1 Introduction
In this paper, we consider the following system:
$$ \begin{align} \dot{x}=( A+ P(t, \epsilon))x,\quad x \in \mathbb{R}^{3}, \end{align} $$
where
$A, P$
are both
$3\times 3$
skew symmetric matrices, A is a constant matrix with eigenvalues
$\pm i \unicode{x3bb} $
and
$0,$
$P(t, \epsilon )$
is analytic quasi-periodic in t with basic frequencies
$\omega =(\omega _{1}, \ldots , \omega _{s}),$
and
$\epsilon $
is a small parameter.
Eliasson [Reference Eliasson13] proved that for most of sufficiently small parameters, the analytic quasi-periodic skew symmetric system (1) can be reduced to a constant system if the eigenvalues
$\pm i \unicode{x3bb} $
and basic frequencies
$\omega $
satisfy the non-resonant conditions:
$$ \begin{align} |\langle k, \omega\rangle|\geq \frac{\tilde{\gamma}}{|k|^{\tilde{\tau}}}\quad \text{for all } 0\neq k\in \mathbb{Z}^{s} \end{align} $$
and
$$ \begin{align*} |\langle k, \omega\rangle \pm \unicode{x3bb}|\geq \frac{\gamma}{|k|^{ \tau}},\quad |\langle k, \omega\rangle \pm 2 \unicode{x3bb}|\geq \frac{\gamma}{|k|^{\tau}} \quad \text{for all } 0\neq k\in \mathbb{Z}^{s}, \end{align*} $$
where
$ \tilde {\gamma }, \gamma> 0$
,
$\tilde {\tau }, \tau> s-1,$
and the non-degeneracy condition:
$$ \begin{align} \frac{d \unicode{x3bb}^{0}(\epsilon) }{d\epsilon}\bigg|_{\epsilon=0} \neq 0, \end{align} $$
where
$\pm i\unicode{x3bb} ^{0} (\epsilon ) $
are the eigenvalues of
$A+[P],$
with
$[P]$
being the average of
$P(t)$
with respect to
$t.$
Moreover, Eliasson [Reference Eliasson13] showed that such a system is generically uniquely ergodic in a topological sense and, in particular, non-reducible.
In this paper, we are mainly concerned with the reducibility problems for the analytic quasi-periodic skew symmetric system (1) without any non-degeneracy condition and with Liouvillean basic frequencies, that is,
$\omega =(1, \alpha ),$
where
$\alpha $
is irrational, which means that the Diophantine condition (2) and the non-degeneracy condition (3) are not necessary for the reducibility of three-dimensional skew symmetric systems with two-dimensional basic frequencies.
We first give a brief review of reducibility history from both perturbative and non-perturbative sides. Consider the system
$$ \begin{align} \dot{x}= A(t) x,\quad x\in\mathbb{R}^{n}, \end{align} $$
where
$A(t) \in gl(n, \mathbb {R})$
is an analytic quasi-periodic matrix with basic frequencies
${\omega = (\omega _{1}, \ldots , \omega _{s}).}$
Here,
$gl(n, \mathbb {R})$
denotes the Lie algebra of
$ n\times n$
real matrices and
$GL(n, \mathbb {R})$
the Lie group of
$ n\times n$
real invertible matrices.
A typical example of quasi-periodic systems comes from the linear Schrödinger equations
$$ \begin{align*} (\mathcal{L} y)(t)= - y"(t)+ q(\theta+ \omega t) y(t)= Ey(t), \end{align*} $$
which is equivalent to the linear systems
$$ \begin{align} \begin{cases} \dot{x}=V_{E, q}(\theta)x,\\ \dot{\theta}=\omega, \end{cases} \end{align} $$
where
$$ \begin{align*} V_{E, q}(\theta) =\left( \begin{array}{@{\,}ccc@{}} 0 & 1\\ q(\theta)-E & 0 \end{array} \!\!\right)\in sl(2,\mathbb{R}). \end{align*} $$
System (4) is said to be reducible if there exists a non-singular quasi-periodic change of variables
$x=\Phi (t)y,$
such that
$\Phi (t)$
,
$\Phi (t)^{-1} $
, and
$\dot {\Phi }(t) $
are quasi-periodic and bounded, and such that it transforms system (4) into a constant system, that is, a linear system with constant coefficients. System (4) is said to be rotations reducible if there exists a quasi-periodic transformation
$x=\Phi (t)y,$
such that system (4) is transformed into a rotation system, that is, a linear system with
$ so(2,\mathbb {R}) (so(3,\mathbb {R}))$
-valued coefficients. Note that the reduced coefficients of the rotation system may not be constant. If
$\Phi (t)$
is
$C^{l} $
(or analytic), we say that system (4) is
$C^{l} $
(or analytic) reducible ( rotations reducible). A weaker concept than reducibility is almost reducibility. System (4) is said to be almost reducible if there exists a sequence of quasi-periodic transformations
$x=\Phi _{n}(t)y,$
where
$\Phi _{n}(t)\in C_{h_{n}}^{\omega }( 2\mathbb {T}^{s}, Gl(n,\mathbb {R}) ),$
a sequence of constant matrices
$A_{n}\in gl(n, \mathbb {R}) $
and sequence of
$F_{n} \in C_{h_{n}}^{\omega }(\mathbb {T}^{s}, gl(n,\mathbb {R}) )$
converging to zero, such that system (4) is transformed by the change of variables
$x=\Phi _{n}(t)y$
into
$\dot {y} = (A_{n}+ F_{n}(\theta ) ) y. $
System (4) is called weak almost reducible if
$h_{n}\rightarrow 0,$
and strong almost reducible if
$h_{n}\rightarrow h_{*}> 0. $
Definiton 1. (Perturbative and non-perturbative reducibility)
If the smallness of the perturbation depends on the Diophantine constants
$(\tilde {\gamma }, \tilde {\tau })$
of
$\omega $
in the Diophantine condition (2), we call it perturbative reducibility; If the smallness of the perturbation does not depend on the Diophantine constants
$(\tilde {\gamma }, \tilde {\tau })$
of
$\omega $
in the Diophantine condition (2), we call it non-perturbative reducibility.
Perturbative reducibility. Dinaburg and Sinai [Reference Dinaburg and Sinai11] used the KAM scheme to obtain that system (5) is reducible for a positive Lebesgue measure set of
$E> E^{*}(q,\omega ), $
where
$\omega $
satisfies the Diophantine condition. Bogoljubov, Mitropoliskii, and Samoilenko [Reference Bogoljubov, Mitropoliskii and Samoilenko6] gave a quite general reducibility result, namely for higher dimensional analytic quasi-periodic linear systems close to a constant, under the same Diophantine condition on the frequency and a condition of separated spectrum. Rüssmann [Reference Rüssmann30] extended Dinaburg and Sinai’s result to the case of the Brjuno condition. For other reducibility results under weaker non-resonant conditions, we refer to [Reference Chavaudret and Marmi9, Reference Lopes Dias24–Reference Lopes Dias and Pedro Gaivão26, Reference Pöschel28, Reference Wang, Xu and Zhang34, Reference Zhang and Xu37] and the references therein. Later, Moser and Pöschel [Reference Moser and Pöschel27] extended the result [Reference Dinaburg and Sinai11] to a class of rotation numbers which is rational with respect to the basic frequency by a crucial resonance-cancellation technique. The reducibility of quasi-periodic systems with coefficients in
$gl(n, \mathbb {R})$
was considered by Jorba and Simó [Reference Jorba and Simó19, Reference Jorba and Simó20]. Finally, the breakthrough belongs to Eliasson [Reference Eliasson12] who proved weak almost reducibility for all energies E and full measure reducibility for quasi-periodic Schrödinger equations. More precisely, Eliasson [Reference Eliasson12] proved that system (5) is reducible for almost all
$E> E^{*}(q,\omega ),$
where
$\omega $
satisfies the Diophantine condition. Chavaudret [Reference Chavaudret8] and Leguil et al [Reference Leguil, You, Zhao and Zhou23] obtained strong almost reducibility results. Krikorian [Reference Krikorian21] obtained the full measure reducibility result for linear systems with coefficients in the Lie algebra of a compact semi-simple Lie group. Her and You [Reference Her and You17] and Chavaudret [Reference Chavaudret7] established the full measure reducibility with coefficients in other groups. Recently, the Brjuno condition has been weakened to multiple Liouvillean frequencies [Reference Cheng, Si and Si10, Reference Wang, Cheng and Si32, Reference Xu, Si and Si35]. For the latest reducibility results of infinite dimensional quasi-periodic systems, readers may refer to [Reference Bambusi3–Reference Bambusi, Grébert, Maspero and Robert5, Reference Eliasson and Kuksin15] and the references therein.
Non-perturbative reducibility. Puig [Reference Puig29] used the localization method to obtain the non-perturbative version of Eliasson’s reducibility theorem. A more precise extension of Eliasson’s reducibility was given by Avila and Jitomirskaya [Reference Avila and Jitomirskaya2] by further developing the techniques in localization. In [Reference Avila and Jitomirskaya2], some beautiful results on the continuity of the spectrum of the discrete quasi-periodic Schrödinger operators are deduced from the non-perturbative reducibility. Avila, Fayad, and Krikorian [Reference Avila, Fayad and Krikorian1] first introduced the method of a CD bridge to prove that for any analytic
$SL(2, \mathbb {R})$
cocycle which is close to constant, for any
$\alpha \in \mathbb {R}\setminus \mathbb {Q},$
the cocycles are analytic rotations reducible (analytic conjugacy to a cocycle with values in
$SO(2, \mathbb {R}))$
for full measure rotation number. Zhou and Wang [Reference Zhou and Wang41] used periodic approximation to study the reducibility of quasi-periodic
$GL(d, \mathbb {R})$
cocycles with Liouvillean frequencies, which can be viewed as a generalization of [Reference Avila, Fayad and Krikorian1] to high dimension. For continuous linear systems, Hou and You [Reference Hou and You18] considered a quasi-periodic linear differential system with two frequencies in
$sl(2,\mathbb {R})$
and obtained weak almost reducibility for all rotation numbers, all frequencies
$\omega =(1, \alpha )\in \mathbb {T}^{2} $
, and small analytic perturbations. Meanwhile, Hou and You [Reference Hou and You18] proved full measure rotations reducibility results that can be transferred to the cocycle case by the local embedding theorem of You and Zhou [Reference You and Zhou36]. Wang, You, and Zhou [Reference Wang, You and Zhou33] proved the existence of response solutions for quasi-periodically forced harmonic oscillators with forcing frequencies
$\omega =(1, \alpha ).$
For the latest reducibility results on the Liouvillean frequencies, we refer to [Reference Krikorian, Wang, You and Zhou22, Reference Zhang and Xu38–Reference Zhang, Xu and Xu40] and the references therein.
From the development of the reduction theory mentioned above, the Diophantine condition can be weakened. In fact, the non-degeneracy conditions can also be weakened or even removed. We mainly start a discussion about the KAM results without non-degeneracy conditions. In [Reference Rüssmann31], Rüssmann considered an analytic Hamiltonian system with two degrees of freedom and obtained the convergence of the transformation into a normal form restricting to the zero-energy surface if the formal Birkhoff normal form is divisible by its quadratic part and the frequencies satisfy the Brjuno condition. Fayad and Krikorian [Reference Fayad and Krikorian16] proved that if F is a smooth diffeomorphism of the annulus having the intersection property, then any given F-invariant curve on which the rotation number of F is Diophantine is accumulated by a positive measure set of smooth invariant curves on which F is smoothly conjugated to rotation maps. The remarkable feature of the result is that it does not require any twist condition. In [Reference Eliasson, Fayad and Krikorian14], the KAM stability of the torus is studied under different hypotheses on the Hamiltonian function and the frequencies.
In this paper, we try to find the optimal non-resonant conditions and non-degeneracy conditions for the reducibility of quasi-periodic systems. So, we will consider the reducibility for the three-dimensional skew symmetric system (1) without any non-degeneracy condition and with Liouvillean basic frequencies
$\omega =(1, \alpha )$
with
$\alpha $
being irrational.
The quasi-periodic systems in [Reference Avila, Fayad and Krikorian1, Reference Hou and You18] are two-dimensional, so one can introduce the rotation number naturally and make good use of the property of rotation number. However, for the three-dimensional skew symmetric system (1), we do not know how to define the rotation number, which yields that there are some obstructions in applying the methods in [Reference Avila, Fayad and Krikorian1, Reference Hou and You18] to the high dimensional situation. In [Reference Zhang, Xu and Xu40], Zhang, Xu, and Xu obtained rotation reducibility for three-dimensional skew symmetric systems with a special form, which indicates that some suitable non-degeneracy conditions are satisfied. Moreover, all the results about Liouvillean frequencies in [Reference Wang, You and Zhou33, Reference Zhang and Xu38, Reference Zhang, Xu, Wu and Xu39, Reference Zhou and Wang41] require the system to satisfy some non-degeneracy conditions. So, the main difficulty in this paper is how to deal with the lack of non-degeneracy conditions.
In the previous conclusions [Reference Wang, You and Zhou33, Reference Zhang and Xu38–Reference Zhou and Wang41], the non-degeneracy conditions are needed to control the measure of small parameters, and so
$P(t,\epsilon )$
at least depends, in a
$C^{1}$
-smooth way, on the parameter
$\epsilon .$
In our paper, there is no non-degeneracy condition, so
$P(t,\epsilon )$
can be continuously dependent on the parameter
$\epsilon .$
2 Main results
Before stating our results, we first give some notation and definitions. Usually, denote by
$\mathbb {Z}$
and
$\mathbb {Z}_{+}$
the sets of integers and positive integers, respectively. Denote by
$so(3, \mathbb {R})$
the set of
$3\times 3$
skew symmetric real matrices.
