilde{y}}_{\Lambda }{\left|\right|}_{\rho -{\rho }_1}{e}^{(z-z_1)[\Lambda ]}\hfill \\ & \le & \epsilon \sum _{\Lambda \in \tau }\frac{\mathrm{\Gamma }({\rho }_1){\Delta }^4([\Lambda ])}{\alpha }\left|\right|h_{\Lambda }{\left|\right|}_{\rho }{e}^{z[\Lambda ]-z_1[\Lambda ]}\hfill \\ & \le & \epsilon \frac{\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }|\parallel h_{\Lambda }{\parallel |}_{z,\rho }.\hfill \end{array}$$

Thus, from

$${|\parallel h\parallel |}_{z,\rho }\le \left|\right|B^{-1}{\left|\right||\parallel g\parallel |}_{z,\rho },$$

$${|\parallel X\parallel |}_{z-z_1,\rho -{\rho }_1}\le {\left|\right|B\left|\right||\parallel y\parallel |}_{z-z_1,\rho -{\rho }_1},$$

we have

$${|\parallel X\parallel |}_{z-z_1,\rho -{\rho }_1}\le \epsilon \frac{c\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }{|\parallel g\parallel |}_{z,\rho }.$$

□

The following lemma is very useful in proving Theorem 1, in order to perform a step of the inductive procedure.

Lemma 4.

Consider the equation

$$\dot{P}(t)=AP(t)-P(t)A+M(t),$$

(12)

where A is a $2 m\times 2 m$ Hamiltonian matrix with eigenvalues ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{2m}$. Suppose $|{\lambda }_i|\ge \sigma$ and $|{\lambda }_i-{\lambda }_j|\ge \sigma$ for $i\ne j,$ and $i,j=1,\cdots ,2m.$ Furthermore, $M(t)={\sum }_{\Lambda \in \tau }{M}_{\Lambda }(t)$ is analytic almost periodic on $D_{\rho }$ with frequencies $\omega =({\omega }_1,{\omega }_2,\cdots )$ and has finite spatial structure $(\tau ,[·])$. Then, $\overline{M}=0$, and

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i+{\lambda }_j|\ge \frac{\alpha }{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}$$

for all $k\in {\mathbb{Z}}^{\mathbb{N}}\setminus \{0\}.$ Set $0<{\rho }_1<\rho$ and $0<z_1<z.$ Then equation (12) has a unique analytic almost periodic Hamiltonian solution $P(t)$ with $\overline{P}=0$, where $P(t)$ has the same frequencies and spatial structure as $M(t)$, and satisfies

$${|\parallel P\parallel |}_{z-z_1,\rho -{\rho }_1}\le c\frac{\mathrm{\Gamma }(\overline{z})\mathrm{\Gamma }(\overline{\rho })}{\alpha }{|\parallel M\parallel |}_{z,\rho }.$$

Proof. 

Choose the matrix S such that $S^{-1}AS=D=diag({\lambda }_1,\cdots ,{\lambda }_{2n})$, make the change of variable $P(t)=SW(t)S^{-1}$, and define $R(t)=S^{-1}M(t)S$. Equation (12) becomes

$$\dot{W}(t)=DW(t)-W(t)D+R(t).$$

(13)

Let $W={\sum }_{\Lambda \in \tau }{W}_{\Lambda }$ and $R={\sum }_{\Lambda \in \tau }{R}_{\Lambda }$, where

$$R_{\Lambda }=(r_{\Lambda }^{ij}),r_{\Lambda }^{ij}=\sum _{suppk\subset \Lambda }{r}_{\Lambda k}^{ij}{e}^{\sqrt{-1}\langle k,\omega \rangle t},$$

$$W_{\Lambda }=(w_{\Lambda }^{ij}),w_{\Lambda }^{ij}=\sum _{suppk\subset \Lambda }{w}_{\Lambda k}^{ij}{e}^{\sqrt{-1}\langle k,\omega \rangle t}.$$

Substitute these into (13). We have $w_{\Lambda 0}^{ij}=0$ and

$$w_{\Lambda k}^{ij}=\frac{r_{\Lambda k}^{ij}}{\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i+{\lambda }_j},k\ne 0.$$

Since $M(t)$ and $R(t)$ are analytic on $D_{\rho }$, we have

$$\begin{array}{ccc}\hfill \left|\right|w_{\Lambda }^{ij}{\left|\right|}_{\rho -{\rho }_1}& \le & \sum _{suppk\subset \Lambda }\frac{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}{\alpha }\left|r_{\Lambda k}^{ij}\right|e^{(\rho -{\rho }_1)\left|k\right|}\hfill \\ & \le & \frac{\mathrm{\Gamma }({\rho }_1){\Delta }^4([\Lambda ])}{\alpha }\left|\right|r_{\Lambda }^{ij}{\left|\right|}_{\rho }.\hfill \end{array}$$

Thus,

$$\left|\right|W_{\Lambda }{\left|\right|}_{\rho -{\rho }_1}\le \frac{\mathrm{\Gamma }({\rho }_1){\Delta }^4([\Lambda ])}{\alpha }\left|\right|R_{\Lambda }{\left|\right|}_{\rho }.$$

By Definition 5, we have

$$\begin{array}{ccc}\hfill {|\parallel W\parallel |}_{z-z_1,\rho -{\rho }_1}& =& \sum _{\Lambda \in \tau }\left|\right|W_{\Lambda }{\left|\right|}_{\rho -{\rho }_1}{e}^{(z-z_1)[\Lambda ]}\hfill \\ & \le & \sum _{\Lambda \in \tau }\frac{\mathrm{\Gamma }({\rho }_1){\Delta }^4([\Lambda ])}{\alpha }\left|\right|R_{\Lambda }{\left|\right|}_{\rho }{e}^{z[\Lambda ]-z_1[\Lambda ]}\hfill \\ & \le & \frac{\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }{|\parallel R\parallel |}_{z,\rho }.\hfill \end{array}$$

Since,

$${|\parallel P\parallel |}_{z-z_1,\rho -{\rho }_1}\le {\left|\right|S\left|\right||\parallel W\parallel |}_{z-z_1,\rho -{\rho }_1}\left|\right|S^{-1}\left|\right|,$$

$${|\parallel R\parallel |}_{z,\rho }\le \left|\right|S^{-1}{\left|\right||\parallel M\parallel |}_{z,\rho }\left|\right|S\left|\right|.$$

Hence,

$${|\parallel P\parallel |}_{z-z_1,\rho -{\rho }_1}\le c\frac{\mathrm{\Gamma }(z_1)\mathrm{\Gamma }({\rho }_1)}{\alpha }{|\parallel M\parallel |}_{z,\rho }.$$

From now on, the symbol c is used to denote different constants.