A function
$f(t)$
is called quasi-periodic with basic frequencies
$\omega _{1}, \ldots , \omega _{s} $
if
${f(t)= F(\theta ) = F(\theta _{1}, \ldots , \theta _{s}),}$
where F is
$2\pi $
-periodic in all its arguments and
$\theta _{j}= \omega _{j}t $
for
$\mathrm{j} = 1, \ldots, \mathrm{s} $
. Let
$\omega =(\omega _{1}, \ldots , \omega _{s}). $
Thus,
$f(t)= F(\omega t).$
Denote a strip domain of complex space
$\mathbb {C}^{s}$
by
$$ \begin{align*} D(r)= \{\theta= (\theta_{1}, \ldots, \theta_{s} )\in \mathbb{C}^{s}: |\mbox{Im}\theta_{j}|\leq r, j=1, \ldots, s\}. \end{align*} $$
Furthermore, if
$F(\theta )$
is analytic with respect to
$\theta $
on
$D(r),$
we say that
$f(t)$
is analytic quasi-periodic on
$D(r).$
Denote by
$$ \begin{align*}[f]=\frac{1}{(2\pi)^{s}} \int_{\mathbb{T}^{s}} F(\theta)\, d\theta \end{align*} $$
the average of
$f.$
Similarly, a function matrix
$P(t)=(P_{ij})_{n\times n} $
is called analytic quasi-periodic on
$D(r)$
if all
$P_{ij}(t)$
are analytic quasi-periodic on it. Denote by
$[P]= ([P_{ij}])_{n \times n}$
the average of
$P.$
Let
$U_{\delta }= [0, \delta ].$
Denote by
$C^{a,m}(\mathbb {T}_{r}^{n}\times U_{\delta })(m=0,1)$
the set which consists of all
$f(t,\epsilon )$
such that f is analytic quasi-periodic in t on
$\mathbb {T}_{r}^{n} $
with the basic frequencies
$\omega $
and
$C^{m}$
-smooth in
$\epsilon $
on
$U_{\delta }.$
Let
$f(t,\epsilon )= \sum _{k \in \mathbb {Z}^{n}} f_{k}(\epsilon )e^{i\langle k, \omega \rangle t } $
, and define
$$ \begin{align*} \|f\|_{r, U_{\delta}}= \sum_{k\in\mathbb{Z}^{n}} |f_{k}|_{U_{\delta}} e^{|k|r} \end{align*} $$
and
$$ \begin{align*} \|\partial_{\epsilon}f\|_{r, U_{\delta}}= \sum_{k\in\mathbb{Z}^{n}} |\partial_{\epsilon}f_{k}|_{U_{\delta}} e^{|k|r}, \end{align*} $$
where
$|f_{k}|_{U_{\delta }}= \sup \nolimits _{\epsilon \in U_{\delta }} |f_{k}(\epsilon )|,$
$|\partial _{\epsilon }f_{k}|_{U_{\delta }}= \sup \nolimits _{\epsilon \in U_{\delta }} |\partial _{\epsilon }f_{k}(\epsilon )|,$
$|k|= \sum \nolimits _{i=1}^{n} |k_{i}| $
for
${k= (k_{1},\ldots , k_{n}).}$
If
$ f\in C^{m}( U_{\delta })(m=0,1), $
define
$$ \begin{align*} \|f\|_{C^{m} (U_{\delta})}= \sum_{ 0\leq l\leq m} |\partial^{(l)}_{\epsilon}f|_{U_{\delta}}. \end{align*} $$
For a function matrix
$P(t, \epsilon )= (P_{ij}(t, \epsilon ))_{n\times n},$
similarly define a norm by
$$ \begin{align*} \|P\|_{r, U_{\delta}}= \max_{ 1\leq i\leq d } \sum_{j=1}^{d} \|P_{ij}\|_{r, U_{\delta}}. \end{align*} $$
We have
$\| P_{1}P_{2}\|_{r, U_{\delta }}\leq \|P_{1}\|_{r, U_{\delta }} \|P_{2}\|_{r,U_{\delta }}. $
For any
$K>0,$
we define the truncating operators
$ \mathcal {T}_{K}$
as
$$ \begin{align*} \mathcal{T}_{K}f= \sum_{k \in \mathbb{Z}^{s}, |k|< K} f_{k}(\epsilon)e^{i\langle k, \omega\rangle t } \end{align*} $$
and
$ \mathcal {R}_{K} $
as
$$ \begin{align*} \mathcal{R}_{K}f= \sum_{k \in \mathbb{Z}^{s}, |k|\geq K} f_{k}(\epsilon)e^{i\langle k, \omega\rangle t }. \end{align*} $$
Denote
$$ \begin{align} J_{1}=\left( \begin{array}{@{}ccc@{\,}} 0 & 1 & 0\\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\!. \end{align} $$
Let
$\alpha \in (0,1)$
be irrational and denote by
${p_{n}}/{q_{n}}$
the nth convergence of
$\alpha .$
Define
$$ \begin{align} \beta(\alpha)=\limsup_{n\rightarrow\infty}\frac{\ln q_{n+1}}{q_{n}}, \end{align} $$
then
$\beta (\alpha )$
measures how Liouvillean is
$\alpha $
. Notice that
$\beta (\alpha )$
has an equivalent definition:
$$ \begin{align} \beta(\alpha)= \limsup_{|k|\rightarrow\infty}\frac{1}{|k|}\ln\frac{1}{|e^{ik\alpha}-1|}. \end{align} $$
Suppose that
$0 \leq \beta (\alpha )< r,$
then there exists an
$r_{*}\in (0,r)$
such that
$ 0\leq \beta (\alpha )< r_{*}< r.$
Theorem 2. Let
$\omega =(1,\alpha )$
with
$\alpha \in \mathbb {R} \setminus \mathbb {Q}$
,
$ 0<r_{*}< r, \tau>2.$
Consider the following system:
$$ \begin{align} \dot{x}=( A+ P(t, \epsilon))x,\quad x \in \mathbb{R}^{3}, \end{align} $$
where
$A, P \in so(3, \mathbb {R})$
,
$A= \unicode{x3bb} J_{1}$
is a constant matrix with eigenvalues
$\pm i \unicode{x3bb} $
and
$0$
,
$ P(t,\epsilon )\in C^{a,m}(\mathbb {T}_{r}^{2} \times U_{\delta }) (m=0, 1)$
with basic frequencies
$\omega =(1, \alpha )$
with
$\alpha $
being irrational, and
$\epsilon $
is a small parameter. Suppose that
$(\omega , \unicode{x3bb} )$
satisfy the following non-resonant condition:
$$ \begin{align} |\langle k, \omega\rangle \pm \unicode{x3bb} | \geq \frac{\gamma_{0}}{(|k|+1)^{\tau} }\quad \text{for all } k \in\mathbb{Z}^{2}, \end{align} $$
where
$\gamma _{0}> 0, \tau > 2. $
Then, we have the following two conclusions.
-
(1) If
$\|P\|_{r, U_{\delta }} \rightarrow 0$
and
$\|\partial _{\epsilon } P\|_{r,U_{\delta }}\leq M$
(m=1) as
$\delta \rightarrow 0,$
then there exist a sufficiently small
$\epsilon _{0}$
, which depends on
$r, r_{*}, \gamma _{0}, \tau $
, and
$M,$
but not on
$\alpha ,$
and a non-empty subset
$ E \subset (0, \epsilon _{0})$
such that for any
$ \epsilon \in E$
, the system (9) is rotations reducible. Moreover, if
$m=1,$
the set
$E\cap (0, \delta )$
has a positive Lebesgue measure for any
$\delta> 0.$
If
$m=0,$
the set
$E\cap (0, \delta )$
has cardinality number of the continuum for any
$\delta> 0.$
-
(2) Furthermore, if
$\beta (\alpha ) =0, $
then for the same
$\epsilon _{0}$
and
$\epsilon \in E$
in conclusion (1), the system (1) is reducible. If
$0 < \beta (\alpha )<r,$
then there exists a sufficiently small
${\epsilon _{0}=\epsilon _{0}( r, r_{*}, \gamma _{0}, \tau , M, \beta (\alpha ) )}$
and a non-empty subset
$ E \subset (0, \epsilon _{0}) $
such that for any
$ {\epsilon \in E}$
, the system (9) is reducible. Moreover, if
$m=1,$
the set
$E\cap (0, \delta )$
has a positive Lebesgue measure for any
$\delta> 0.$
If
$m=0,$
the set
$E\cap (0, \delta )$
has cardinality number of the continuum for any
$\delta> 0.$
Remark 3. When
$\beta (\alpha )=0,$
the smallness of
$\epsilon _{0}$
does not depend on
$\alpha ;$
therefore, we not only weaken the Diophantine condition (2) to Liouvillean frequencies and remove the non-degeneracy condition (3), but also improve the results in [Reference Eliasson13] to be non-perturbative in the case of two-dimensional basic frequencies.
Remark 4. In the
$C^{1}$
-smooth case, we obtain the positive measure reducibility of the system (9). However, we cannot have the precise measure estimates of parameters, so further research work is to investigate the full measure reducibility in the context of real analytic dependence on the parameters.
Remark 5. The conclusion of the theorem still holds, even removing the algebraic structure of
$so(3, \mathbb {R}),$
that is, consider the following system:
$$ \begin{align*} \dot{x}=( A+ P(t, \epsilon))x,\quad x \in \mathbb{R}^{3}, \end{align*} $$
where A is a constant matrix with eigenvalues
$\pm i \unicode{x3bb} $
and
$0$
,
$ P(t,\epsilon )\in C^{a,m}(\mathbb {T}_{r}^{2} \times U_{\delta }) (m=0, 1)$
with basic frequencies
$\omega =(1, \alpha )$
with
$\alpha $
being irrational, and
$\epsilon $
is a small parameter. Suppose that
$(\omega , \unicode{x3bb} )$
satisfy the non-resonant condition:
$$ \begin{align*} |\langle k, \omega\rangle \pm 2\unicode{x3bb} | \geq \frac{\gamma_{0}}{(|k|+1)^{\tau} }\quad \text{for all } k \in\mathbb{Z}^{2}, \end{align*} $$
where
$\gamma _{0}> 0, \tau > 2, $
then we can obtain the same conclusion.
Before giving the detailed proof of our theorem, we first explain the strategies and techniques of the proof. An essential idea of the proof is to construct an infinite number of quasi-periodic transformations to eliminate the first-order non-resonant terms. Since A has eigenvalues
$\pm i \unicode{x3bb} $
and
$0,$
the corresponding small divisors are
$ \langle k, \omega \rangle $
and
$ \langle k, \omega \rangle \pm \unicode{x3bb} $
in the process of eliminating the first-order non-resonant terms on the diagonal and the off-diagonal. Due to the occurrence of Liouvillean frequencies, the main difficulty is that we cannot eliminate the terms on the diagonal, that is, the terms on the diagonal will be retained. We will put these terms into the linear principal part. Thus, the linear principal part in our proof will be variable coefficients, which yields that the homological equation is an equation with variable coefficients. We overcome this problem by the method of modified iterations to obtain an approximation solution of the homological equation. Therefore, the proof of Theorem 2 is divided into two parts. First, we prove that system (9) can be reduced to a system with non-constant coefficients,
$$ \begin{align*} \dot{y} =( A+ B_{*}(t, \epsilon)) y, \end{align*} $$
where
$$ \begin{align*} B_{*}(t,\epsilon) = \left( \begin{array}{@{}ccc@{\kern1pt}} 0 & \Xi^{*}(t,\epsilon) & 0 \\ -\Xi^{*}(t,\epsilon) & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\!. \end{align*} $$
The proof of the first part can be achieved by infinite steps of KAM iteration. Second, we eliminate the non-resonant terms containing t on the diagonal of
$B_{*}(t,\epsilon )$
and transform the above system into a system with constant coefficients,
$$ \begin{align*} \dot{y}= (A+ [B_{*}]) y= A_{*}(\epsilon) y, \end{align*} $$
where
$$ \begin{align*} A_{*}(\epsilon) = \left(\begin{array}{@{}ccc@{\kern1pt}} 0 & \Lambda^{*}(\epsilon) & 0\\ -\Lambda^{*}(\epsilon) & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\!. \end{align*} $$
This process can be obtained by only one step if
$ 0 \leq \beta (\alpha )< r_{*},$
where
$ 0< r_{*} <r.$
To overcome the lack of non-degeneracy conditions, we will develop a new technique of the KAM iteration to separate the non-degeneracy condition from the KAM iteration. The key lies in an explicit extension of small divisors to the entire definition domain of the parameters. Specifically, we give a formal reducibility theorem, which means the constructed transformations and KAM iteration are well defined for all the parameters. However, only for those parameters on
$\Pi _{n}$
where the small divisor conditions hold, the constructed transformation is a true one. The non-resonant sets have the properties
$ \Pi _{n+1}\subset \Pi _{n} \subset \cdots \subset \Pi _{0} \text { for all } n\in \mathbb {Z}_{+}$
and
$\Pi _{\infty } \subset \bigcap _{n\geq 1 } \Pi _{n},$
where the definition of
$\Pi _{\infty }$
is given in conclusion (3) of Theorem 6, so we only need to estimate the measure of
$\Pi _{\infty }.$
3 Proof of Theorem 2
In this section, we give the proof of Theorem 2, which consists of three parts: formal reducibility theorem in §3.1; elimination of the non-resonant terms on the diagonal in §3.2; and measure estimates in §3.3.
3.1 Formal reducibility theorem
We state a formal reducibility theorem, which is used to prove conclusion (1) of Theorem 2, except for the measure estimates, that is, the result of rotations reducibility.
Theorem 6. (Formal reducibility theorem)
Let
$\omega =(1,\alpha )$
with
$\alpha \in \mathbb {R} \setminus \mathbb {Q}$
,
$ 0<r_{*}< r, \tau>2.$
Consider the system
$$ \begin{align*} \dot{x}= (A+ P(t,\epsilon))x, x\in \mathbb{R}^{3}, \end{align*} $$
where
$A, P \in so(3, \mathbb {R})$
, A is a constant matrix with eigenvalues
$\pm i \unicode{x3bb} $
and
$0$
,
$\epsilon $
is a small parameter, and
$ P(t,\epsilon )\in C^{a,m}(\mathbb {T}_{r}^{2} \times U_{\delta }) (m=0, 1)$
with basic frequencies
$\omega =(1, \alpha ),$
satisfying that
$P(t,0)=0, $
and for all
$\epsilon \in [0,\delta ],$
$$ \begin{align*} \|P(\cdot,\epsilon)\|_{r}\leq \mathcal{E},\quad \|\partial_{\epsilon} P(\cdot,\epsilon) \|_{r}\leq M\quad \mbox{if } m=1, \end{align*} $$
where
$\partial _{\epsilon } P= {\partial P}/{\partial \epsilon } $
and
$M> 0$
is a constant. Suppose that the eigenvalues
$\pm i \unicode{x3bb} $
and the basic frequencies
$\omega =(1, \alpha )$
satisfy the non-resonant condition (10).
Let
$ {0 <\gamma \leq \gamma _{0}/2.}$
Then, there exists a sufficiently small
$\mathcal {E}_{0}> 0,$
which depends on
$ r, r_{*}, \gamma , \gamma _{0},\tau $
, and
$M,$
but not on
$\alpha ,$
such that if
$\mathcal {E} \leq \mathcal {E}_{0},$
the following conclusions hold.
-
(1) There exists a matrix function
$\Phi ^{*}(t,\epsilon ),$
which is analytic quasi-periodic in t with basic frequencies
$\omega $
and
$C^{m}$
-smooth in
$\epsilon ,$
such that
$\Phi ^{*}\in C^{a,m}({\mathbb {T}}^{2}_{r_{*}} \times U_{\delta } ) $
with where the definitions of
$$ \begin{align*} \|\Phi^{*}(\cdot,\epsilon)-I\|_{r_{*}}\leq \mathcal{E}^{{14}/{15}},\quad \|\partial_{\epsilon}\Phi^{*}(\cdot,\epsilon)\|_{r_{*}} \leq c \mathcal{L}_{0}^{- {1}/{240}} \gamma_{0}^{-\mathcal{A}\tau+2} Q_{n_{0}+1}^{18\tau} M, \end{align*} $$
$\mathcal {L}_{0}$
and
$Q_{n_{0}}$
are given in equations (54) and (53), respectively.