Now, we verify that $P={\sum }_{\Lambda \in \tau }{P}_{\Lambda }$ is Hamiltonian. Since A and $M={\sum }_{\Lambda \in \tau }{M}_{\Lambda }$ are Hamiltonian, we have

$$A=J{A}_J,M=J{M}_J$$

where $A_J$ and $M_J$ are symmetric. Let $P_J=J^{-1}P$. If $P_J$ is symmetric, then P is Hamiltonian. Now, we demonstrate that $P_J$ is symmetric. Substitute $P=J{P}_I$ into Equation (12). We have

$$\dot{P_J}=A_{J}J{P}_J-P_{J}J{A}_J+M_J,$$

(14)

Transposing Equation (14), we obtain

$${\dot{P_J}}^T=A_{J}J{P}_J^T-P_J^{T}J{A}_J+M_J.$$

Obviously, $J{P}_J$ and $J{P}_J^T$ are all solutions of (12). Furthermore, $\overline{J{P}_J}=\overline{J{P}_J^T}=0.$ From the uniqueness of solution of (12) with $\overline{P}=0$, it follows that $J{P}_J=J{P}_J^T$; hence, P is Hamiltonian. □

Lemma 5.

Consider the following Hamiltonian system:

$$\dot{x}=(A+\epsilon Q(t))x+\epsilon g(t)+h(x,t),x\in {\mathbb{R}}^{2n},$$

(15)

where A is a $2n\times 2n$ matrix that can be diagonalized with the eigenvalues ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{2n}$, and $|{\lambda }_i|\ge \sigma$, $\sigma >0$ is a constant. Assume that $Q(t)={\sum }_{\Lambda \in \tau }{Q}_{\Lambda }(t),$$g(t)={\sum }_{\Lambda \in \tau }{g}_{\Lambda }(t)$, and $h(x,t)={\sum }_{\Lambda \in \tau }{h}_{\Lambda }(x,t)$ are analytic almost periodic on $D_{\rho }$. Their frequencies are $\omega =({\omega }_1,{\omega }_2,\cdots )$, and they have the spatial structure $(\tau ,[·])$. Suppose that $h(x,t)$ is analytic about x on $B_a(0)$, where $\left|\right|D_{xx}h(x)\left|\right|\le M,\forall x\in B_a(0).$ Furthermore,

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i|\ge \frac{\alpha }{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}$$

holds for all $k\in {\mathbb{Z}}^{\mathbb{N}}\setminus \{0\}$, and the constant $\alpha >0$. Let $0<{\rho }_1<\rho$ and $0<z_1<z.$ Then, there exists a symplectic transformation $x=y+\underline{x}$ that transforms (15) into

$$\dot{y}=(A+\epsilon \overline{\widehat{Q}}(t))y+{\epsilon }^2\widehat{g}(t)+\widehat{h}(y,t),$$

satisfing

$$|\parallel \widehat{Q}{\parallel |}_{z-z_1,\rho -{\rho }_1}\le {|\parallel Q\parallel |}_{z,\rho }+M\frac{c\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }{|\parallel g\parallel |}_{z,\rho }$$

and

$$|\parallel \widehat{g}{\parallel |}_{z-z_1,\rho -{\rho }_1}\le \frac{c\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }{|\parallel Q\parallel |}_{z,\rho }{|\parallel g\parallel |}_{z,\rho }+Mc{\left(\frac{\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }\right)}^2{|\parallel g\parallel |}_{z,\rho }^2,$$

where $a_1=a_0-|\parallel \underline{x}{\parallel |}_{z-z_1,\rho -{\rho }_1},y\in B_{a_1}(0)$.

Proof. 

The solution of Equation $\dot{x}=Ax+\epsilon g(t)$ is denoted by $\underline{x}$. From Lemma 3, it follows that

$$|\parallel \underline{x}{\parallel |}_{z-z_1,\rho -{\rho }_1}\le \epsilon \frac{c\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }{|\parallel g\parallel |}_{z,\rho }.$$

By the symplectic transformation $x=y+\underline{x}$, Equation (15) is transformed into

$$\dot{y}=(A+\epsilon \overline{\widehat{Q}}(t))y+{\epsilon }^2\widehat{g}(t)+\widehat{h}(y,t),$$

where

$$\widehat{Q}=Q(t)+\frac{1}{\epsilon }{D}_{x}h(\underline{x},t),$$

$$\widehat{g}=\frac{1}{{\epsilon }^2}h(\underline{x},t)+\frac{1}{\epsilon }Q(t)\underline{x}$$

and

$$\widehat{h}=h(\underline{x}+y,t)-h(\underline{x},t)-D_{x}h(\underline{x},t)y.$$

From Lemmas 1 and 3, it follows that

$$\begin{array}{ccc}\hfill |\parallel \widehat{Q}{\parallel |}_{z-z_1,\rho -{\rho }_1}& \le & {|\parallel Q\parallel |}_{z,\rho }+\frac{M}{\epsilon }|\parallel \underline{x}{\parallel |}_{z-z_1,\rho -{\rho }_1}\hfill \\ & \le & {|\parallel Q\parallel |}_{z,\rho }+M\frac{c\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }{|\parallel g\parallel |}_{z,\rho }\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill |\parallel \widehat{g}{\parallel |}_{z-z_1,\rho -{\rho }_1}& \le & \frac{1}{{\epsilon }^2}\frac{M}{2}|\parallel \underline{x}{\parallel |}_{z-z_1,\rho -{\rho }_1}^2+\frac{1}{\epsilon }{\parallel Q\parallel |}_{z-z_1,\rho -{\rho }_1}|\parallel \underline{x}{\parallel |}_{z-z_1,\rho -{\rho }_1}\hfill \\ & \le & \frac{c\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }{|\parallel Q\parallel |}_{z,\rho }{|\parallel g\parallel |}_{z,\rho }+Mc{\left(\frac{\mathrm{\Gamma }({\rho }_1)\mathrm{\Gamma }(z_1)}{\alpha }\right)}^2{|\parallel g\parallel |}_{z,\rho }^2.\hfill \end{array}$$

The results are obtained. □

3. Proof of Theorem 1

3.1. The First KAM Step

In the first step, we will change A in the Equation (1) from the case with multiple eigenvalues into the case with different eigenvalues, and the $\epsilon$ of $\epsilon Q(t)$ and $\epsilon g(t)$ become ${\epsilon }^2$.

First of all, for Equation (1), by the symplectic transformation $x={\underline{x}}_0+y$, where ${\underline{x}}_0$ is the solution of

$${\dot{\underline{x}}}_0=A{\underline{x}}_0+\epsilon g(t).$$

Hamiltonian system (1) is changed into

$$\dot{y}=(A+\epsilon \widehat{Q}(t))y+{\epsilon }^2\widehat{g}(t)+\widehat{h}(y,t).$$

(16)

Here,

$$\widehat{Q}=Q(t)+\frac{1}{\epsilon }{D}_{x}h({\underline{x}}_0,t),$$

$$\widehat{g}=\frac{1}{{\epsilon }^2}h({\underline{x}}_0,t)+\frac{1}{\epsilon }Q(t){\underline{x}}_0(t),$$

$$\widehat{h}=h({\underline{x}}_0+y,t)-h({\underline{x}}_0,t)-D_{x}h({\underline{x}}_0,t)y.$$

By the assumptions of Theorem 1 and Lemma 3, we have

$$|\parallel {\underline{x}}_0{\parallel |}_{z_0-{\overline{z}}_1,{\rho }_0-{\overline{\rho }}_1}\le \epsilon \frac{c\mathrm{\Gamma }({\overline{\rho }}_1)\mathrm{\Gamma }({\overline{z}}_1)}{{\alpha }_0}{|\parallel g\parallel |}_{z,\rho },$$