-
(2) There exists a t-dependent matrix
$B_{*}(\omega t, \epsilon ), $
which is quasi-periodic in t with basic frequencies
$\omega $
and
$C^{m}$
-smooth in
$\epsilon ,$
such that
$B_{*}(\omega t, \epsilon )\in C^{a,m}({\mathbb {T}}^{2}_{r_{*}} \times U_{\delta } ). $
Moreover,
$B_{*}(\omega t, \epsilon )$
has the normal form as equation (60) and satisfies
$$ \begin{align*} B_{*}(\omega t,0)=0,\quad \|B_{*}(\cdot, \epsilon)\|_{r_{*}} \leq 4\mathcal{E},\quad \| \partial_{\epsilon}B_{*}(\cdot,\epsilon)\|_{r_{*}} \leq 4M,\quad \epsilon\in [0,\delta]. \end{align*} $$
-
(3) Let
$\pm i \unicode{x3bb} ^{*}(\epsilon )$
and
$0$
be the eigenvalues of
$A+[B_{*}(\omega t, \epsilon )]$
, and define
$$ \begin{align*} \Pi_{\infty}= \bigg\{\epsilon\in [0, \delta] : |\langle k, \omega\rangle \pm \unicode{x3bb}^{*}(\epsilon)|\geq \frac{\gamma}{(|k|+1)^{3\tau}} \text{ for all } k\in \mathbb{Z}^{2}\bigg\}. \end{align*} $$
If
$\epsilon \in \Pi _{\infty },$
then the transformation
$x=\Phi ^{*}(t,\epsilon ) y$
reduces the system (9) into
$$ \begin{align*} \dot{x}=(A + B_{*}(\omega t, \epsilon))x,\quad x\in \mathbb{R}^{3}, \end{align*} $$
where
$B_{*}(\omega t,\epsilon )$
has the normal form as equation (60).
We first use Theorem 6 to prove Theorem 2 and delay the proof of Theorem 6 until later.
3.2 Elimination of the non-resonant terms on the diagonal
This section is devoted to conclusion (2) of Theorem 2, except for the measure estimates, which is the reducibility to a constant system.
According to Theorem 6, for any
$\epsilon \in \Pi _{\infty } $
, the system (9) can be reduced to a rotation system with non-constant coefficients:
$$ \begin{align} \dot{x} = (A+ B_{*}(t,\epsilon))x,\quad x \in \mathbb{R}^{3}, \end{align} $$
where
$$ \begin{align*} B_{*}(t,\epsilon) = \left(\kern-1pt \begin{array}{@{}ccc@{\kern1pt}} 0 & \Xi^{*}(t, \epsilon) & 0 \\ -\Xi^{*}(t, \epsilon) & 0 & 0 \\ 0 & 0 &0 \\ \end{array} \right)\!. \end{align*} $$
For simplicity, we omit the dependence on
$\epsilon $
and write
$B_{*}(t,\epsilon )$
as
$B_{*}(t).$
We now eliminate all the non-resonant terms containing t on the diagonal. First, notice that
$\beta (\alpha )$
has an equivalent definition (8), which implies that if
$ 0\leq \beta (\alpha )<r_{*}, $
the equation
$$ \begin{align*} \frac{d H(t)}{dt}=B_{*}(t)-[B_{*}] \end{align*} $$
has an analytic solution, where
$$ \begin{align*} H(t)= \left( \begin{array}{@{}ccc@{}} 0 & \sum\limits_{ 0 \neq k \in \mathbb{Z}^{2}} \frac{\Xi^{*}(k) e^{i\langle k, \omega\rangle t}}{i\langle k, \omega\rangle } & 0 \\ -\sum\limits_{ 0 \neq k \in \mathbb{Z}^{2}} \frac{\Xi^{*}(k) e^{i\langle k,\omega\rangle t}}{i\langle k, \omega\rangle } & 0 & 0\\ 0& 0 &0 \\ \end{array} \right)\!. \end{align*} $$
We first provide an intuitive proof. By the equivalent definition (8), there exists a
${K \in \mathbb {Z}_{+},}$
such that for any
$|k|\geq K$
,
${1}/{|\langle k, \omega \rangle |}\leq e^{|k|\beta (\alpha )}< e^{|k| ({\beta (\alpha )+r_{*}})/{2}}.$
Then,
$$ \begin{align*} \|H_{12}\|_{({r_{*}-\beta(\alpha)})/{4}} &= \bigg(\sum_{ 0< |k| <K } + \sum_{ |k|\geq K }\bigg) \bigg| \frac{\Xi^{*}(k)}{\langle k, \omega\rangle} \bigg| e^{|k| ({r_{*}-\beta(\alpha)})/{4}}\\ &\leq \|\Xi^{*}\|_{r_{*}} \bigg(\sum_{ 0<|k|< K} \frac{e^{-|k|r_{*}}}{|\langle k, \omega\rangle|} e^{|k| ({r_{*}-\beta(\alpha)})/{4}}\\ &\quad + \sum_{ |k|\geq K } e^{-|k|r_{*}} e^{|k| ({\beta(\alpha)+r_{*}})/{2}} e^{|k|({r_{*}-\beta(\alpha)})/{4}}\bigg)\\ &\leq \|\Xi^{*}\|_{r_{*}} \bigg(\sum_{ 0<|k|< K} \!\frac{1}{|\langle k, \omega\rangle|} e^{-|k|(({3r_{*}}/{4}) + ({\beta(\alpha)}/{4}) )}+\!\! \sum_{ |k|\geq K } e^{-|k|({r_{*}-\beta(\alpha)})/{4} }\bigg)\\ &\leq c\|\Xi^{*}\|_{r_{*}}, \end{align*} $$
where c depends on
$r_{*}$
and
$\beta (\alpha ).$
In the following, we give a detailed proof. In fact, by equation (7), there exists an
${N\in \mathbb {Z}_{+},}$
such that for any
$ n\geq N,$
we have
$ q_{n+1} < e^{\beta (\alpha ) q_{n}}< e^{(({\beta (\alpha )+r_{*}})/{2}) q_{n}}. $
Then,
$$ \begin{align*} \|H_{12}\|_{({r_{*}-\beta(\alpha)})/{4}} &= \bigg(\sum_{ 0<|k| <q_{N}} + \sum_{ q_{N}\leq |k| <q_{N+1}} + \cdots+ \sum_{ q_{n}\leq |k| <q_{n+1}}+ \cdots \bigg)\\ &\quad\times \bigg| \frac{\Xi^{*}(k)}{\langle k, \omega\rangle} \bigg| e^{|k| ({r_{*}-\beta(\alpha)})/{4}} \\ &\leq \|\Xi^{*}\|_{r_{*}} \bigg( 2 q_{N} \sum_{ 0<|k| <q_{N}} e^{- |k| ({3r_{*}}/{4} + {\beta(\alpha)}/{4})} + 2 q_{N+1}\\ &\quad\times \sum_{ q_{N}\leq |k| <q_{N+1}} e^{- |k| ({3r_{*}}/{4} + {\beta(\alpha)}/{4} )} \\ &\quad + \cdots+ 2 q_{n+1}\sum_{ q_{n}\leq |k| <q_{n+1}} e^{- |k| ({3r_{*}}/{4} + {\beta(\alpha)}/{4} )} + \cdots \bigg) \\ &\leq \|\Xi^{*}\|_{r_{*}} \bigg( 2 q_{N} \sum_{ 0<|k| <q_{N}} e^{- |k| ({3r_{*}}/{4} + {\beta(\alpha)}{4} )}+ 2 q_{N+1} e^{- (({\beta(\alpha)+r_{*}})/{2}) q_{N} } \\ &\quad\times \sum_{ q_{N}\leq |k| <q_{N+1}} e^{- |k| ({r_{*}-\beta(\alpha)})/{4}} \\ &\quad + \cdots+ 2 q_{n+1} e^{- (({\beta(\alpha)+r_{*}})/{2}) q_{n} } \sum_{ q_{n}\leq |k| <q_{n+1}} e^{- |k| ({r_{*}-\beta(\alpha)})/{4}} + \cdots\bigg)\\ &\leq \|\Xi^{*}\|_{r_{*}} \bigg( 2 q_{N} \sum_{ 0<|k| <q_{N}} e^{- |k| ({3r_{*}}/{4} + {\beta(\alpha)}/{4} )}+ 2 \\ &\quad \times \sum_{|k|\geq q_{N}} e^{- |k| ({r_{*}-\beta(\alpha)})/{4}}\bigg) \leq c\|\Xi^{*}\|_{r_{*}}, \end{align*} $$
where c depends on
$r_{*}$
and
$\beta (\alpha )$
,
$ \{q_{n}\}$
is the sequence of denominators of the best rational approximations for
$\alpha ,$
the first equality is due to the definition of the norm of quasi-periodic functions, the first inequality is from equations (12)–(13) and
$|\Xi ^{*}(k)|\leq \|\Xi ^{*}\|_{r_{*}} e^{-|k|r_{*}},$
and the third inequality is according to the fact
$q_{n+1} < e^{(({\beta (\alpha )+r_{*}})/{2}) q_{n}} \text { for all } n\geq N.$
Let
$x= e^{H(t)}x_{*}.$
Then, the system (11) becomes
$$ \begin{align*} e^{H(t)} \frac{d H(t)}{dt} x_{*} + e^{H(t)} \frac{d x_{*}}{dt}= (A+B_{*}(t) ) e^{H(t)}x_{*}. \end{align*} $$
Because the commutation of
$e^{H(t)}$
and
$A, B_{*}(t),$
it follows that
$$ \begin{align*} \frac{d x_{*}}{dt}= (A+[B_{*}] ) x_{*} =A_{*} x_{*}, \end{align*} $$
where
$$ \begin{align*} A_{*}= \left( \begin{array}{@{}ccc@{\kern1pt}} 0 & \Lambda^{*} & 0 \\ -\Lambda^{*} & 0 & 0\\ 0& 0 & 0 \end{array} \right)\!,\quad \Lambda^{*}=\unicode{x3bb} + [\Xi^{*}]. \end{align*} $$
Moreover, for any
$ \epsilon \in \Pi _{\infty },$
we have
$$ \begin{align*} \|A_{*}- A \|_{ \Pi_{\infty}} \leq 4 \mathcal{E}. \end{align*} $$
3.3 Measure estimates
In this section, we prove
$\Pi _{\infty }$
in Theorem 6 includes many sufficiently small parameters, which corresponds to the measure estimates in Theorem 2. For convenience, we first prove a lemma about measure estimates.
Lemma 7. Suppose
$\omega =(1,\alpha )$
and
$\pm i \unicode{x3bb} $
satisfy the non-resonant condition (10). Let
$ 0 <\gamma \leq \gamma _{0}/2,$
$\tau '>2\tau +2$
, and
$f_{k}(\epsilon )= \langle k, \omega \rangle \pm \unicode{x3bb} +\epsilon .$
Let
$\rho $
be an arbitrarily small positive constant and set
$$ \begin{align*} E_{\pm}= \bigg\{\epsilon\in (0, \pm \rho) : |f_{k}(\epsilon)|\geq \frac{\gamma}{(|k|+1)^{\tau'}} \text{ for all } k \in \mathbb{Z}^{2} \bigg\}, \end{align*} $$
where the notation
$(0,-\rho )$
denotes the interval
$(-\rho , 0)$
for simplicity. Then,
$$ \begin{align*} \mbox{meas} ((0, \pm\rho)\setminus E_{\pm} )\leq c \rho^{2} \frac{\gamma}{\gamma_{0}^{2}}, \end{align*} $$
where c is a constant.
Proof. Let
$R_{k}= \{\epsilon \in (0,\rho ) : |f_{k}(\epsilon )|< {\gamma }/{(|k|+1)^{\tau '}}\}. $
If
$ 0< \epsilon \leq \gamma _{0}/ 2(|k|+1)^{\tau }, $
by equation (10), it follows that
$$ \begin{align*} |f_{k}(\epsilon)|\geq \frac{\gamma_{0}}{ 2 (|k|+1)^{\tau}} \geq \frac{\gamma }{(|k|+1)^{\tau'}}. \end{align*} $$
If
$ \gamma _{0}/ 2(|k|+1)^{\tau } <\epsilon < \rho , $
we have
$$ \begin{align*} \mbox{meas}(R_{k}) \leq \frac{ 2\gamma}{(|k|+1)^{\tau'}} \leq \bigg(\frac{ 2(|k|+1)^{\tau} \rho}{ \gamma_{0}} \bigg)^{2} \frac{ 2\gamma}{(|k|+1)^{\tau'}} = \frac{ 8 \rho^{2} \gamma}{\gamma_{0}^{2} (|k|+1)^{\tau'-2\tau} }. \end{align*} $$
Since
$\tau '-2\tau> 2, $
it follows that
$$ \begin{align*} \mbox{meas}((0, \rho)\setminus E_{+} ) \leq \sum_{k\in\mathbb{Z}^{2}} \mbox{meas}(R_{k}) \leq c \rho^{2} \frac{ \gamma}{\gamma_{0}^{2}}, \end{align*} $$
where
$c= 8 \sum _{k\in \mathbb {Z}^{2}} ({1}/{(|k|+1)^{\tau '-2\tau }}).$
Here,
$E_{-}$
can be considered similarly and we omit the details.
Next, we summarize the measure estimates of Theorem 2 in the following lemma.
Lemma 8. (Measure estimates)
If
$m=1,$
that is,
$P(t,\epsilon )$
is differentiable in the parameter
$\epsilon $
on
$U_{\delta },$
the set
$\Pi _{\infty }$
has a positive Lebesgue measure for any
$\delta> 0.$
If
$m=0,$
that is,
$P(t,\epsilon )$
is continuous in the parameter
$\epsilon $
on
$U_{\delta },$
the set
$\Pi _{\infty }$
has cardinality number of the continuum for any
$\delta> 0.$
Proof. The proof is divided into two parts: differentiable case and continuous case. At first, let
$ \gamma =\gamma _{0}/2,$
$\tau '=3\tau .$
By assumption, there is a constant
$M>0$
such that
${\|\partial _{\epsilon } P\|\leq M.}$
Let
$\mathcal {E}_{0}$
be the sufficiently small number as given in Theorem 6. By assumption, when
$\delta $
is sufficiently small, we have
$\|P\|_{r}\leq \mathcal {E}_{0}$
for all
$\epsilon \in [0,\delta ].$
Moreover, we have
${P(t,0)\equiv 0.}$
In the differentiable case, we will prove
$\mbox {meas}(\Pi _{\infty })> 0.$
Let
${\unicode{x3bb} ^{*}(\epsilon )= \unicode{x3bb} + [\Xi ^{*}]}$
and
${\nu (\epsilon ) = [\Xi ^{*}].}$
Noting that
$[\Xi ^{*}](0)= 0, $
then
$\nu (\epsilon )$
is differentiable on
$[0,\delta ]$
and
$\nu (0)=0.$
Let
$$ \begin{align*} \rho_{0}=\max_{0 \leq\epsilon\leq\delta }\nu(\epsilon)= \nu(\epsilon_{0}),\quad 0\leq \epsilon \leq\delta. \end{align*} $$
Without loss of generality, we assume
$\rho _{0}> 0, \epsilon _{0} > 0.$
Since
$\nu (\epsilon )$
is continuous, we have
$$ \begin{align*} [0, \rho_{0}] \subset \{\nu(\epsilon) : \epsilon\in [0, \epsilon_{0}] \}. \end{align*} $$
Recall
$\gamma =\gamma _{0}^{2}/2$
and
$\tau '= 3\tau>2\tau +2.$
By Lemma 7,
$E_{\rho _{0}}= E_{+}$
with
$\rho =\rho _{0}$
is non-empty and
$\mbox {meas}(E_{\rho _{0}})> 0.$
Let
$$ \begin{align*} \Pi_{\epsilon_{0}} =\{ \epsilon\in [0, \epsilon_{0}]: \nu(\epsilon)\in E_{\rho_{0}}\}. \end{align*} $$
Thus,
$\nu (\Pi _{\epsilon _{0}})= E_{\rho _{0}}$
and
$\mbox {meas}(\nu (\Pi _{\epsilon _{0}}))> 0.$
Since
$\nu (\epsilon )$
is differentiable and
$|\nu '(\epsilon )|\leq c, \text {for all } \epsilon \in [0, \delta ],$
we have
$$ \begin{align*} \mbox{meas}(\nu(\Pi_{\epsilon_{0}})) \leq \int_{\Pi_{\epsilon_{0}}} |\nu'(\epsilon)|\, d\epsilon\leq c \cdot\mbox{meas}(\Pi_{\epsilon_{0}}) \end{align*} $$
and
$\mbox {meas}(\Pi _{\epsilon _{0}})> 0.$
Since
$\Pi _{\epsilon _{0}}\subset \Pi _{\infty },$
we have
$\mbox {meas}(\Pi _{\infty })> 0.$
In the continuous case, we will prove the set
$\Pi _{\infty }$
has cardinality number of the continuum. By the above discussion of the differentiable case, we have
$\nu (\Pi _{\epsilon _{0}})= E_{\rho _{0}}.$
Because the mapping
$\nu : \Pi _{\epsilon _{0}}\rightarrow E_{\rho _{0}} $
is continuous and surjective, the cardinality of the set
$\Pi _{\epsilon _{0}}$
is not less than that of the set
$E_{\rho _{0}}.$
Since
$\mbox {meas}(E_{\rho _{0}})> 0,$
both
$E_{\rho _{0}}$
and
$\Pi _{\epsilon _{0}}$
have cardinality number of the continuum. Noting that
$\Pi _{\epsilon _{0}} \subset \Pi _{\infty },$
$\Pi _{\infty }$
also has cardinality number of the continuum. Thus, Theorem 2 is proved completely.