(17)

where $0<{\overline{z}}_1<\frac{1}{2}{z}_0$ and $0<{\overline{\rho }}_1<\frac{1}{2}{\rho }_0$. Define the average of $\widehat{Q}$ by $\overline{\widehat{Q}}$. Equation (16) is changed into

$$\dot{y}=(A_1+\epsilon \tilde{Q}(t))y+{\epsilon }^2\tilde{g}(t)+\tilde{h}(y,t),$$

(18)

where

$$A_1=A+\epsilon \overline{\widehat{Q}},\widehat{Q}-\overline{\widehat{Q}}=\tilde{Q},\widehat{g}=\tilde{g},\widehat{h}=\tilde{h}.$$

From the assumptions of Theorem 1, we see that the eigenvalues of $A_1$ are ${\lambda }_1^{+},{\lambda }_2^{+},\cdots ,{\lambda }_{2n}^{+}$, which satisfy $|{\lambda }_i^{+}-{\lambda }_j^{+}|\ge 2\delta \epsilon >0$ and $|{\lambda }_i^{+}|\ge 2\delta \epsilon >0,i,j=1,\cdots ,2n,i\ne j.$

Introduce the transformation $y=e^{\epsilon P_0(t)}{x}_1$. By this symplectic transformation, system (18) is changed into

$$\begin{array}{ccc}\hfill {\dot{x}}_1& =& (e^{-\epsilon P_0(t)}(A_1+\epsilon \tilde{Q}-\epsilon {\dot{P}}_0)e^{\epsilon P_0(t)}\hfill \\ & & +e^{-\epsilon P_0(t)}(\epsilon {\dot{P}}_0{e}^{\epsilon P_0(t)}-\frac{d}{dt}(e^{\epsilon P_0(t)}))x_1\hfill \\ & & +e^{-\epsilon P_0(t)}{\epsilon }^2\tilde{g}(t)+e^{-\epsilon P_0(t)}\tilde{h}(e^{\epsilon P_0(t)}{x}_1,t),\hfill \end{array}$$

(19)

where $x_1\in B_{a_1}(0).$

Expand $e^{\epsilon P_0}$ and $e^{-\epsilon P_0}$ into

$$e^{\epsilon P_0}=I+\epsilon P_0+B,e^{-\epsilon P_0}=I-\epsilon P_0+\tilde{B},$$

where

$$B=\frac{{(\epsilon P_0)}^2}{2!}+\frac{{(\epsilon P_0)}^3}{3!}+\cdots ,\tilde{B}=\frac{{(\epsilon P_0)}^2}{2!}-\frac{{(\epsilon P_0)}^3}{3!}+\cdots .$$

System (19) is rewritten as follows:

$$\begin{array}{ccc}\hfill {\dot{x}}_1& =& ((I-\epsilon P_0+\tilde{B})(A_1+\epsilon \tilde{Q}-\epsilon {\dot{P}}_0)(I+\epsilon P_0+B)\hfill \\ & & +e^{-\epsilon P_0(t)}(\epsilon {\dot{P}}_0{e}^{\epsilon P_0(t)}-\frac{d}{dt}(e^{\epsilon P_0(t)})))x_1\hfill \\ & & +e^{-\epsilon P_0(t)}{\epsilon }^2\tilde{g}(t)+e^{-\epsilon P_0(t)}\tilde{h}(e^{\epsilon P_0(t)}{x}_1,t)\hfill \\ & =& (A_1+\epsilon \tilde{Q}-\epsilon {\dot{P}}_0+\epsilon A_1{P}_0-\epsilon P_0{A}_1+Q^{(1)})x_1\hfill \\ & & +e^{-\epsilon P_0(t)}{\epsilon }^2\tilde{g}(t)+e^{-\epsilon P_0(t)}\tilde{h}(e^{\epsilon P_0(t)}{x}_1,t),\hfill \end{array}$$

(20)

where

$$\begin{array}{ccc}\hfill Q^{(1)}& =& -{\epsilon }^2{P}_0(\tilde{Q}-{\dot{P}}_0)+{\epsilon }^2(\tilde{Q}-{\dot{P}}_0)P_0-{\epsilon }^2{P}_0(A_1+\epsilon \tilde{Q}-\epsilon {\dot{P}}_0)P_0\hfill \\ & & -\epsilon P_0(A_1+\epsilon \tilde{Q}-\epsilon {\dot{P}}_0)B+(A_1+\epsilon \tilde{Q}-\epsilon {\dot{P}}_0)B\hfill \\ & & +\tilde{B}(A_1+\epsilon \tilde{Q}-\epsilon {\dot{P}}_0)e^{\epsilon P_0}+e^{-\epsilon P_0}\left(\epsilon {\dot{P}}_0{e}^{\epsilon P_0}-\frac{d}{dt}{e}^{\epsilon P_0}\right).\hfill \end{array}$$

We would like to have

$$\tilde{Q}-{\dot{P}}_0+A_1{P}_0-P_0{A}_1=0,$$

which is equivalent to

$${\dot{P}}_0=A_1{P}_0-P_0{A}_1+\widehat{Q}-\overline{\widehat{Q}}.$$

(21)

According to Lemma 4, if

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i^{+}+{\lambda }_j^{+}|\ge \frac{{\alpha }_1}{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}$$

for all $k\in {\mathbb{Z}}^{\mathbb{N}}\setminus \{0\},$ where ${\alpha }_1=\frac{\alpha }{4}$, then Equation (21) has a unique analytic almost periodic solution $P(t)$, its frequencies are $\omega$, and it has a spatial structure $(\tau ,[·])$, which satisfies

$$|\parallel P_0{\parallel |}_{z_0-2{\overline{z}}_1,{\rho }_0-2{\overline{\rho }}_1}\le c\frac{\mathrm{\Gamma }({\overline{z}}_1)\mathrm{\Gamma }({\overline{\rho }}_1)}{{\alpha }_1}|\parallel \widehat{Q}{\parallel |}_{z_0-{\overline{z}}_1,{\rho }_0-{\overline{\rho }}_1}.$$

(22)

System (20) becomes

$${\dot{x}}_1=(A_1+{\epsilon }^2{Q}_1(t))x_1+{\epsilon }^2{g}_1(t)+h_1(x_1,t),$$

(23)

where

$${\epsilon }^2{Q}_1=Q^{(1)},g_1(t)=e^{-\epsilon P_0}\tilde{g},h_1(x_1,t)=e^{-\epsilon P_0}\tilde{h}(e^{\epsilon P_0}{x}_1,t).$$

Since $\tilde{Q}-{\dot{P}}_0=P_0{A}_1-A_1{P}_0,$

$$\begin{array}{ccc}\hfill {\epsilon }^2{Q}_1& =& Q^{(1)}=-{\epsilon }^2{P}_0(P_0{A}_1-A_1{P}_0)+{\epsilon }^2(P_0{A}_1-A_1{P}_0)P_0\hfill \\ & & -{\epsilon }^2{P}_0(A_1+\epsilon P_0{A}_1-\epsilon A_1{P}_0)P_0\hfill \\ & & -\epsilon P_0(A_1+\epsilon P_0{A}_1-\epsilon A_1{P}_0)B+(A_1+\epsilon P_0{A}_1-\epsilon A_1{P}_0)B\hfill \\ & & +\tilde{B}(A_1+\epsilon P_0{A}_1-\epsilon A_1{P}_0)e^{\epsilon P_0}+e^{-\epsilon P_0}\left(\epsilon {\dot{P}}_0{e}^{\epsilon P_0}-\frac{d}{dt}{e}^{\epsilon P_0}\right).\hfill \end{array}$$

Thus, the symplectic transformation is $T_0{x}_1={\underline{x}}_0+e^{\epsilon P}{x}_1={\varphi }_0(t)+{\psi }_0(t)x_1.$ If

$$|\parallel P_0{\parallel |}_{z_0-2{\overline{z}}_1,{\rho }_0-2{\overline{\rho }}_1}\le \frac{1}{2},$$

then by (17) and (22), we have

$$|\parallel {\varphi }_0{\parallel |}_{z_0-2{\overline{z}}_1,{\rho }_0-2{\overline{\rho }}_1}\le \epsilon \frac{c\mathrm{\Gamma }({\overline{\rho }}_1)\mathrm{\Gamma }{(\overline{z})}_1}{{\alpha }_0}{|\parallel g\parallel |}_{z,\rho }.$$

$$|\parallel {\psi }_0-{I\parallel |}_{z_0-2{\overline{z}}_1,{\rho }_0-2{\overline{\rho }}_1}\le \epsilon \frac{c\mathrm{\Gamma }({\overline{z}}_1)\mathrm{\Gamma }({\overline{\rho }}_1)}{{\alpha }_1}|\parallel \widehat{Q}{\parallel |}_{z_0-{\overline{z}}_1,{\rho }_0-{\overline{\rho }}_1}.$$

Hence, under the symplectic transformation $x=T_0{x}_1$, system (1) becomes Hamiltonian system (23).

3.2. The mth KAM Step

The first step has been completed. That is, $A_1$ has $2n$ different eigenvalues, and ${\epsilon }^2{Q}_1(t)$ and ${\epsilon }^2{g}_1(t)$ are smaller perturbations. In the mth step, consider the Hamiltonian system

$${\dot{x}}_m=(A_m+{\epsilon }^{2^m}{Q}_m(t))x_m+{\epsilon }^{2^m}{g}_m(t)+h_m(x_m,t),m\ge 1,$$

(24)

where $x_m\in B_{a_m}(0),Q_m,g_m,h_m$ are analytic almost periodic on $D_{{\rho }_m}$, with frequencies $\omega$ and the same spatial structure $(\tau ,[·])$. $A_m$ has $2n$ different eigenvalues ${\lambda }_1^m,\cdots ,{\lambda }_{2n}^m$ with

$$|{\lambda }_i^m|\ge \delta \epsilon ,|{\lambda }_i^m-{\lambda }_j^m|\ge \delta \epsilon ,i\ne j,1\le i,j\le 2n,$$

where we denote ${\lambda }_i^1={\lambda }_i^{+},i=1,\cdots ,2n.$

By the symplectic transformation $x_m={\underline{x}}_m+y$, where ${\underline{x}}_m$ is solution of ${\dot{\underline{x}}}_m=A_m{\underline{x}}_m+{\epsilon }^{2^m}g(t)$ on $D_{{\rho }_m-{\overline{\rho }}_{m+1}}$, Hamiltonian system (24) becomes

$$\dot{y}=(A_m+{\epsilon }^{2^m}{\widehat{Q}}_m(t))y+{\epsilon }^{2^{m+1}}\widehat{g}(t)+\widehat{h}(y,t),$$

(25)

where

$${\widehat{Q}}_m=Q_m(t)+\frac{1}{{\epsilon }^{2^m}}{D}_x{h}_m({\underline{x}}_m,t),$$

$${\widehat{g}}_m=\frac{1}{{\epsilon }^{2^{m+1}}}{h}_m({\underline{x}}_m,t)+\frac{1}{{\epsilon }^{2^m}}{Q}_m(t){\underline{x}}_m(t),$$

$${\widehat{h}}_m=h_m({\underline{x}}_m+y,t)-h_m({\underline{x}}_m,t)-D_x{h}_m({\underline{x}}_m,t)y.$$

By Lemma 3, if

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i^m|\ge \frac{{\alpha }_m}{{\Delta }^4(|k|){\Delta }^4(\left[k\right])},k\in {\mathbb{Z}}^{\mathbb{N}},$$

we have

$$|\parallel {\underline{x}}_m{\parallel |}_{z_m-{\overline{z}}_{m+1},{\rho }_m-{\overline{\rho }}_{m+1}}\le {\epsilon }^{2^m}\frac{c\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}|\parallel g_m{\parallel |}_{z_m,{\rho }_m}.$$

(26)

Define the average of ${\widehat{Q}}_m$ by ${\overline{\widehat{Q}}}_m$. Equation (25) is changed into

$$\dot{y}=(A_{m+1}+{\epsilon }^{2^m}{\tilde{Q}}_m(t))y+{\epsilon }^{2^{m+1}}{\tilde{g}}_m(t)+{\tilde{h}}_m(y,t),$$

(27)

where

$$A_{m+1}=A_m+{\epsilon }^{2^m}{\overline{\widehat{Q}}}_m,{\widehat{Q}}_m-{\overline{\widehat{Q}}}_m={\tilde{Q}}_m,{\widehat{g}}_m={\tilde{g}}_m,{\widehat{h}}_m={\tilde{h}}_m.$$

Denote the eigenvalues of $A_{m+1}$ by ${\lambda }_1^{m+1},{\lambda }_2^{m+1},\cdots ,{\lambda }_{2n}^{m+1}$.

In making the change of variables $y=e^{{\epsilon }^{2^m}{P}_m(t)}{x}_{m+1}$, where $P_m(t)$ is to be determined later, by the symplectic transformation, Hamiltonian system (27) becomes the new system

$$\begin{array}{ccc}\hfill {\dot{x}}_{m+1}& =& (e^{-{\epsilon }^{2^m}{P}_m(t)}(A_{m+1}+{\epsilon }^{2^m}{\tilde{Q}}_m-{\epsilon }^{2^m}{\dot{P}}_m)e^{{\epsilon }^{2^m}{P}_m(t)}\hfill \\ & & +e^{-{\epsilon }^{2^m}{P}_m(t)}({\epsilon }^{2^m}{\dot{P}}_m{e}^{{\epsilon }^{2^m}{P}_m(t)}-\frac{d}{dt}(e^{{\epsilon }^{2^m}{P}_m(t)})))x_{m+1}\hfill \\ & & +e^{-{\epsilon }^{2^m}{P}_m(t)}{\epsilon }^{2^{m+1}}{\tilde{g}}_m(t)\hfill \\ & & +e^{-{\epsilon }^{2^m}{P}_m(t)}{\tilde{h}}_m(e^{{\epsilon }^{2^m}{P}_m(t)}{x}_{m+1},t),\hfill \end{array}$$

(28)

where $x_{m+1}\in B_{a_{m+1}}(0).$

Expand $e^{{\epsilon }^{2^m}{P}_m}$ and $e^{-{\epsilon }^{2^m}{P}_m}$ into

$$e^{{\epsilon }^{2^m}{P}_m}=I+{\epsilon }^{2^m}{P}_m+B_m,e^{-{\epsilon }^{2^m}{P}_m}=I-{\epsilon }^{2^m}{P}_m+{\tilde{B}}_m,$$

where

$$B_m=\frac{{({\epsilon }^{2^m}{P}_m)}^2}{2!}+\frac{{({\epsilon }^{2^m}{P}_m)}^3}{3!}+\cdots ,{\tilde{B}}_m=\frac{{({\epsilon }^{2^m}{P}_m)}^2}{2!}-\frac{{({\epsilon }^{2^m}{P}_m)}^3}{3!}+\cdots .$$