4 Preliminaries
The motivation of this section is to introduce the concept of the CD bridge, which first appeared in [Reference Avila, Fayad and Krikorian1], and give some useful estimates. For detailed proofs, we refer to [Reference Avila, Fayad and Krikorian1].
4.1 Continued fraction expansion
Let
$\alpha \in (0,1)$
be irrational. Define
$a_{0}=0, \alpha _{0}=\alpha ,$
and inductively for
$k\geq 1,$
$$ \begin{align*} a_{k}=[\alpha_{k-1}^{-1}],\quad \alpha_{k}=\alpha_{k-1}^{-1}-a_{k}= \bigg\{\frac{1}{\alpha_{k-1}}\bigg\}, \end{align*} $$
where
$[\cdot ]$
denotes its integral part and
$\{\cdot \}$
denotes its fractional part.
We define
$ p_{0}=0, p_{1}=1, q_{0}=1, q_{1}=a_{1},$
and inductively,
$$ \begin{align*} p_{k}=a_{k}p_{k-1}+p_{k-2}, \end{align*} $$
$$ \begin{align*} q_{k}=a_{k}q_{k-1}+q_{k-2}. \end{align*} $$
The sequence
$(q_{n})$
is the sequence of denominators of the best rational approximations for
$\alpha ,$
since it satisfies
$$ \begin{align} \text{ for all } 1 \leq k < q_{n},\quad \|k\alpha\|_{\mathbb{T}}\geq \|q_{n-1}\alpha\|_{\mathbb{T}}, \end{align} $$
and
$$ \begin{align} \frac{1}{q_{n}+ q_{n+1}} \leq \|q_{n}\alpha\|_{\mathbb{T}}\leq \frac{1}{q_{n+1}}, \end{align} $$
where we use the norm
$$ \begin{align*} \|x\|_{\mathbb{T}}=\inf_{p\in\mathbb{Z}}|x-p|. \end{align*} $$
4.2 CD bridge
For any
$\alpha \in \mathbb {R}\setminus \mathbb {Q},$
we fix a particular subsequence
$(q_{n_{k}})$
in the sequence of the denominators of
$\alpha ,$
which will be denoted by
$(Q_{k})$
for simplicity. Denote the sequence
$(q_{n_{k}+1})$
by
$(\bar {Q}_{k})$
and denote
$(p_{n_{k}})$
by
$(P_{k}). $
Definiton 9. (CD bridge [Reference Avila, Fayad and Krikorian1])
Let
$0<\mathcal {A}\leq \mathcal {B}\leq \mathcal {C}.$
We say that the pair of denominators
$(q_{l}, q_{n})$
forms a
$\mbox {CD}(\mathcal {A}, \mathcal {B}, \mathcal {C})$
bridge, if:
-
(1)
$q_{i+1}\leq q_{i}^{\mathcal {A}} \text { for all } i=l, \ldots , n-1;$
-
(2)
$q_{l}^{\mathcal {B}} \leq q_{n}\leq q_{l}^{\mathcal {C}}. $
Lemma 10. [Reference Avila, Fayad and Krikorian1]
For any
$\mathcal {A}\geq 1, $
there exists a subsequence
$(Q_{k})$
such that
$Q_{0}=1$
and for each
$k\geq 0,$
$Q_{k+1}\leq \bar {Q}_{k}^{\mathcal {A}^{4}},$
and either
$\bar {Q}_{k}\geq Q_{k}^{\mathcal {A}},$
or the pairs
$(\bar {Q}_{k-1}, Q_{k})$
and the pairs
$(Q_{k}, Q_{k+1})$
are both
$\mbox {CD}(\mathcal {A}, \mathcal {A}, \mathcal {A}^{3})$
bridges.
Lemma 11. [Reference Avila, Fayad and Krikorian1]
For
$0< r_{*}<r, \tau>2, \tilde {c}< 1,$
there exists
$C_{1}=C_{1}(r_{*}, \tau , \tilde {c})$
such that if
$f \in C^{a,m}(\mathbb {T}_{r}^{2} \times \Pi ) (m=0, 1),$
where
$\Pi $
is the parameter domain, its definition is given in equation (16), then the equation
$$ \begin{align*} \partial_{\omega}g (t, \epsilon) = - \mathcal{T}_{Q_{n+1}} f(t, \epsilon) + [f ] \end{align*} $$
has a solution with
$$ \begin{align*} \|g\|_{r(1-\eta ), \Pi}\leq C_{1}(r_{*}, \tau, \tilde{c}) \|f - [f]\|_{r, \Pi} \bigg(\frac{\bar{Q}_{n}}{Q_{n}^{\mathcal{A}^{4}}}+ \bar{Q}_{n}^{{1}/{\mathcal{A}}} \bigg). \end{align*} $$
5 Proof of Theorem 6
In the proof, we will make a formal convergent KAM iteration without any non-degeneracy condition. For this purpose, we will develop a new technique of the KAM iteration to separate the non-degeneracy condition from the KAM iteration. The key lies in an explicit extension of small divisors to the parameter definition domain. The extension of small divisors always works even when the small divisors condition does not hold for every
$\epsilon \in [0,\delta ].$
Thus, the constructed transformation and the new perturbation are well defined for all the parameters. However, only for those parameters for which the small divisor conditions hold, the new system comes exactly from the original one under the transformation. Otherwise, we only obtain a formal new system, which may have no relation with the previous system and thus cannot provide any useful information.
5.1 Small divisors extension
Now, define a
$C^{\infty }$
-smooth function
$\varphi (t)$
such that
$$ \begin{align*} \varphi(t)= \begin{cases} 0,\quad |t| \leq \frac{1}{2}, \\ 1,\quad |t|\geq 1. \end{cases} \end{align*} $$
For
$h>0,$
let
$\varphi _{h}(t)= \varphi ({t}/{h}).$
Then,
$ \varphi _{h}(t)\in C^{\infty }( \mathbb {R}) $
with
$$ \begin{align} \bigg|\frac{d^{l} }{dt^{l} }\varphi_{h}(t) \bigg|\leq \frac{c_{l}}{h^{^{l}}}\quad \text{ for all } t\in\mathbb{R}, \text{ for all } l\geq 1, \end{align} $$
where
$c_{l}$
is a constant depending on
$l.$
Let
$h= {\gamma }/{(|k|+1)^{ 3\tau }}$
and
$$ \begin{align} t_{k}^{\pm}(\epsilon)= \langle k, \omega\rangle \pm (\unicode{x3bb}+[\Xi]),\quad g_{k}^{\pm}(\epsilon)= \frac{\varphi_{h}(t_{k}^{\pm}(\epsilon))}{t_{k}^{\pm}(\epsilon)}. \end{align} $$
Define
$$ \begin{align} \Pi= \bigg\{ \epsilon\in [0, \delta]: |\langle k, \omega\rangle \pm (\unicode{x3bb}+[\Xi)|\geq\frac{\gamma}{(|k|+1)^{3\tau}} \text{ for all } |k| < K \bigg\}. \end{align} $$
Then,
$g_{k}^{\pm }(\epsilon )$
is well defined on
$[0, \delta ]$
and
$$ \begin{align*} g_{k}^{\pm}(\epsilon)=\frac{1}{\langle k, \omega\rangle \pm(\unicode{x3bb}+[\Xi])}\quad \text{ for all } \epsilon\in \Pi. \end{align*} $$
Note that even if
$\Pi = \emptyset ,$
the above extension is still valid. Furthermore, if
$[\Xi ](\epsilon )\in C^{m}[0, \delta ], $
by equation (14), it follows that
$g_{k}^{\pm }(\epsilon ) \in C^{m}[0, \delta ]$
with
$$ \begin{align*} \bigg|\frac{d^{m}}{d\epsilon^{m}} g_{k}^{\pm}(\epsilon)\bigg|\leq \frac{c}{h^{m+1}} \quad \text{ for all } \epsilon\in [0, \delta], \end{align*} $$
where c depends on
$\|[\Xi ]\|_{C^{m}[0,\delta ]}.$
5.2 Homological equation
In this section, we mainly show how to use the method of modified iterations to solve the homological equation
$$ \begin{align} \frac{dW}{dt}+ W(A + B(t,\epsilon)+ b(t,\epsilon) ) - (A+ B(t,\epsilon)+ b(t,\epsilon)) W = P, \end{align} $$
where
$W, P $
are all
$3 \times 3 $
skew symmetric matrices,
$B(t,\epsilon ), b(t,\epsilon ) $
have the same form as
$A,$
that is,
$$ \begin{align*} B(t,\epsilon) = \left( \begin{array}{@{}ccc@{\kern1pt}} 0 & \Xi(t,\epsilon) & 0 \\ -\Xi(t,\epsilon) & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\!,\quad b(t,\epsilon) = \left( \begin{array}{@{}ccc@{\kern1pt}} 0 & \Delta(t,\epsilon) & 0 \\ -\Delta(t,\epsilon) & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\!. \end{align*} $$
The modified iterations consist of three steps: first, we use a rotation transformation to remove a part of non-resonant terms and transform
$B(t,\epsilon )$
into
$[B(t,\epsilon )]_{t},$
where
$[B(t,\epsilon )]_{t}$
is the average of
$B(t,\epsilon )$
with respect to
$t;$
second, we derive an approximate solution of the transformed homological equations; and third, we use the inverse of the rotation transformation to ensure that the final transformation is a nearly identical transformation.
Lemma 12. Suppose W satisfies the homological equation (17), where
$A, B, b, P$
are all skew symmetric matrices, then W is a skew symmetric matrix.
Proof. Let
$W=\sum _{k} W_{k} e^{i \langle k, \omega \rangle t}$
,
$ P= \sum _{k} P_{k} e^{i\langle k, \omega \rangle t}.$
Then, comparing the Fourier coefficients of the above homological equation, we have
$$ \begin{align} \partial_{\omega} W_{k} + W_{k} (A+B+b) - (A+B+b)W_{k}= P_{k}. \end{align} $$
Transposing on both sides of the equation gives
$$ \begin{align} \partial_{\omega} W_{k}^{T} + (A+B+b)^{T}W_{k}^{T} - W_{k}^{T} (A+B+b)^{T} = P_{k} ^{T}. \end{align} $$
Due to
$ A, B, b, P$
all being skew symmetric matrices, equation (19) is equivalent to
$$ \begin{align*} \partial_{\omega} W_{k}^{T} - (A+B+b)W_{k}^{T} + W_{k}^{T} (A+B+b) = - P_{k}, \end{align*} $$
that is,
$$ \begin{align} \partial_{\omega}( -W_{k}^{T}) + (-W_{k}^{T} ) (A+B+b) - (A+B+b)( -W_{k}^{T}) = P_{k}. \end{align} $$
By equations (18), (20), and the uniqueness of the solutions of the equations, we have
$ W_{k}= - W_{k}^{T}. $
Therefore, W is a skew symmetric matrix.
Before stating the following lemma, we first give the definition of extended solutions and some appropriate parameter sequences.
Definiton 13. (Extended solutions)
Consider the homological equation (17), which is defined on the non-resonant parameter domain
$\Pi \subset U_{\delta },$
on which small divisor conditions are satisfied, and
$U_{\delta }= [0, \delta ]$
is the initial parameter domain, on which small divisor conditions may not hold. By the method of small divisors extension, we can obtain a formal solution
$\tilde {W}(t,\epsilon ),$
which is defined on the entire parameter domain
$U_{\delta }.$
If
$\epsilon \in \Pi ,$
the formal solution
$\tilde {W}(t, \epsilon )$
is a true solution, that is,
$ \tilde {W}(t, \epsilon )= W(t,\epsilon ) $
$ \text { for all } \epsilon \in \Pi .$
Then,
$\tilde {W}(t, \epsilon )$
is called an extended solution of
$W(t,\epsilon ).$
For
$\tau> 2,$
we define
$$ \begin{align*} \mathcal{A}= \tau+3, \mathcal{M}= \frac{\mathcal{A}^{4}}{2}, \end{align*} $$
and let
$( Q_{n})$
be the selected subsequence of
$\alpha $
in Lemma 10 with this given
$\mathcal {A}.$
At each step, we fix a certain n and then use
$Q_{n}$
to define the related quantities, which will be allowed to vary along the steps of KAM iteration. For
$ r, \gamma> 0,$
we define
$$ \begin{align*} \eta= \frac{\tilde{c}}{ Q_{n}^{{1}/{ 2\mathcal{A}^{4}}}},\quad \mathcal{L}= e^{- c_{0}\gamma r (({ \bar{Q}_{n}}/{ Q_{n}^{\mathcal{M}}}) + \bar{Q}_{n}^{{1}/{\mathcal{M}^{1/4}}})}, \end{align*} $$
where
$ c_{0}= {\tilde {c}}/({4^{5}\cdot 10^{\tau }})$
, and
$0<\tilde {c}<1$
is a constant, which will be defined in equation (52).