Then, system (28) can be rewritten as follows:

$$\begin{array}{ccc}\hfill {\dot{x}}_{m+1}& =& ((I-{\epsilon }^{2^m}{P}_m+{\tilde{B}}_m)(A_{m+1}+{\epsilon }^{2^m}\tilde{Q}-{\epsilon }^{2^m}{\dot{P}}_m)(I+{\epsilon }^{2^m}{P}_m+B_m)\hfill \\ & & +e^{-{\epsilon }^{2^m}{P}_m(t)}({\epsilon }^{2^m}{\dot{P}}_m{e}^{{\epsilon }^{2^m}{P}_m(t)}-\frac{d}{dt}(e^{{\epsilon }^{2^m}{P}_m(t)})))x_{m+1}\hfill \\ & & +e^{-{\epsilon }^{2^m}{P}_m(t)}{\epsilon }^{2^{m+1}}{\tilde{g}}_m(t)+e^{-{\epsilon }^{2^m}{P}_m(t)}{\tilde{h}}_m(e^{{\epsilon }^{2^m}{P}_m(t)}{x}_{m+1},t)\hfill \\ & =& (A_{m+1}+{\epsilon }^{2^m}{\tilde{Q}}_m-{\epsilon }^{2^m}{\dot{P}}_m+{\epsilon }^{2^m}{A}_{m+1}{P}_m-{\epsilon }^{2^m}{P}_m{A}_{m+1}+Q_m^{(1)})x_{m+1}\hfill \\ & & +e^{-{\epsilon }^{2^m}{P}_m(t)}{\epsilon }^{2^{m+1}}{\tilde{g}}_m(t)+e^{-{\epsilon }^{2^m}{P}_m(t)}{\tilde{h}}_m(e^{{\epsilon }^{2^m}{P}_m(t)}{x}_{m+1},t),\hfill \end{array}$$

(29)

where

$$\begin{array}{ccc}\hfill Q_m^{(1)}& =& -{\epsilon }^{2^{m+1}}{P}_m({\tilde{Q}}_m-{\dot{P}}_m)+{\epsilon }^{2^{m+1}}({\tilde{Q}}_m-{\dot{P}}_m)P_m\hfill \\ & & -{\epsilon }^{2^{m+1}}{P}_m(A_{m+1}+{\epsilon }^{2^m}{\tilde{Q}}_m-\epsilon {\dot{P}}_m)P_m\hfill \\ & & -{\epsilon }^{2^m}{P}_m(A_{m+1}+{\epsilon }^{2^m}{\tilde{Q}}_m-{\epsilon }^{2^m}{\dot{P}}_m)B_m\hfill \\ & & +(A_{m+1}+{\epsilon }^{2^m}{\tilde{Q}}_m-{\epsilon }^{2^m}{\dot{P}}_m)B_m\hfill \\ & & +{\tilde{B}}_m(A_{m+1}+{\epsilon }^{2^m}{\tilde{Q}}_m-{\epsilon }^{2^m}{\dot{P}}_m)e^{{\epsilon }^{2^m}{P}_m}\hfill \\ & & +e^{-{\epsilon }^{2^m}{P}_m}\left({\epsilon }^{2^m}{\dot{P}}_m{e}^{{\epsilon }^{2^m}{P}_m}-\frac{d}{dt}{e}^{{\epsilon }^{2^m}{P}_m}\right).\hfill \end{array}$$

We would like to have

$${\tilde{Q}}_m-{\dot{P}}_m+A_{m+1}{P}_m-P_m{A}_{m+1}=0,$$

which is equivalent to

$${\dot{P}}_m=A_{m+1}{P}_m-P_m{A}_{m+1}+{\widehat{Q}}_m-{\overline{\widehat{Q}}}_m.$$

(30)

By Lemma 4, if

$$|{\lambda }_i^{m+1}|\ge \delta \epsilon ,|{\lambda }_i^{m+1}-{\lambda }_j^{m+1}|\ge \delta \epsilon ,i\ne j,1\le i,j\le 2n$$

and

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i^{m+1}+{\lambda }_j^{m+1}|\ge \frac{{\alpha }_{m+1}}{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}$$

for $k\in {\mathbb{Z}}^{\mathbb{N}}\setminus \{0\},$ then Equation (30) has a unique almost periodic Hamiltonian solution $P_m(t)$. Furthermore,

$$|\parallel P_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\le c\frac{\mathrm{\Gamma }({\overline{z}}_{m+1})\mathrm{\Gamma }({\overline{\rho }}_{m+1})}{{\alpha }_{m+1}}|\parallel {\widehat{Q}}_m{\parallel |}_{z_m-{\overline{z}}_{m+1},{\rho }_m-{\overline{\rho }}_{m+1}}.$$

(31)

System (29) becomes

$${\dot{x}}_{m+1}=(A_{m+1}+{\epsilon }^{2^{m+1}}{Q}_{m+1}(t))x_{m+1}+{\epsilon }^{2^{m+1}}{g}_{m+1}(t)+h_{m+1}(x_{m+1},t),$$

(32)

where

$${\epsilon }^{2^{m+1}}{Q}_1=Q_m^{(1)},g_{m+1}(t)=e^{-{\epsilon }^{2^m}{P}_m}{\tilde{g}}_m,$$

$$h_{m+1}(x_{m+1},t)=e^{-{\epsilon }^{2^m}{P}_m}{\tilde{h}}_m(e^{{\epsilon }^{2^m}{P}_m}{x}_{m+1},t).$$

Since ${\tilde{Q}}_m-{\dot{P}}_m=P_m{A}_{m+1}-A_{m+1}{P}_m,$

$$\begin{array}{ccc}\hfill {\epsilon }^{2^{m+1}}{Q}_{m+1}& =& Q_m^{(1)}=-{\epsilon }^{2^{m+1}}{P}_m(P_m{A}_{m+1}-A_{m+1}{P}_m)\hfill \\ & & +{\epsilon }^{2^{m+1}}(P_m{A}_{m+1}-A_{m+1}{P}_m)P_m\hfill \\ & & -{\epsilon }^{2^{m+1}}{P}_m(A_{m+1}+{\epsilon }^{2^m}{P}_m{A}_{m+1}-{\epsilon }^{2^m}{A}_{m+1}{P}_m)P_m\hfill \\ & & -{\epsilon }^{2^m}{P}_m(A_{m+1}+{\epsilon }^{2^m}{P}_m{A}_{m+1}-{\epsilon }^{2^m}{A}_{m+1}{P}_m)B_m\hfill \\ & & +(A_{m+1}+{\epsilon }^{2^m}{P}_m{A}_{m+1}-{\epsilon }^{2^m}{A}_{m+1}{P}_m)B_m\hfill \\ & & +{\tilde{B}}_m(A_{m+1}+{\epsilon }^{2^m}{P}_m{A}_{m+1}-{\epsilon }^{2^m}{A}_{m+1}{P}_m)e^{{\epsilon }^{2^m}{P}_m}\hfill \\ & & +e^{-{\epsilon }^{2^m}{P}_m}\left({\epsilon }^{2^m}{\dot{P}}_m{e}^{{\epsilon }^{2^m}{P}_m}-\frac{d}{dt}{e}^{{\epsilon }^{2^m}{P}_m}\right).\hfill \end{array}$$

(33)

Hence, the symplectic changes of variables are

$$T_m{x}_{m+1}={\underline{x}}_m+e^{{\epsilon }^{2^m}{P}_m}{x}_{m+1}={\varphi }_m(t)+{\psi }_m(t)x_{m+1}.$$

If $|\parallel P_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\le \frac{1}{2},$ by (26) and (31), we have

$$|\parallel {\varphi }_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\le {\epsilon }^{2^m}\frac{c\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}|\parallel g_m{\parallel |}_{z_m,{\rho }_m}.$$

$$|\parallel {\psi }_m-{I\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\le {\epsilon }^{2^m}\frac{c\mathrm{\Gamma }({\overline{z}}_{m+1})\mathrm{\Gamma }({\overline{\rho }}_{m+1})}{{\alpha }_{m+1}}|\parallel {\widehat{Q}}_m{\parallel |}_{z_m-{\overline{z}}_{m+1},{\rho }_m-{\overline{\rho }}_{m+1}}.$$

Under the symplectic transformation $x_m=T_m{x}_{m+1}$, system (24) becomes system (32).

3.3. Iteration

In this section, we prove the convergence of the iteration as $m\to \infty$.

From the arbitrariness of $\overline{z}$ and $\overline{\rho }$, we set $z_m,{\rho }_m$ as follows:

$$z_m=z-\sum _{\nu =1}^{m}2{\overline{z}}_{\nu },{\rho }_m=\rho -\sum _{\nu =1}^{m}2{\overline{\rho }}_{\nu },$$

where ${\overline{z}}_{\nu }\downarrow 0$ and ${\overline{\rho }}_{\nu }\downarrow 0$ satisfy

$$\sum _{\nu =1}^{\infty }{\overline{z}}_{\nu }=\frac{1}{4}z,\sum _{\nu =1}^{\infty }{\overline{\rho }}_{\nu }=\frac{1}{4}\rho .$$

Moreover, we choose

$${\alpha }_m=\frac{{\alpha }_0}{{(m+1)}^2},a_{m+1}=\frac{a_m-|\parallel {\underline{x}}_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}}{e^{{\epsilon }^{2^m}}|\parallel P_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}}.$$

If $|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}\le \frac{1}{2},$ we have

$$a_{m+1}\ge \frac{a_m-|\parallel {\underline{x}}_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}}{1+2{\epsilon }^{2^m}|\parallel P_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}}.$$

(34)

If $\epsilon$ is small enough, from [8], it follows that

$$a_{\infty }=\underset{m\to \infty }{lim}{a}_m\ge \sigma >0.$$

By Lemma 5, we have

$$|\parallel {\widehat{Q}}_m{\parallel |}_{z_m-{\overline{z}}_{m+1},{\rho }_m-{\overline{\rho }}_{m+1}}\le |\parallel Q_m{\parallel |}_{z_m,{\rho }_m}+K_m\frac{c\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}|\parallel g_m{\parallel |}_{z_m,{\rho }_m}.$$

(35)

Thus, by (31) and (35), we have

$$\begin{array}{ccc}\hfill & & |\parallel P_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\hfill \\ & \le & \frac{c\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_{m+1}}\left(|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}+K_m\frac{c\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}|\parallel g_m{\parallel |}_{z_m,{\rho }_m}\right).\hfill \end{array}$$

From $K_m$ being convergent (see below), it follows that there exists $c_0>1$ such that $K_m\le c_0$. Thus, we have

$$\begin{array}{ccc}\hfill & & |\parallel P_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\hfill \\ & \le & c_{0}c{\left(\frac{\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_{m+1}}\right)}^2\left(|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}+|\parallel g_m{\parallel |}_{z_m,{\rho }_m}\right)\hfill \\ & \le & c{\left(\frac{\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_{m+1}}\right)}^2\left(|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}+|\parallel g_m{\parallel |}_{z_m,{\rho }_m}\right).\hfill \end{array}$$

(36)

We first estimate $|\parallel g_{m+1}{\parallel |}_{z_{m+1},{\rho }_{m+1}}.$ By Lemma 5, we have

$$\begin{array}{ccc}\hfill & & |\parallel g_{m+1}{\parallel |}_{z_{m+1},{\rho }_{m+1}}\hfill \\ & \le & \frac{c\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}|\parallel g_m{\parallel |}_{z_m,{\rho }_m}\hfill \\ & & +c{\left(\frac{\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}\right)}^2|\parallel g_m{\parallel |}_{z_m,{\rho }_m}^2\hfill \\ & \le & c{\left(\frac{\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}\right)}^2\left(|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}|\parallel g_m{\parallel |}_{z_m,{\rho }_m}+|\parallel g_m{\parallel |}_{z_m,{\rho }_m}^2\right).\hfill \end{array}$$

(37)

Now, we estimate $|\parallel Q_{m+1}{\parallel |}_{z_{m+1},{\rho }_{m+1}}.$ If $|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}\le \frac{1}{2}$, it follows that

$$\begin{array}{ccc}\hfill & & |\parallel e^{\pm {\epsilon }^{2^m}{P}_m}{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\hfill \\ & \le & 1+|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}+\frac{|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}^2}{2!}+\cdots \hfill \\ & \le & 2.\hfill \end{array}$$

Moreover, if $|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_m-{\overline{z}}_{m+1},{\rho }_m-{\overline{\rho }}_{m+1}}\le \frac{1}{2},$ by

$$\begin{array}{ccc}& & |\parallel \frac{d}{dt}{({\epsilon }^{2^m}{P}_m)}^n{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\hfill \\ & \le & n|\parallel {\epsilon }^{2^m}{\dot{P}}_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}(|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}{)}^{n-1}\hfill \end{array}$$

for $n\in {\mathbb{Z}}^{+}$, we have

$$\begin{array}{ccc}& & |\parallel {\epsilon }^{2^m}{\dot{P}}_m{e}^{{\epsilon }^{2^m}{P}_m}-\frac{d}{dt}{e}^{{\epsilon }^{2^m}{P}_m}{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\hfill \\ & \le & 4|\parallel {\epsilon }^{2^m}{\dot{P}}_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}.\hfill \end{array}$$

So,

$$\begin{array}{ccc}\hfill & & |\parallel e^{-{\epsilon }^{2^m}{P}_m}({\epsilon }^{2^m}\dot{P_m}{e}^{{\epsilon }^{2^m}{P}_m}-\frac{d}{dt}{e}^{{\epsilon }^{2^m}{P}_m}){\parallel |}_{z_{m+1},{\rho }_{m+1}}\hfill \\ & \le & c{\epsilon }^{2^{m+1}}(|\parallel P_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}^2+|\parallel P_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}\hfill \\ & & +\parallel P_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}{K}_m\frac{c\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}|\parallel g_m{\parallel |}_{z_m,{\rho }_m}).\hfill \end{array}$$

(38)