Lemma 14. Let
$ \tau> 2, 0< r_{*}< r, 0<\tilde {c} < 1. $
Then, there exist
$C_{3}= C_{3}(\tau )$
and
${\varepsilon _{1}\kern1.4pt{=}\kern1.4pt \varepsilon _{1}(\tau , r_{*}, \tilde {c}) }$
such that for any
$ 0 \kern1.4pt{<}\kern1.4pt\sigma \kern1.4pt{<}\kern1.4pt r_{*} \kern1.4pt{<}\kern1.4pt\tilde {r} \kern1.4pt{\leq}\kern1.4pt r(1-\eta ),$
if
$ {B(t,\epsilon ) \in C^{a,m}(\mathbb {T}^{2}_{r}\times U_{\delta })}$
with
$\mathcal {R}_{Q_{n+1}} B(t, \epsilon )=0$
,
$b(t, \epsilon ), P(t,\epsilon )\in C^{a,m}(\mathbb {T}^{2}_{\tilde {r}}\times U_{\delta })$
with
$P(t,0)=0,$
$$ \begin{align} \|b\|_{\tilde{r}, U_{\delta}}< \frac{ \gamma^{\mathcal{A}\tau+2}}{ 2 C_{3} Q_{n+1}^{18\tau}}, \end{align} $$
and
$$ \begin{align} \|B\|_{r, U_{\delta}}\cdot \bigg(\frac{\bar{Q}_{n}}{Q_{n}^{\mathcal{A}^{4}}}+ \bar{Q}_{n}^{{1}/{\mathcal{A}}}\bigg) \leq \varepsilon_{1} \gamma \bigg(\frac{\bar{Q}_{n}}{Q_{n}^{M}} + \bar{Q}_{n}^{{1}/{{M}^{1/4}} } \bigg), \end{align} $$
then the following conclusions hold.
-
(1) For any
$\epsilon \in U_{\delta }=[0, \delta ],$
the homological equation (17) has an extended and approximate solution
$ \tilde {W}(t,\epsilon ),$
which satisfies
$$ \begin{align*} \|\tilde{W}\|_{\tilde{r}, U_{\delta}}\leq C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau} \mathcal{L}^{- {1}/{240}} \|P\|_{\tilde{r}, U_{\delta}}, \end{align*} $$
$\tilde {W}(t,0)\equiv 0.$
Moreover, the truncated remainder
$P_{e}= (P_{e, ij})_{ 3\times 3 }$
with where
$$ \begin{align*} P_{e, ij} = e^{-i \mathcal{B}(t)} \mathcal{R}_{K} (e^{i \mathcal{B}(t) }( P_{ij}(t) +i \Delta(t)\tilde{ W}_{ij} )), \end{align*} $$
$\partial _{\omega } \mathcal {B}(t)= - \Xi (t)+ [\Xi ],$
satisfies (23)In the differentiable case where
$$ \begin{align} \|P_{e}\|_{\tilde{r}- \sigma, U_{\delta}}\leq \mathcal{L}^{- {1}/{240} } e^{-K \sigma} ( \|P\|_{\tilde{r}, U_{\delta}} + 2 \|b\|_{\tilde{r}, U_{\delta}} \|\tilde{W}_{ij}\|_{\tilde{r}, U_{\delta}}). \end{align} $$
$m=1,$
except for the above conditions, if
$\|A\|_{C^{1}[0,\delta ]}\leq c, \|\partial _{\epsilon }B\|_{r, U_{\delta }}\leq c, $
we have and
$$ \begin{align*} &\|\partial_{\epsilon}\tilde{W}\|_{{\tilde{r}, U_{\delta}}}\leq \mathcal{L}^{- {1}/{240}}C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau} \\ &\quad\times ( 2C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau}\|P\|_{\tilde{r},U_{\delta}} + \|\partial_{\epsilon}P\|_{\tilde{r},U_{\delta}}) \end{align*} $$
$$ \begin{align*} \|\partial_{\epsilon}P_{e}\|_{{\tilde{r}, U_{\delta}}}&\leq C_{3}(\tau) \mathcal{L}^{- {1}/{240} } e^{-K \sigma} (\|P\|_{\tilde{r},U_{\delta}} + 2 \|b\|_{\tilde{r}, U_{\delta}}\|\tilde{W}\|_{\tilde{r}, U_{\delta}}\\ &\quad + \|\partial_{\epsilon}P\|_{\tilde{r}, U_{\delta}} + 2 \|\partial_{\epsilon}b\|_{\tilde{r}, U_{\delta}} \|\tilde{W}\|_{\tilde{r}, U_{\delta}} + 2 \|b\|_{\tilde{r}, U_{\delta}}\|\partial_{\epsilon}\tilde{W}\|_{\tilde{r}, U_{\delta}}). \end{align*} $$
-
(2) Let
$ \pm i (\unicode{x3bb} + [\Xi (\epsilon )]) $
be the eigenvalues of
$A+[B]$
and define
$$ \begin{align*} \Pi=\bigg\{\epsilon\in [0, \delta] : |\langle k, \omega \rangle \pm (\unicode{x3bb} + [\Xi(\epsilon)]) | \geq \frac{\gamma}{(|k|+1)^{3\tau}} \text{ for all } |k|< K \bigg\}. \end{align*} $$
If
$\epsilon \in \Pi , $
the extended and approximate solution
$ \tilde {W}(t, \epsilon )$
is a true and approximate solution, that is,
$ \tilde {W}(t, \epsilon )= W(t,\epsilon ) $
$ \text { for all } \epsilon \in \Pi , $
and
$W(t,\epsilon ) $
satisfies the above corresponding estimates.
Proof. Let
$$ \begin{align*} R = \left( \begin{array}{@{}ccc@{}} ({1-i})/{2} & ({1+i})/{2} \\ ({1+i})/{2} & ({1-i})/{2} \end{array} \!\!\right),\quad S = \left( \begin{array}{@{}ccc@{}} R & 0 \\ 0 & 1 \end{array} \!\!\right)\!. \end{align*} $$
Then, by
$$ \begin{align*} R^{-1} \left( \begin{array}{@{}ccc@{}} 0 & 1 \\ -1 & 0 \end{array} \!\!\right) R= \left( \begin{array}{@{}ccc@{}} i & 0 \\ 0 & -i \end{array} \!\!\right)\!, \end{align*} $$
we have
$$ \begin{align*} S^{-1} A S = \left( \begin{array}{@{}ccc@{}} \unicode{x3bb} i& 0 & 0 \\ 0 & -\unicode{x3bb} i & 0 \\ 0 & 0 & 0 \end{array} \right)\!. \end{align*} $$
For simplicity, denote
$$ \begin{align*} \tilde{A}= S^{-1} A S,\quad \tilde{B}(t)= S^{-1} B(t) S,\quad \tilde{b}(t) = S^{-1} b(t) S, \end{align*} $$
$$ \begin{align*} Z = S^{-1} W S,\quad H = S^{-1} PS. \end{align*} $$
Then, equation (17) is equivalent to
$$ \begin{align} \frac{ d Z}{ dt} + Z ( \tilde{A}+ \tilde{B}(t) + \tilde{b}(t)) - ( \tilde{A} + \tilde{B}(t) + \tilde{b}(t)) Z= H. \end{align} $$
Suppose that P has the following form:
$$ \begin{align*} P(t) = \left( \begin{array}{@{}ccc@{}} 0 & a^{1}(t) & a^{2}(t) \\ -a^{1}(t) & 0 & a^{3}(t) \\ -a^{2}(t) & - a^{3}(t) & 0 \end{array} \right)\!, \end{align*} $$
then
$$ \begin{align*} H= S^{-1} P S= \left( \begin{array}{@{}ccc@{}} a^{1} i & 0 & H_{13} \\ 0 & -a^{1} i & H_{23} \\ H_{31} & H_{32} & 0 \end{array} \right)\!, \end{align*} $$
where
$$ \begin{align*} H_{13}= \frac{1}{2} ( a^{2}+ a^{3} ) + \frac{i}{2} (a^{2} -a^{3} ),\quad H_{23} = \frac{1}{2} ( a^{2}+ a^{3} ) + \frac{i}{2} (a^{3} -a^{2} ) , \end{align*} $$
$$ \begin{align*} H_{31}= - \frac{1}{2} ( a^{2}+ a^{3} ) + \frac{i}{2} (a^{2} -a^{3} ) ,\quad H_{32} = - \frac{1}{2} ( a^{2}+ a^{3} ) + \frac{i}{2} (a^{3} -a^{2} ). \end{align*} $$
Therefore, the homological equation (24) can be written in the form of components:
$$ \begin{align*} \frac{d Z_{13}}{dt} - i (\unicode{x3bb} + \Xi(t) + \Delta(t) )Z_{13} =H_{13}, \end{align*} $$
$$ \begin{align*} \frac{d Z_{23}}{dt} + i (\unicode{x3bb} + \Xi(t) + \Delta(t) ) Z_{23} = H_{23}, \end{align*} $$
$$ \begin{align*} \frac{d Z_{31}}{dt} + i (\unicode{x3bb} + \Xi(t) + \Delta(t) ) Z_{31} = H_{31}, \end{align*} $$
$$ \begin{align*} \frac{d Z_{32}}{dt}- i (\unicode{x3bb} + \Xi(t) + \Delta(t) ) Z_{32} = H_{32}. \end{align*} $$
Note that the above homological equations have four cases. Let
$$ \begin{align*} \frac{d\mathcal{B}(t)}{ dt} = - \Xi (t) + [\Xi] \end{align*} $$
and
$$ \begin{align*} H^{\prime}_{13}(t) = e^{ i\mathcal{B}(t) } H_{13}(t),\quad H^{\prime}_{32} (t) = e^{ i\mathcal{B}(t ) } H_{32}(t), \end{align*} $$
$$ \begin{align*} H^{\prime}_{23} (t) = e^{- i\mathcal{B}(t ) } H_{23}(t),\quad H^{\prime}_{31} (t) = e^{ -i\mathcal{B}(t ) } H_{31}(t). \end{align*} $$
Note,
$Z^{\prime }_{ij} $
has a similar definition. Then, the homological equation (24) is transformed into
$$ \begin{align} \frac{d Z^{\prime}_{ij} }{d t} \mp i (\unicode{x3bb} + [\Xi] + \Delta(t)) Z^{\prime}_{ij}= H^{\prime}_{ij}. \end{align} $$
Instead of solving equation (25), we first solve the approximation equation
$$ \begin{align} \mathcal{T}_{K} \bigg(\frac{d Z^{\prime}_{ij} }{d t} \mp i (\unicode{x3bb} + [\Xi] + \Delta(t)) Z^{\prime}_{ij}\bigg) = \mathcal{T}_{k} H^{\prime}_{ij}. \end{align} $$
If we write
$$ \begin{align*} Z^{\prime}_{ij}= \sum_{k\in \mathbb{Z}^{2}, |k|< K } {Z'}_{ij}^{k} e^{i \langle k, \omega\rangle t},\quad \mathcal{T}_{K} H^{\prime}_{ij}= \sum_{k\in \mathbb{Z}^{2}, |k|< K } {H'}_{ij}^{k} e^{i\langle k, \omega\rangle t } , \end{align*} $$
and compare the Fourier coefficients of equation (26), then for
$|k| < K, $
we have
$$ \begin{align} i ( \langle k, \omega\rangle \mp (\unicode{x3bb} + [\Xi] ) {Z'}_{ij}^{k} - i \sum_{|k_{1}|< K} \Delta^{k-k_{1} } {Z'}_{ij}^{k_{1}}={H'}_{ij}^{k}. \end{align} $$
View equation (27) as a matrix equation
$$ \begin{align*} ( D + F ) \mathcal{ W} = \mathcal{P}, \end{align*} $$
where
$$ \begin{align*} D = \mbox{diag}( \ldots, i (\langle k, \omega\rangle \mp\Omega(\epsilon), \ldots)_{|k|< K}, \end{align*} $$
$$ \begin{align*} \Omega(\epsilon)= (\unicode{x3bb} + [\Xi]) , \end{align*} $$
$$ \begin{align*} F = ( -i \Delta^{k_{1}-k_{2}})_{|k_{1}|, |k_{2}| < K}, \end{align*} $$
$$ \begin{align*} \mathcal{W}= ( {Z'}_{ij}^{k} )^{T}_{|k|< K},\quad \mathcal{P}= ( {H'}_{ij}^{k} )^{T}_{|k|< K}. \end{align*} $$
If we denote
$ \Gamma _{\tilde {r} }= \mbox {diag}( \ldots , e^{|k| \tilde {r} }, \ldots ) _{|k|< K},$
then
$$ \begin{align} (D+ \Gamma_{\tilde{r}} F \Gamma_{\tilde{r}}^{-1} ) \Gamma_{\tilde{r}} \mathcal{W}= \Gamma_{\tilde{r}} \mathcal{P}. \end{align} $$
By [Reference Wang, You and Zhou33, Lemma 3.2], we have that for
$ 0<\gamma <1, \tau>2, $
there exists
$c_{2}=c_{2}(\tau )$
such that if
$\Omega (\epsilon ) $
satisfies the following non-resonant condition:
$$ \begin{align*} |\sqrt{-1}\langle k, \omega\rangle -\Omega(\epsilon) |\geq \frac{\gamma}{(|k|+1)^{ \tau}}\quad \text{for all } |k|< K, \end{align*} $$
where the definition of K is given in equation (32), then for
$|k|< K, $
we have
$$ \begin{align*} |\sqrt{-1}\langle k, \omega\rangle - \Omega(\epsilon)|\geq c_{2}(\tau) \gamma^{({\mathcal{A}\tau}/{2})+1 } Q_{n+1}^{-3\tau}. \end{align*} $$
Since
$$ \begin{align*} |\langle k, \omega \rangle - \Omega(\epsilon) |\geq \frac{ \gamma}{(|k|+1)^{3\tau}}\quad \text{for all } |k| < K, \end{align*} $$
then we have
$$ \begin{align*} \|D^{-1}\|_{\Pi} &= \max _{|k|< K}\sup_{\epsilon\in \Pi} \frac{1} { |\langle k, \omega\rangle - \Omega(\epsilon)|} \\ & \leq C_{3}(\tau) \gamma^{ -( \mathcal{A}\tau +2 )} Q_{n+1}^{ 18 \tau}/2 \leq \frac{1}{4 \|b\|_{\tilde{r},\Pi}}, \end{align*} $$
where
$\|D\|_{\Pi }=\max _{i} \sup _{\epsilon \in \Pi } \sum _{j} |D_{ij}(\epsilon )|$
and
$D_{ij}$
is the
$(i,j)$
th element of the matrix
$D.$
By equation (15), to extend
$D^{-1}$
in
$\epsilon $
from
$\Pi $
to
$[0, \delta ],$
we define
$$ \begin{align*} \tilde{D}^{-1}(\epsilon)=\mbox{diag }( \ldots, g_{k}^{1}(\epsilon), \ldots )_{|k|< K} \end{align*} $$
or
$$ \begin{align*} \tilde{D}^{-1}(\epsilon)=\mbox{diag }( \ldots, g_{k}^{2}(\epsilon), \ldots )_{|k|< K}, \end{align*} $$