From the representations of $B_m$ and ${\tilde{B}}_m$, we have

$$\begin{array}{ccc}\hfill & & |\parallel B_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}},|\parallel {\tilde{B}}_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\hfill \\ & \le & 2|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}.\hfill \end{array}$$

(39)

Then from (33) and (38), it follows that

$$\begin{array}{ccc}& & |\parallel Q_{m+1}{\parallel |}_{z_m-2{\overline{z}}_{m+1},{\rho }_m-2{\overline{\rho }}_{m+1}}\hfill \\ & \le & \frac{c\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}(|\parallel P_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}^2\hfill \\ & & +|\parallel P_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}+|\parallel P_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}|\parallel g_m{\parallel |}_{z_m,{\rho }_m}).\hfill \end{array}$$

Then by (36), we have

$$\begin{array}{ccc}& & |\parallel Q_{m+1}{\parallel |}_{z_{m+1},{\rho }_{m+1}}\hfill \\ & \le & c{\left(\frac{\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_{m+1}}\right)}^5(|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}^2+|\parallel g_m{\parallel |}_{z_m,{\rho }_m}^2\hfill \\ & & +|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}|\parallel g_m{\parallel |}_{z_m,{\rho }_m}).\hfill \end{array}$$

Set

$$C_m={\left[{(m+2)}^{2^{-m+2}}{(m+1)}^{2^{-m+1}}{m}^{2^{-m}}\cdots 2^{-2}\right]}^5$$

and

$${\mathrm{\Phi }}_m(z)={\Pi }_{\nu =1}^{m+1}{\left[\mathrm{\Gamma }({\overline{z}}_{\nu })\right]}^{5^{-\nu }},{\mathrm{\Phi }}_m(\rho )={\Pi }_{\nu =1}^{m+1}{\left[\mathrm{\Gamma }({\overline{\rho }}_{\nu })\right]}^{5^{-\nu }}.$$

From [18], $C_m,{\mathrm{\Phi }}_m(z),$ and ${\mathrm{\Phi }}_m(\rho )$ are all convergent as $m\to \infty$.

Let

$$C_1=max\{\frac{c}{{\alpha }_0^5},1\},\gamma ={max\{|\parallel Q\parallel |}_{z_0,{\rho }_0}{,|\parallel g\parallel |}_{z_0,{\rho }_0}\}$$

and

$$M=max\{1,\underset{m}{sup}(C_1{C}_m{\mathrm{\Phi }}_m(z){\mathrm{\Phi }}_m(\rho )\}\gamma .$$

Then, we have

$$|\parallel Q_{m+1}{\parallel |}_{z_{m+1},{\rho }_{m+1}}<M^{2^{m+2}},|\parallel g_{m+1}{\parallel |}_{z_{m+1},{\rho }_{m+1}}<M^{2^{m+2}}.$$

If $0<\epsilon M^2<1$, then

$$\underset{m\to \infty }{lim}{\epsilon }^{2^m}{Q}_m=0,\underset{m\to \infty }{lim}{\epsilon }^{2^m}{g}_m=0.$$

Moreover, by (35), we obtain

$$\begin{array}{ccc}\hfill \left|\right|A_{m+1}-A_m\left|\right|& \le & {\epsilon }^{2^m}\left|\right|{\overline{\widehat{Q}}}_m\left|\right|\hfill \\ & \le & {\epsilon }^{2^m}(|\parallel Q_m{\parallel |}_{z_m,{\rho }_m}+\frac{c{K}_m\mathrm{\Gamma }({\overline{\rho }}_{m+1})\mathrm{\Gamma }({\overline{z}}_{m+1})}{{\alpha }_m}|\parallel Q_m{\parallel |}_{z_m,{\rho }_m})\hfill \\ & \le & {(\epsilon M^2)}^{2^m}.\hfill \end{array}$$

Thus, ${lim}_{m\to \infty }\left|\right|A_{m+1}-A_m\left|\right|=0.$ That is, $A_m$ is convergent when $m\to \infty$. Let

$$A_m\to B(m\to \infty ).$$

Furthermore, if ${(\epsilon M^2)}^{2^m}<\frac{\delta \epsilon }{(6n-1){\beta }_m^2}$, we have

$$\left|\right|A_{m+1}-A_m\left|\right|\le \frac{\delta \epsilon }{(6n-1){\beta }_m^2},foranym\ge 1,$$

where ${\beta }_m=max\{||S_m\left|\right|,\left|\right|S_m^{-1}\left|\right|\}$, and $S_m$ is the nonsingular matrix in Lemma 2 satisfing

$$S_m^{-1}{A}_m{S}_m=diag({\lambda }_1^m,\cdots ,{\lambda }_{2n}^m).$$

Therefore, from Lemma 2, it follows that the eigenvalues ${\lambda }_1^{m+1},\cdots ,{\lambda }_{2n}^{m+1}$ of $A_{m+1}$ are different. Moreover,

$$|{\lambda }_i^{m+1}-{\lambda }_j^{m+1}|\ge \epsilon \delta ,1\le i,j\le 2n,i\ne j,$$

and

$$|{\lambda }_j^{m+1}|\ge \epsilon \delta ,j=1,\cdots ,2n.$$

Next we present the proof of the above inequalities:

$$\begin{array}{ccc}\hfill |{\lambda }_i^{m+1}-{\lambda }_j^{m+1}|& \ge & |{\lambda }_i^1-{\lambda }_j^1|-\sum _{l=1}^m\left(|{\lambda }_i^{l+1}-{\lambda }_i^l|+|{\lambda }_j^{l+1}-{\lambda }_j^l|\right)\hfill \\ & \ge & |{\lambda }_i^1-{\lambda }_j^1|-2\sum _{l=1}^m\left|\right|A_{l+1}-A_l\left|\right|\hfill \\ & \ge & |{\lambda }_i^1-{\lambda }_j^1|-2\sum _{l=1}^m{(\epsilon M^2)}^{2^l}\hfill \\ & \ge & 2\delta \epsilon -2\sum _{l=1}^m{(\epsilon M^2)}^{2^l}\hfill \\ & \ge & 2\delta \epsilon -4{(\epsilon M^2)}^2.\hfill \end{array}$$

Thus, if $\epsilon <\frac{\delta }{4M^4}$, we have

$$|{\lambda }_i^{m+1}-{\lambda }_j^{m+1}|\ge \epsilon \delta ,1\le i,j\le 2n,i\ne j.$$

Similarly, we obtain

$$|{\lambda }_i^{m+1}|\ge \epsilon \delta ,i=1,\cdots ,2n.$$

Then,

$$\underset{m\to \infty }{lim}|\parallel {\varphi }_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}=0,\underset{m\to \infty }{lim}|\parallel {\psi }_m-I{\parallel |}_{z_{m+1},{\rho }_{m+1}}=0.$$

Let $T^m=T_0\circ T_1\circ \cdots \circ T_{m-1}.$ Thus, $T^m$ is convergent on $D_{\frac{z_0}{2},\frac{{\rho }_0}{2}}$. Assume that $T^m\to T$ as $m\to \infty$. By (32), we have

$$|\parallel D_{x_{m+1}{x}_{m+1}}{h}_{m+1}{\parallel |}_{z_{m+1},{\rho }_{m+1}}\le \frac{(1+2|\parallel {\epsilon }^{2^m}{\parallel |}_{z_{m+1},{\rho }_{m+1}}{)}^2}{1-2|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}}{K}_m.$$

(40)

If $|\parallel {\epsilon }^{2^m}{P}_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}\le \frac{1}{4}$, by (40), we have $K_m\le {(\frac{9}{2})}^m{K}_0.$ Since, $\frac{1}{1-x}\le 1+2x$, if $0\le x\le \frac{1}{2}$,

$$K_{m+1}\le (1+4{\epsilon }^{2^m}|\parallel P_m{\parallel |}_{z_{m+1},{\rho }_{m+1}}{)}^3{K}_m.$$

(41)

Since $K_m\le {(\frac{9}{2})}^m{K}_0$, then by (41), we obtain the convergence of $K_m.$ Let $K_m\to K_{\infty }(m\to \infty ).$ Hence,

$$\underset{m\to \infty }{lim}{h}_m(x_m,t)=h_{\infty }(y,t)=O(y^2),$$

as $y\to 0$.