where
$$ \begin{align*} g_{k}^{1}(\epsilon)=\frac{\varphi_{h}(t_{k}^{1}(\epsilon)) }{t_{k}^{1}(\epsilon)},\quad g_{k}^{2}(\epsilon)=\frac{\varphi_{h}(t_{k}^{2}(\epsilon)) }{t_{k}^{2}(\epsilon)}. \end{align*} $$
Then,
$\tilde {D}^{-1}(\epsilon )$
is well defined for all
$\epsilon \in U_{\delta }.$
In particular, for
$\epsilon \in \Pi ,$
we have
$\tilde {D}^{-1}(\epsilon )= D^{-1}(\epsilon ).$
Meanwhile, since the
$ (k_{1}, k_{2})$
th element of
$\Gamma _{\tilde {r} } F \Gamma _{\tilde {r} }^{-1} $
is
$ -i e^{(|k_{1}|- |k_{2}|) \tilde {r}} \Delta ^{k_{1}- k_{2}},$
we obtain that
$$ \begin{align*} \|\Gamma_{\tilde{r} } F \Gamma_{\tilde{r} }^{-1}\|_{U_{\delta}} &\leq \max_{|k_{1}| < K } \sum_{|k_{2}| < K} | e^{(|k_{1}|- |k_{2}|) \tilde{r}} \Delta^{k_{1}- k_{2}}|_{U_{\delta}}\\ &\leq \max_{|k_{1}| < K } \sum_{|k_{2}| < K} |\Delta^{k_{1}- k_{2}}|_{U_{\delta} } e^{|k_{1}- k_{2}|\tilde{r}} \leq 2 \| b\|_{\tilde{r}, U_{\delta} }. \end{align*} $$
Thus, if
$\|b\|_{\tilde {r}, U_{\delta } }< {\gamma ^{\mathcal {A}\tau +2 }}/{ 2 C_{3}(\tau ) Q_{n+1}^{18\tau }},$
we have
$$ \begin{align*} \| D^{-1} \Gamma_{\tilde{r} } F \Gamma_{\tilde{r} }^{-1}\|_{\Pi} \leq \| D^{-1}\|_{ \Pi} \|\Gamma_{\tilde{r} } F \Gamma_{\tilde{r}} \|_{\Pi }<\tfrac{1}{2}, \end{align*} $$
which implies
$D+ \Gamma _{\tilde {r} } F \Gamma _{\tilde {r} }^{-1} $
has a bounded inverse:
$$ \begin{align*} \|(D+ \Gamma_{\tilde{r} } F \Gamma_{\tilde{r} }^{-1})^{-1}\|_{\Pi} &= \|(I+ D^{-1}\Gamma_{\tilde{r} } F \Gamma_{\tilde{r} }^{-1})^{-1} D^{-1} \|_{\Pi} \\ &\leq \|D^{-1}\|_{\Pi} \frac{1}{ 1- \|D^{-1}\Gamma_{\tilde{r} } F \Gamma_{\tilde{r}}^{-1}\|_{\Pi}} \\ &\leq C_{3}(\tau)\gamma ^{- (\mathcal{A}\tau +2 )}Q_{n+1}^{ 18\tau}. \end{align*} $$
To extend
$(D+ \Gamma _{\tilde {r} } F \Gamma _{\tilde {r} }^{-1})^{-1} $
in
$\epsilon $
from
$\Pi $
to
$[0,\delta ],$
we define
$(I+ \tilde {D}^{-1}\Gamma _{\tilde {r} } F \Gamma _{\tilde {r} }^{-1})^{-1} \tilde {D}^{-1}, $
and then
$(I+ \tilde {D}^{-1}\Gamma _{\tilde {r} } F \Gamma _{\tilde {r} }^{-1})^{-1}\tilde {D}^{-1} $
is well defined for all
$\epsilon \in U_{\delta }.$
In particular, for
$\epsilon \in \Pi $
, we have
$(I+ \tilde {D}^{-1}\Gamma _{\tilde {r} } F \Gamma _{\tilde {r} }^{-1})^{-1}\tilde {D}^{-1}= (D+ \Gamma _{\tilde {r}} F \Gamma _{\tilde {r} }^{-1})^{-1}.$
It follows that
$$ \begin{align} \|{\tilde{Z'}}_{ij}\|_{\tilde{r}, U_{\delta} }&= \sum_{|k|< K }| {\tilde{Z'}}_{ij}^{k} |_{U_{\delta}} e^{|k|\tilde{r} }= \|\Gamma_{\tilde{r}} \mathcal{W}\|_{U_{\delta}} \nonumber\\ &\leq \|(I+ \tilde{D}^{-1}\Gamma_{\tilde{r} } F \Gamma_{\tilde{r}}^{-1})^{-1} \tilde{D}^{-1}\|_{U_{\delta}} \| \Gamma_{\tilde{r}}\mathcal{P} \|_{U_{\delta}} \nonumber\\ &\leq \|(I+ \tilde{D}^{-1}\Gamma_{\tilde{r} } F \Gamma_{\tilde{r} }^{-1})^{-1} \tilde{D}^{-1}\|_{U_{\delta}} \|{\tilde{H}}_{ij}\|_{\tilde{r}, U_{\delta}}. \end{align} $$
Let
$\tilde {Z}_{ij}(\theta )= e^{\mp i ( \mathcal {B}(\theta )) } \tilde {Z'}_{ij}(\theta ).$
Then, by Lemma 11, we have
$$ \begin{align*} \|\mathcal{B}\|_{r(1-\eta), U_{\delta}}\leq C_{1}( r_{*}, \tau, \tilde{c}) \|\Xi - [\Xi]\|_{r,U_{\delta}} \bigg( \frac{\bar{Q}_{n}}{Q_{n}^{\mathcal{A}^{4}}} + \bar{Q}_{n}^{{1}/{\mathcal{A}} }\bigg). \end{align*} $$
Therefore, by the assumption (22), if
$\varepsilon _{1}< {c_{0}(\tau , \tilde {c}) r}/{ 960 C_{1}(r_{*}, \tau , \tilde {c}) },$
we get
$$ \begin{align*} \|e^{\mp i \mathcal{B}} \|_{r(1-\eta),U_{\delta}} &\leq e^{ \|\mathcal{B} \|_{r(1-\eta), U_{\delta}}} \leq e^{2 C_{1}\|B\|_{r,U_{\delta}}( ({\bar{Q}_{n}}/{Q_{n}^{\mathcal{A}^{4}}}) +\bar{Q}_{n}^{{1}/{\mathcal{A}}})}\\ &\leq e^{2 C_{1} \varepsilon_{1} \gamma ( ({\bar{Q}_{n}}/{Q_{n}^{M}}) + \bar{Q}_{n}^{{1}/{M^{1/4}} } )} < \mathcal{L}^{- {1}/{480} }. \end{align*} $$
By estimate (29), we have
$$ \begin{align*} \|\tilde{Z}_{ij}\|_{\tilde{r},U_{\delta}}\leq \mathcal{L}^{-{1}/{480}} \|\tilde{Z'}_{ij} \|_{\tilde{r},U_{\delta}}\leq C_{3}(\tau) \gamma^{- \mathcal{A}\tau+2} Q_{n+1}^{18\tau} \mathcal{L}^{- {1}/{240}}\|H_{ij}\|_{\tilde{r},U_{\delta}} \end{align*} $$
and
$$ \begin{align*} \|\tilde{Z}\|_{ \tilde{r},U_{\delta}}\leq C_{3}(\tau) \gamma^{- \mathcal{A}\tau+2} Q_{n+1}^{18\tau} \mathcal{L}^{-{1}/{240}}\|H\|_{\tilde{r},U_{\delta}}. \end{align*} $$
Hence,
$\tilde {W}= S \tilde {Z}S^{-1}$
satisfies
$$ \begin{align*} \|\tilde{W}\|_{ \tilde{r},U_{\delta}}\leq C_{3}(\tau) \gamma^{- \mathcal{A}\tau+2} Q_{n+1}^{18\tau} \mathcal{L}^{-{1}/{240}}\|P\|_{\tilde{r},U_{\delta}}. \end{align*} $$
Moreover, one can verify that the error term
$P_{e}= (P_{e, ij})_{3 \times 3} $
with
$$ \begin{align*} P_{e, ij} = e^{-i \mathcal{B}(t) } \mathcal{R}_{K} (e^{ i \mathcal{B}(t) }( P_{ij}(t) +i \Delta(t) W_{ij} )) \end{align*} $$
satisfies estimate (23).
In the differentiable case where
$m=1,$
except for the above conditions, suppose
$\|A\|_{C^{1}[0,\delta ]}\leq c, \|\partial _{\epsilon }B\|_{r, U_{\delta }}\leq c.$
Seeking derivative on both sides of the equations (28) and
$\tilde {Z}_{ij}(t)= e^{\mp i \mathcal {B}(t)} {\tilde {Z'}}_{ij}(t)$
, respectively, we have
$$ \begin{align*} \|\partial_{\epsilon}\tilde{W}\|_{{\tilde{r}, U_{\delta}}}\!\leq \mathcal{L}^{- {1}/{240}}C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau} ( 2C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau}\|P\|_{\tilde{r},U_{\delta}} + \|\partial_{\epsilon}P\|_{\tilde{r},U_{\delta}}). \end{align*} $$
Similarly,
$$ \begin{align*} \|\partial_{\epsilon}P_{e}\|_{{\tilde{r}, U_{\delta}}}&\leq C_{3}(\tau) \mathcal{L}^{- {1}/{240} } e^{-K \sigma} (\|P\|_{\tilde{r}, U_{\delta}} + 2 \|b\|_{\tilde{r}, U_{\delta}}\|\tilde{W}\|_{\tilde{r}, U_{\delta}}\\ &\quad + \|\partial_{\epsilon}P\|_{\tilde{r}, U_{\delta}} + 2 \|\partial_{\epsilon}b\|_{\tilde{r}, U_{\delta}} \|\tilde{W}\|_{\tilde{r}, U_{\delta}} + 2 \|b\|_{\tilde{r}, U_{\delta}}\|\partial_{\epsilon}\tilde{W}\|_{\tilde{r}, U_{\delta}}). \end{align*} $$
Thus, the proof of Lemma 14 is finished.
5.3 Elimination of non-resonant terms on the off-diagonal
In this section, we use the method of small divisors extension and CD bridge to prove formal reducibility Theorem 6, which consists of KAM step, setting of parameters and iteration, and convergence of the KAM scheme. As all KAM steps can be carried out inductively, below, we only describe one step of KAM iteration in more detail. Compared with the classic KAM theory, one KAM step in our proof will be finished by a family of subiterations. Moreover, the steps of the subiteration will go to
$\infty .$
5.3.1 KAM step
Suppose that we are now in a certain step and, in what follows, the quantities without subscripts refer to those of this step, while the quantities with subscripts ‘+’ denote the corresponding ones of the next step. Thus, we consider the following system:
$$ \begin{align} \frac{d x}{dt} = ( A+B(t, \epsilon)+ P(t, \epsilon)) x, x\in \mathbb{R}^{3}, \end{align} $$
where
$ A= \unicode{x3bb} J_{1}, B(t, \epsilon )= \Xi (t, \epsilon ) J_{1}, $
$B(t,\epsilon ), P(t,\epsilon )\in C^{a,m}(\mathbb {T}^{2}_{r} \times U_{\delta }) $
and satisfy
$\|P\|_{r,U_{\delta }} \leq \mathcal {E}, \|\partial _{\epsilon }P\|_{r,U_{\delta }} \leq M.$
For
$\tau> 2,$
we define
$$ \begin{align} \mathcal{A}= \tau+3, \mathcal{M}= \frac{\mathcal{A}^{4}}{2}, \end{align} $$
and let
$( Q_{n})$
be the selected subsequence of
$\alpha $
in Lemma 10 with this given
$\mathcal {A}.$
For the detailed properties of this sequence, refer to Lemma 10 and the definition of CD bridge in §4. At each step, we fix a certain n and then use
$Q_{n}$
to define the related quantities, which will be allowed to vary along the steps of KAM iteration. For
$ r, \gamma> 0,$
we define
$$ \begin{align*} \eta= \frac{\tilde{c}}{ Q_{n}^{{1}/{ 2\mathcal{A}^{4}}}},\quad \mathcal{L}= e^{- c_{0}\gamma r (({ \bar{Q}_{n}}/{ Q_{n}^{\mathcal{M}}}) + \bar{Q}_{n}^{{1}/{{\mathcal{M}}^{1/4}}})}, \end{align*} $$
$$ \begin{align*} r_{+}=r(1-\eta)^{2},\quad \mathcal{L}_{+}= e^{-c_{0}\gamma r_{+}(({\bar{Q}_{n+1} }/{Q_{n+1}^{\mathcal{M}}}) + \bar{Q}_{n+1}^{{1}/{\mathcal{M}^{1/4}}})}, \end{align*} $$
$$ \begin{align} \mathcal{E}_{+}= \mathcal{L}_{+} \mathcal{E},\quad K= \bigg[\frac{\gamma}{4 \cdot 10^{\tau}} \max \bigg\{ \frac{ \bar{Q}_{n+1}}{Q_{n+1}^{\tau}}, \bar{Q}_{n+1}^{{3}/{\mathcal{A}}} \bigg\} \bigg], \end{align} $$
where
$[\cdot ]$
denotes its integer part,
$ 0<\tilde {c}<1$
is a constant which will be defined in equation (52), and
$c_{0}= {\tilde {c}}/({ 4^{5} \cdot 10^{\tau } }).$
Lemma 15. We consider the linear quasi-periodic system (30). Let
$ \tau>2, 0< r_{*}<r, 0\kern1.4pt{<}\kern1.4pt\tilde {c}\kern1.4pt{<}\kern1.4pt 1. $
There exist
$\varepsilon _{0}= \varepsilon _{0}(\tau , r_{*}, \tilde {c}), \varepsilon _{1}= \varepsilon _{1}(\tau , r_{*}, \tilde {c}), J\kern1.4pt{=}\kern1.4pt J(\tau ) $
, and
${T_{0}=T_{0}(\tau , r_{*}, \tilde {c} ) }$
such that for
$B(t,\epsilon )\in C^{a,m} ( \mathbb {T}^{2}_{r} \times U_{\delta }) $
, with
$\mathcal {R}_{Q_{n+1}} B(t, \epsilon )=0,$
and
$ P(t, \epsilon )\in C^{a,m} ( \mathbb {T}^{2}_{r} \times U_{\delta }), $
if
$$ \begin{align*} \bar{Q}_{n+1}\geq T_{0}\cdot \gamma^{-\mathcal{A}/2 }, \end{align*} $$
$$ \begin{align*} \|B\|_{r, U_{\delta}} \bigg( \frac{\bar{Q}_{n}}{Q_{n}^{\mathcal{A}^{4}}}+ \bar{Q}_{n}^{{1}/{\mathcal{A}}}\bigg) \leq \varepsilon_{1} \gamma \bigg( \frac{\bar{Q}_{n}}{Q_{n}^{\mathcal{M}}}+ \bar{Q}_{n}^{{1}/{\mathcal{M}^{1/4}}}\bigg), \end{align*} $$
and
$$ \begin{align} \|P\|_{r, U_{\delta}}\leq \mathcal{E} \leq \varepsilon_{0} \gamma^{J} \mathcal{L}, \end{align} $$
then the following conclusions hold.
-
(1) For any
$\epsilon \in U_{\delta }=[0, \delta ],$
there exists an orthogonal transformation
$ x= \Phi x_{+},$
which is real analytic in t on
$D(s_{+})$
and
$C^{m}$
-smooth in
$\epsilon $
on
$U_{\delta }$
such that (34)
$$ \begin{align} \| \Phi-\mathrm{id}\|_{r_{+}, U_{\delta}} \leq \frac{\mathcal{E}^{{14}/{15}}}{4}. \end{align} $$
-
(2) There exist two matrix functions
$B_{+}(t,\epsilon ), P_{+}(t,\epsilon )\in C^{a,m} ( \mathbb {T}^{2}_{r_{+}} \times U_{\delta }), $
where
$B_{+}(t, \epsilon ) = \Xi _{+}(\theta ) J_{1},$
with
$\mathcal {R}_{ Q_{n+2}} B_{+}(t, \epsilon )=0$
, and
$P_{+}(t, \epsilon )$
is a new perturbation, which satisfy the estimates (35)
$$ \begin{align} \| P_{+}\|_{r_{+}, U_{\delta}}\leq \mathcal{E}_{+},\quad \|B_{+}- B\|_{r_{+}, U_{\delta}}\leq 2 \mathcal{E}. \end{align} $$
Furthermore, in the differentiable case where
$m=1,$
except for the above conditions, if
$\|A\|_{C^{1}[0,\delta ]}\leq c, \|\partial _{\epsilon }B\|_{r, U_{\delta }}\leq c, \|\partial _{\epsilon }P\|_{r, U_{\delta }}\leq M, $
we have and
$$ \begin{align*} \|\partial_{\epsilon}\Phi\|_{{r_{+}, U_{\delta}}}\leq 2 e\mathcal{L}^{- {1}/{240}}C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau} ( 2C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau}\mathcal{E} + M ) \end{align*} $$
where the definition of
$$ \begin{align*} \|\partial_{\epsilon}P_{+}\|_{{r_{+}, U_{\delta}}}\leq M_{+}, \end{align*} $$
$M_{+}$
is as follows:
$$ \begin{align*} M_{+}=\tilde{M}_{L}\quad\mbox{with}\quad \tilde{M}_{k+1}=c \tilde{\mathcal{E}}_{k}^{{1}/{15}} \tilde{M}_{k}+ c\tilde{\mathcal{E}}_{k}^{{16}/{15}}\quad(k=0, \ldots, L-1), \end{align*} $$
and L is the number of steps in the KAM iteration.
$$ \begin{align*} \tilde{M}_{0}=M,\quad \tilde{\mathcal{E}}_{k}= \tilde{\mathcal{E}}_{0}^{({16}/{15})^{k}},\quad \tilde{\mathcal{E}}_{0}= \mathcal{E}, \end{align*} $$
-
(3) Let
$ \pm i (\unicode{x3bb} + [\Xi ])$
be the eigenvalues of
$A+ [B]$
and define
$$ \begin{align*} \Pi= \bigg\{ \epsilon\in [0,\delta] : | \langle k, \omega \rangle \pm (\unicode{x3bb} + [\Xi ] ) | \geq \frac{ \gamma}{(|k|+1)^{3\tau} } \text{ for all } |k| < K\bigg\}. \end{align*} $$
If
$\epsilon \in \Pi ,$
the orthogonal transformation
$ x= \Phi x_{+}$
reduces the system (30) into that is, the extended and orthogonal transformation is a true one and the new quasi-periodic system comes exactly from the original one under the orthogonal transformation.
$$ \begin{align*} \frac{d x_{+}}{dt} = (A + B_{+}(t, \epsilon) + P_{+}(t, \epsilon) ) x_{+},\quad x_{+}\in\mathbb{R}^{3}, \end{align*} $$
In the case
$B(t,\epsilon )=0$
in system (30), that is, for the first step, the same conclusions are true if one replaces equation (33) by
$$ \begin{align*} \| P\|_{r, U_{\delta}} \leq \mathcal{E} \leq \min\bigg\{\varepsilon_{0} \gamma^{J},\frac{1}{ Q_{n_{0}}^{120\tau}} \bigg\}. \end{align*} $$
Proof. Before giving the proof, we first collect some useful estimates. Let
$$ \begin{align*} \tilde{\mathcal{E}}_{0}= \mathcal{E},\quad\tilde{r}_{0}= r(1-\eta),\quad \tilde{M}_{0}=M, \end{align*} $$
and define inductively the following sequences:
$$ \begin{align*} \tilde{\mathcal{E}}_{m}=\tilde{\mathcal{E}}_{m-1}^{{16}/{15}}= \tilde{\mathcal{E}}_{0}^{({16}/{15})^{m}},\quad \tilde{M}_{m}=c \tilde{\mathcal{E}}_{m-1}^{{1}/{15}} \tilde{M}_{m-1}+ c\tilde{\mathcal{E}}_{m-1}^{{16}/{15}}. \end{align*} $$
Let
$m_{0}= \min \{ m\in \mathbb {Z}_{+}: K\tilde {\mathcal {E}}_{m}^{{1}/{15}} < 1 \}, $
and we define
$$ \begin{align*} \sigma_{m}= \begin{cases} \displaystyle\frac{\eta}{2^{m+3} },\kern6pt\qquad m< m_{0}, \\- \displaystyle\frac{2 \ln \tilde{\mathcal{E}}_{m}}{15 K \tilde{r}_{0} },\quad m \geq m_{0}, \end{cases} \end{align*} $$
and
$\tilde {r}_{m}= \tilde {r}_{m-1}- 2\tilde {r}_{0}\sigma _{m-1}. $
As in [Reference Zhang, Xu and Xu40], there exist
$J(\tau )= [120\tau \mathcal {A}^{5}], \varepsilon _{0}= \varepsilon _{0}(\tau , r_{*}, \tilde {c}), $
and
$T_{0}= T_{0}(\tau , r_{*}, \tilde {c}) $
such that if
$$ \begin{align*} \tilde{\mathcal{E}}_{0} \leq \varepsilon_{0}\gamma^{J(\tau)} \mathcal{L},\quad\bar{Q}_{n+1}\geq T_{0} \gamma ^{- \mathcal{A}/2}, \end{align*} $$
then we have the following useful estimates:
$$ \begin{align} \tilde{\mathcal{E}}_{0} \leq\min \bigg\{\frac{1}{Q_{n+1}^{120\tau}} , \bigg( \frac{\gamma^{\mathcal{A}\tau+2 }}{ 16 C_{3}(\tau)}\bigg)^{30} \bigg\}, \end{align} $$
$$ \begin{align} e^{- K \tilde{r}_{0}\sigma_{m} }\leq \tilde{\mathcal{E}}_{m}^{{2}/{15}},\quad\tilde{\mathcal{E}}_{m} \leq (\tilde{r}_{0} \sigma_{m} )^{15}, \end{align} $$
where
$C_{3}(\tau )$
is the constant in Lemma 14. For detailed proofs, we refer to [Reference Zhang, Xu and Xu40].
We now prove Lemma 15 by induction. For simplicity, we omit the dependence on
$\epsilon $
and write
$\|\cdot \|_{r,U_{\delta }} $
as
$\|\cdot \|_{r}.$
Consider
$$ \begin{align} \frac{d u}{dt} = ( A+B(t, \epsilon)+ P(t, \epsilon)) u,\quad u\in \mathbb{R}^{3}, \epsilon\in [0, \delta], \end{align} $$
where
$u=x, A= \unicode{x3bb} J_{1}, B(t,\epsilon )= \Xi (t,\epsilon ) J_{1}, $
$B(t,\epsilon ) \in C^{a,m}(\mathbb {T}^{2}_{r} \times U_{\delta }) (m=0,1) $
with
$\mathcal {R}_{Q_{n+1}} B(t,\epsilon )=0, $
$P(t,\epsilon )\in C^{a,m}(\mathbb {T}^{2}_{r} \times U_{\delta })(m=0,1) $
with
$\|P\|_{r}\leq \tilde {\mathcal {E}}_{0}$
and
$\|\partial _{\epsilon }P\|_{r}\leq \tilde {M}_{0}$
if
$m=1.$
Assume that for
$j=1, \ldots , \nu , $
one can find a quasi-periodic skew symmetric matrix
$\tilde {W}_{j-1}$
such that
$\phi _{j-1}: u_{j-1}=e^{ \tilde {W}_{j-1}} u_{j}$
is an orthogonal transformation and changes the system
$$ \begin{align*} \frac{d u_{j-1}}{dt} =(A + B(t, \epsilon) + b_{j-1}(t,\epsilon) + P_{j-1}(t,\epsilon)) u_{j-1} \end{align*} $$
into
$$ \begin{align*} \frac{d u_{j}}{dt} = (A+ B(t,\epsilon) + b_{j}(t,\epsilon ) + P_{j}(t,\epsilon) ) u_{j}, \end{align*} $$
satisfying
$ \mathcal {R}_{ Q_{n+2}} b_{j}=0, $
and
$$ \begin{align*} \| P_{j}\|_{\tilde{r}_{j}}\leq\tilde{\mathcal{E}}_{j}, \end{align*} $$
$$ \begin{align} \|{b}_{j}- {b}_{j-1} \|_{\tilde{r}_{j-1}}\leq\tilde{\mathcal{E}}_{j-1}, \end{align} $$
$$ \begin{align*} \|\tilde{W}_{j-1}\|_{\tilde{r}_{j-1} }\leq C_{3}(\tau) \gamma^{-(\mathcal{A}\tau+2) } Q_{n+1}^{18\tau} \mathcal{L}^{- {1}/{240}} \tilde{\mathcal{E}}_{j-1}. \end{align*} $$
In the differentiable case,
$$ \begin{align*} \|\partial_{\epsilon}P_{j}\|_{\tilde{r}_{j} } \leq \tilde{M}_{j} \end{align*} $$
and
$$ \begin{align*} \|\partial_{\epsilon} \tilde{W}_{j-1}\|_{\tilde{r}_{j-1} }\leq C_{3}(\tau) \gamma^{-(\mathcal{A}\tau+2) } Q_{n+1}^{18\tau} \mathcal{L}^{- {1}/{240}} ( 2 C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau} \tilde{\mathcal{E}} + \tilde{ M} ). \end{align*} $$
For
$j=\nu +1,$
one wants to find a matrix function
$\tilde {W}_{\nu }$
such that the orthogonal transformation
$\phi _{\nu }: u_{\nu }= e^{ \tilde {W}_{\nu }} u_{\nu +1}$
reduces the system
$$ \begin{align} \frac{d u_{\nu}}{dt} = (A+ B(t, \epsilon)+ {b}_{\nu}(t, \epsilon) + P_{\nu}(t, \epsilon))u_{\nu} \end{align} $$
into the desired form
$$ \begin{align*} \frac{d u_{\nu+1}}{dt} = (A+ B(t, \epsilon)+ {b}_{\nu+1}(t, \epsilon) + P_{\nu+1}(t, \epsilon))u_{\nu+1}, \end{align*} $$
and satisfies the corresponding estimates.