Thus, under the transformation

$$x=Ty=\psi (t)y+\phi (t),$$

Hamiltonian system (1) is changed into system (8).

3.4. Measure Estimate

Firstly, we prove that the following non-resonant conditions

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i^m|\ge \frac{{\alpha }_m}{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}$$

(42)

and

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i^{m+1}+{\lambda }_j^{m+1}|\ge \frac{{\alpha }_{m+1}}{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}$$

(43)

hold for most small $\epsilon \in (0,{\epsilon }_0),$ where $k\in {\mathbb{Z}}^{\mathbb{N}}\setminus \{0\}$.

By Theorem B in [13], there exist ${\epsilon }_0$ and non-empty set $E^{\ast }\subset (0,{\epsilon }_0)$ such that for $\epsilon \in E^{\ast }$, we have

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i^{m+1}+{\lambda }_j^{m+1}|\ge \frac{{\alpha }_m}{2{\Delta }^4(|k|){\Delta }^4(\left[k\right])},$$

$$\underset{{\epsilon }_0\to 0}{lim}\frac{meas(E^{\ast })}{{\epsilon }_0}=1.$$

Obviously, (43) holds.

In the same way as above, we can obtain

$$|\langle k,\omega \rangle \sqrt{-1}-{\lambda }_i^m|\ge \frac{{\alpha }_m}{{\Delta }^4(|k|){\Delta }^4(\left[k\right])},$$

$$\underset{{\epsilon }_0\to 0}{lim}\frac{meas(E^{\ast })}{{\epsilon }_0}=1.$$

That is, (42) holds.

Thus, there exists a non-empty Cantor subset $E^{\ast }\subset (0,{\epsilon }_0)$ such that for $\epsilon \in E^{\ast }$, there exists an almost periodic symplectic transformation

$$x=\psi (t,\epsilon )y+\phi (t,\epsilon )$$

that changes (1) into

$$\dot{y}=By+h_{\infty }(y,t).$$

where $E^{\ast }$ has a positive Lebesgue measure, and $\psi (t)$ and $\phi (t)$ have the same basic frequencies and spatial structure as $Q(t)$. The matrix B is a real constant, and $h_{\infty }(y,t)=O(y^2)$ as $y\to 0$. Moreover, $meas((0,{\epsilon }_0)\setminus E^{\ast })=o({\epsilon }_0)$ as ${\epsilon }_0\to 0$. Therefore, we have completed the proof of Theorem 1.

4. Application

Now, we apply Theorem 1 to the almost periodic Hill’s equation

$$\ddot{x}+\epsilon a(t)x=0,$$

(44)

where $a(t)$ is an analytic almost periodic function on $D_{\rho }$ with frequencies $\omega =({\omega }_1,{\omega }_2,\cdots )$ and has spatial structure $(\tau ,[·])$. Denote the average of $a(t)$ by $\overline{a}$.

Let $\dot{x}=y$, Equation (44) equivalently becomes of the form

$$\dot{x}=y,\dot{y}=-\epsilon a(t)x.$$

(45)

In order to apply Theorem 1, we rewrite (45) as follows:

$$\dot{Z}=(A+\epsilon Q(t))Z,$$

(46)

where

$$Z=\left(\begin{array}{c}x\\ y\end{array}\right),A=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),Q=\left(\begin{array}{cc}0& 0\\ -a(t)& 0\end{array}\right)$$

and $g(t)\equiv 0,$ and $h(x,t)\equiv 0.$ It is easy to see that A has multiple eigenvalues ${\lambda }_1={\lambda }_2=0$ and that $A+\epsilon \overline{Q}$ has two different eigenvalues ${\mu }_1=i\sqrt{\overline{a}\epsilon }$ and ${\mu }_2=-i\sqrt{\overline{a}\epsilon },$ where $i=\sqrt{-1}.$ Obviously,

$$|{\mu }_i|=\sqrt{\overline{a}}\sqrt{\epsilon }\ge 2\delta \epsilon ,i=1,2,$$

and

$$|{\mu }_1-{\mu }_2|=2\sqrt{\overline{a}}\sqrt{\epsilon }\ge 2\delta \epsilon$$

hold, where $\delta =\frac{1}{2}\sqrt{\overline{a}}$. Thus, by Theorem 1, we have the result as follows.

Theorem 2.

Assume that $a(t)={\sum }_{\Lambda \in \tau }{a}_{\Lambda }(t)$ is analytic almost periodic on $D_{\rho }$ with frequencies $\omega =({\omega }_1,{\omega }_2,\cdots )$ and has spatial structure $(\tau ,[·])$. If $\overline{a}>0$ and

$$|\langle k,\omega \rangle \sqrt{-1}|\ge \frac{{\alpha }_0}{{\Delta }^4(|k|){\Delta }^4(\left[k\right])}$$

(47)

holds for all $k\in {\mathbb{Z}}^{\mathbb{N}}\setminus \{0\}$, where the constant ${\alpha }_0>0$, and Δ is an approximation function.

Then, there exist small enough ${\epsilon }_0>0$ and the non-empty Cantor subset $E_{{\epsilon }_0}\subset (0,{\epsilon }_0)$ such that for $\epsilon \in E_{{\epsilon }_0}$, there exists an almost periodic symplectic transformation that changes (45) into a constant coefficient linear system. In addition, $\frac{meas(E_{{\epsilon }_0})}{{\epsilon }_0}\to 1$ as ${\epsilon }_0\to 0.$

From Theorem 2, we see that, for most small $\epsilon >0,$ Equation (44) is changed into a constant coefficient system. Hence, similar to Xue [12], by an analytic almost periodic transformation, Equation (44) is transformed into

$${\ddot{x}}_{\infty }+b{x}_{\infty }=0,$$

(48)

where $b=\overline{a}\epsilon +O({\epsilon }^2)$, which depends on $\overline{a}$ and $\epsilon$ only. Obviously, Equation (47) is elliptic, so the equilibrium is Lyapunov stable for most small enough $\epsilon$.

5. Conclusions

In this paper, we considered the reducibility of almost-periodic nonlinear Hamiltonian systems and proved that, for most small enough $\epsilon$, system (1) was reduced to a Hamiltonian system with an equilibrium. The result was proved by using some non-resonant conditions, non-degeneracy conditions and the KAM iterations. Application to the almost periodic Hill’s equation was also presented.