Suppose that
$P_{\nu }(t, \epsilon ) = a^{1}_{\nu }(t, \epsilon ) J_{1} + a^{2}_{\nu }(t, \epsilon ) J_{2} + a^{3}_{\nu }(t, \epsilon ) J_{3}, $
where the definition of
$J_{i}(i=1,2,3) $
is given in equation (A.1). First, we divide
$P_{\nu }(t, \epsilon )$
into three different parts:
$$ \begin{align*} P_{\nu} = \mathcal{T}_{K}^{1} P_{\nu}+ \mathcal{T}_{K}^{2} P_{\nu}+ \mathcal{R}_{K} P_{\nu}, \end{align*} $$
where
$$ \begin{align*} \mathcal{T}_{K}^{1} P_{\nu}= \left(\begin{array}{@{}ccc@{}} 0 & 0 & a_{\nu}^{2}(t, \epsilon) \\ 0 & 0 & a_{\nu}^{3} (t, \epsilon) \\ -a_{\nu}^{2}(t, \epsilon) & -a_{\nu}^{3}(t, \epsilon) & 0 \end{array} \right)_{|k| < K} \end{align*} $$
and
$$ \begin{align*} \mathcal{T}_{K}^{2} P_{\nu}= \left( \begin{array}{@{}ccc@{}} 0 & a_{\nu}^{1}(t, \epsilon) & 0 \\ -a_{\nu}^{1} (t, \epsilon) & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) _{|k| < K}, \end{align*} $$
satisfying
$$ \begin{align} \|\mathcal{T}_{K}^{1}P_{\nu}\|_{\tilde{r}_{\nu}} + \|\mathcal{T}_{K}^{2}P_{\nu}\|_{\tilde{r}_{\nu}} \leq 2 \|P_{\nu}\|_{\tilde{r}_{\nu}} \end{align} $$
and
$$ \begin{align*} \|\mathcal{R}_{K} P_{\nu}\|_{\tilde{r}_{\nu+1}}&= \sum_{|k|\geq K }|P_{\nu}(\epsilon)|_{U_{\delta}} e^{|k|(\tilde{r}_{\nu}-2 \tilde{r}_{0}\sigma_{\nu})} \\ &\leq e^{-2 K \tilde{r}_{0}\sigma_{\nu}} \|P_{\nu}\|_{\tilde{r}_{\nu}} \\ &\leq \tilde{\mathcal{E}}_{\nu}^{ {4}/{15}}\| P_{\nu}\|_{\tilde{r}_{\nu}}. \end{align*} $$
In the differentiable case,
$$ \begin{align} \| \mathcal{T}_{K}^{1}\partial_{\epsilon} P_{\nu}\|_{\tilde{r}_{\nu}} + \|\mathcal{T}_{K}^{2}\partial_{\epsilon} P_{\nu}\|_{\tilde{r}_{\nu}} \leq 2\|\partial_{\epsilon}P_{\nu}\|_{\tilde{r}_{\nu}} \end{align} $$
and
$$ \begin{align*} \|\mathcal{R}_{K}\partial_{\epsilon} P_{\nu}\|_{\tilde{r}_{\nu+1}}&= \sum_{|k|\geq K }|\partial_{\epsilon}P_{\nu}(\epsilon)|_{U_{\delta}} e^{|k|(\tilde{r}_{\nu}- 2\tilde{r}_{0}\sigma_{\nu})} \\ &\leq e^{- 2 K \tilde{r}_{0}\sigma_{\nu}} \|\partial_{\epsilon}P_{\nu}\|_{\tilde{r}_{\nu}} \\ &\leq \tilde{\mathcal{E}}_{\nu}^{{4}/{15}} \|\partial_{\epsilon}P_{\nu}\|_{\tilde{r}_{\nu}}. \end{align*} $$
We rewrite the system (40) as
$$ \begin{align} \frac{d u_{\nu}}{dt}= (A + B(t, \epsilon) + b_{\nu+1}(t, \epsilon) + (\mathcal{T}_{K}^{1} P_{\nu} + \mathcal{R}_{K} P_{\nu} ) ) u_{\nu}, \end{align} $$
where
$b_{\nu +1}(t, \epsilon )=b_{\nu }(t, \epsilon ) + \hat {b}_{\nu }(t, \epsilon ), \hat {b}_{\nu }(t, \epsilon )= \mathcal {T}_{K}^{2} P_{\nu }(t, \epsilon ).$
Let
$\phi _{\nu } : u_{\nu }= e^{ \tilde {W}_{\nu }} u_{\nu +1}, $
where
$ \tilde {W}_{\nu } =\tilde {W}_{\nu } (\omega t, \epsilon )$
is an analytic quasi-periodic skew symmetric matrix, which guarantees
$\phi _{\nu }$
is an orthogonal transformation. Denote
${\hat {A}}_{\nu }(t, \epsilon )= A+B(t, \epsilon )+b_{\nu +1}(t, \epsilon ).$
Then, the system (43) is transformed into
$$ \begin{align} \frac{ du_{\nu+1} }{dt} = \bigg(e^{ -\tilde{W}_{\nu}} ({\hat{A}}_{\nu}(t, \epsilon)+ \mathcal{T}_{K}^{1} P_{\nu} + \mathcal{R}_{K} P_{\nu} ) e^{ \tilde{W}_{\nu}}- e^{ -\tilde{W}_{\nu}} \frac{d}{d t} e^{\tilde{W}_{\nu}} \bigg) u_{\nu+1}. \end{align} $$
By Lemma A.3, the system (44) is still a skew symmetric system. Moreover, the system (44) can be rewritten as
$$ \begin{align*} \frac{ d u_{\nu+1}}{dt} = \bigg ( {\hat{A}}_{\nu} + \bigg({\hat{A}}_{\nu} \tilde{W}_{\nu} - \tilde{W}_{\nu}{\hat{A}}_{\nu} + \mathcal{T}_{K}^{1} P_{\nu}- \frac{ d \tilde{W}_{\nu}}{dt}\bigg) \bigg) u_{\nu+1}+ P_{\nu+1}' u_{\nu+1}, \end{align*} $$
where
$$ \begin{align} P_{\nu+1}^{\prime} &= \mathcal{R}_{K} P_{\nu}- \tilde{W}_{\nu} {\hat{A}}_{\nu} \tilde{W}_{\nu}+ (e^{-\tilde{W}_{\nu}} -I + \tilde{W}_{\nu} ) {\hat{A}}_{\nu} e^{\tilde{W}_{\nu}}+(I-\tilde{W}_{\nu} ){\hat{A}_{\nu}}(e^{\tilde{W}_{\nu}} -I - \tilde{W}_{\nu} ) \nonumber \\ &\quad + ( e^{-\tilde{W}_{\nu}} \kern1.5pt{-}\kern1.5pt I ) (\mathcal{T}_{K}^{1} P_{\nu} \kern1.5pt{+}\kern1.5pt \mathcal{R}_{K} P_{\nu} )e^{\tilde{W}_{\nu}} \kern1.5pt{+}\kern1.5pt (\mathcal{T}_{K}^{1} P_{\nu} \kern1.5pt{+}\kern1.5pt \mathcal{R}_{K} P_{\nu} ) ( e^{\tilde{W}_{\nu}} \kern1.5pt{-}\kern1.5pt I )\kern1.5pt{-}\kern1.5pt (e^{-\tilde{W}_{\nu}} \kern1.5pt{-}\kern1.5pt I) \dot{\tilde{W}}_{\nu} \nonumber \\ &\quad - e^{-\tilde{W}_{\nu}} \frac{d}{dt}(e^{\tilde{W}_{\nu}}-I- \tilde{W}_{\nu}). \end{align} $$
The point is to find
$\tilde {W}_{\nu }$
such that
$$ \begin{align} \frac{ d \tilde{W}_{\nu}}{dt}-{\hat{A}}_{\nu} \tilde{W}_{\nu} + \tilde{W}_{\nu}{\hat{A}}_{\nu} = \mathcal{T}_{K}^{1} P_{\nu}. \end{align} $$
Thanks to the definition of
$b_{\nu +1} $
and equation (41), one gets
$\|b_{\nu +1} - b_{\nu }\|_{\tilde {r}_{\nu }} \leq \|P_{\nu }\|_{\tilde {r}_{\nu }}=\tilde {\mathcal {E}}_{\nu }. $
From equation (39), it follows that
$$ \begin{align} \|b_{\nu +1}\|_{\tilde{r}_{\nu}}\leq \|b_{\nu+1}- b_{\nu}\|_{\tilde{r}_{\nu}}+ \sum_{j=1}^{\nu} \|b_{j}-b_{j-1}\|_{\tilde{r}_{j-1}} \leq 2 \tilde{\mathcal{E}}_{0}. \end{align} $$
Then, by equation (36), we have
$$ \begin{align*} \|b_{\nu+1}\|_{\tilde{r}_{\nu}}\leq 2 \tilde{\mathcal{E}}_{0}<\frac{\gamma^{\mathcal{A}\tau+2}}{ 2C_{3}(\tau) Q_{n+1}^{18\tau} }, \end{align*} $$
which implies that condition (21) is satisfied. Furthermore, since
$K< Q_{n+2},$
it is obvious that
$\mathcal {R}_{Q_{n+2}}(b_{\nu +1}- b_{\nu })=0, $
which implies
$\mathcal {R}_{Q_{n+2}}b_{\nu +1}=0 $
by
$\mathcal {R}_{Q_{n+2}}b_{\nu }=0. $
However, by the assumptions of Lemma 15,
$ B(t)$
satisfies equation (22),
$ {\mathcal {R}_{Q_{n+1} } B(t)=0, }$
then for any
$|k|< K, $
we can apply Lemma 14 to get an extended and approximate solution
$\tilde {W}_{\nu } $
of equation (46) with the error term
$(\mathcal {T}_{K}^{1} P_{\nu })_{e}.$
That is to say, instead of solving equation (46), we first solve the approximation equation
$$ \begin{align*} \frac{ d\tilde{W}_{\nu}}{ dt} - {\hat{A}_{\nu}} \tilde{W}_{\nu} + \tilde{W}_{\nu} {\hat{A}_{\nu}}= \mathcal{T}_{K}^{1} P_{\nu} - (\mathcal{T}_{K}^{1} P_{\nu})_{e}. \end{align*} $$
By Lemma 14, we have
$$ \begin{align*} \| \tilde{W}_{\nu}\|_{\tilde{r}_{\nu}} \leq C_{3}(\tau) \gamma^{-(\mathcal{A}\tau+2)} Q_{n+1}^{18\tau} \mathcal{L}^{- {1}/{240}} \|P_{\nu}\|_{\tilde{r}_{\nu}} \end{align*} $$
and
$$ \begin{align} \|\partial_{\epsilon}\tilde{W}_{\nu}\|_{\tilde{r}_{\nu}}\leq C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau} \mathcal{L}^{- {1}/{240}}( 2C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau}\|P_{\nu}\|_{\tilde{r}_{\nu}} + \|\partial_{\epsilon}P_{\nu}\|_{\tilde{r}_{\nu}}). \end{align} $$
Moreover, the truncated remainder
$(\mathcal {T}_{K}^{1} P_{\nu })_{e} = ((\mathcal {T}_{K}^{1} P_{\nu ,ij})_{e} )_{3 \times 3}$
with
$$ \begin{align*} (\mathcal{T}_{K}^{1} P_{\nu,ij})_{e}= e^{-i \mathcal{B}(t) } \mathcal{R}_{K} ( e^{i \mathcal{B}(t) }( \mathcal{T}_{K}^{1}P_{\nu, ij} + i \Delta_{\nu+1}(t) \tilde{W}_{ij} )), \end{align*} $$
satisfying
$$ \begin{align*} \|(\mathcal{T}_{K}^{1} P_{\nu})_{e}\|_{\hat{r}_{\nu+1}} \leq \mathcal{L}^{- {1}/{240}} e^{-K\tilde{r}_{0}\sigma_{\nu}} ( \|P_{\nu}\|_{\tilde{r}_{\nu}} + 2\|b_{\nu+1} \|_{\tilde{r}_{\nu}} \|\tilde{W}\|_{\tilde{r}_{\nu}}) \end{align*} $$
and
$$ \begin{align} \|\partial_{\epsilon}(\mathcal{T}_{K}^{2} P_{\nu})_{e}\|_{\hat{r}_{\nu+1}}&\leq C_{3}(\tau) \mathcal{L}^{- {1}/{240} } e^{-K \sigma_{\nu}} (\|P_{\nu}\|_{\tilde{r}_{\nu}} + 2 \|b_{\nu+1}\|_{\tilde{r}_{\nu}}\|\tilde{W}_{\nu}\|_{\tilde{r}_{\nu}} \nonumber\\ &\quad+ \|\partial_{\epsilon}P_{\nu}\|_{\tilde{r}} + 2 \|\partial_{\epsilon}b_{\nu+1}\|_{\tilde{r}} \|\tilde{W}_{\nu}\|_{\tilde{r}_{\nu}} + 2 \|b_{\nu+1}\|_{\tilde{r}_{\nu}}\|\partial_{\epsilon}\tilde{W}_{\nu}\|_{\tilde{r}_{\nu}}), \end{align} $$
where
$\hat {r}_{\nu +1}=\tilde {r}_{\nu }-\tilde {r}_{0}\sigma _{\nu }.$
Define
$$ \begin{align*} \Pi= \bigg\{\epsilon\in [0, \delta] ~ \bigg| ~ |\langle k, \omega \rangle \pm (\unicode{x3bb} + [\Xi] ) | \geq \frac{ {\gamma}}{(|k|+1)^{ 3\tau} } \text{ for all } |k|< K \bigg\}, \end{align*} $$
where
$ 0 <{\gamma }\leq \gamma _{0}/2. $
If
$\epsilon \in \Pi , $
the extended and approximate solution
$ \tilde {W}_{\nu }(t; \epsilon )$
is a true and approximate solution, that is,
$ \tilde {W}_{\nu }(t, \epsilon )= W_{\nu }(t,\epsilon ) $
$ \text { for all } \epsilon \in \Pi. $
Then, the system (40) becomes
$$ \begin{align*} \frac{ du_{\nu+1}}{dt}= (A +B(t, \epsilon) + b_{\nu+1}(t, \epsilon)+ P_{\nu+1}(t,\epsilon) ) u_{\nu+1}, \end{align*} $$
where
$b_{\nu +1}(t)=b_{\nu }(t) + \hat {b}_{\nu }(t), \hat {b}_{\nu }(t)= \mathcal {T}_{K}^{2} P_{\nu }$
,
$P_{\nu +1 }= P_{\nu +1 }' +(\mathcal {T}_{K}^{1} P_{\nu })_{e}. $
We now estimate the new perturbation term
$P_{\nu +1}.$
Suppose
$\| P_{\nu }\|_{\tilde {r}_{\nu }} \leq \tilde {\mathcal {E}}_{\nu }$
and
$\tilde {\mathcal {E}}_{0}$
is sufficiently small such that
$$ \begin{align} \|\tilde{W}_{\nu}\|_{\tilde{r}_{\nu} }\leq C_{3}(\tau) \gamma^{- (\mathcal{A} \tau +2) } Q_{n+1}^{18\tau} \mathcal{L}^{- {1}/{240} } \| P_{\nu}\|_{r_{\nu}} \leq \frac{\tilde{\mathcal{E}}_{\nu}^{{14}/{15}} }{32}\leq \frac{1}{2}, \end{align} $$
then by
$ e^{\tilde {W}}= I + \tilde {W}+ ({\tilde {W}^{2}}/{2!}) + \cdots + ({\tilde {W}^{n}}/{ n!}) + \cdots , $
we have
$$ \begin{align*} \|e^{\pm \tilde{W}_{\nu}}\|_{\tilde{r}_{\nu}} \leq \frac{1}{1- \|\tilde{W}_{\nu}\|_{\tilde{r}_{\nu}} } \leq 2. \end{align*} $$
Similarly, we have
$$ \begin{align} \|e^{\pm \tilde{W}_{\nu}}-I \|_{\tilde{r}_{\nu}} \leq 2 C_{3}(\tau) \gamma^{-(\mathcal{A}\tau+2 )} Q_{n+1}^{18\tau} \mathcal{L}^{-{1}/{240}} \tilde{\mathcal{E}}_{\nu} \leq \frac{\tilde{\mathcal{E} }_{\nu}^{{14}/{15}}}{16}, \end{align} $$
$$ \begin{align*} \|e^{\pm \tilde{W}_{\nu}}- I \mp \tilde{ W}_{\nu} \|_{\tilde{r}_{\nu}} \leq 2 C_{3}^{2}(\tau) \gamma^{-2(\mathcal{A}\tau+2 )} Q_{n+1}^{36\tau} \mathcal{L}^{- {1}/{120}} \tilde{\mathcal{E}}_{\nu}^{2} \leq \frac{\tilde{\mathcal{E}}_{\nu}^{{17}/{15}}}{16}. \end{align*} $$
By Cauchy’s estimates and in view of
$$ \begin{align*} \tilde{r}_{\nu}- \tilde{r}_{\nu+1}= 2 \tilde{r}_{0} \sigma_{\nu}, \end{align*} $$
it follows that
$$ \begin{align*} \bigg\|\frac{d \tilde{W}_{\nu} }{dt} \bigg\|_{\tilde{r}_{\nu+1}} \leq \frac{\|\tilde{ W}_{\nu}\|_{\tilde{r}_{\nu}}} { 2 \tilde{r}_{0} \sigma_{\nu}} \leq \frac{ C_{3}(\tau) \gamma^{-(\mathcal{A}\tau+2 )} Q_{n+1}^{18\tau} \mathcal{L}^{-{1}/{240}} \tilde{\mathcal{E}}_{\nu} }{ 2 \tilde{r}_{0}\sigma_{\nu}} \leq \frac{\tilde{\mathcal{E}}_{\nu}^{{13}/{15}}}{16} \end{align*} $$
and
$$ \begin{align*} \bigg\|\frac{ d(e^{ \tilde{W}_{\nu}} -I- \tilde{W}_{\nu} )}{dt} \bigg\|_{\tilde{r}_{\nu+1}} \leq \frac{ 2C_{3}^{2}(\tau) \gamma^{-2(\mathcal{A}\tau+2 )} Q_{n+1}^{36\tau} \mathcal{L}^{- {1}/{120}} \tilde{\mathcal{E}}_{\nu}^{2}}{ 2 \tilde{r}_{0} \sigma_{\nu} } \leq \frac{\tilde{\mathcal{E}}_{\nu}^{{16}/{15}} }{8}. \end{align*} $$
Altogether, combining the definition (45) of
$P_{\nu +1}',$
the estimate of
$(\mathcal {T}_{K}^{1} P_{\nu })_{e}$
, and all the above estimates, we obtain
$$ \begin{align*} \|P_{\nu+1} \|_{\tilde{r}_{\nu+1}} \leq \|P_{\nu+1}'\|_{\tilde{r}_{\nu+1}} + \| (\mathcal{T}_{K}^{1} P_{\nu})_{e} \|_{\tilde{r}_{\nu+1}} \leq \tilde{\mathcal{E}}_{\nu}^{{16}/{15}}=\tilde{\mathcal{E}}_{\nu+1}. \end{align*} $$
In the differentiable case, by equations (48) and (49), we have
$$ \begin{align*} \|\partial_{\epsilon}{P}_{\nu+1} \|_{\tilde{r}_{\nu+1}} \leq c \tilde{\mathcal{E}}_{\nu}^{{1}/{15}} \tilde{M}_{\nu} + c\tilde{\mathcal{E}}_{\nu}^{{16}/{15}}= \tilde{M}_{\nu+1}. \end{align*} $$
Notice that there exists an
$L \geq m_{0}$
such that
$\tilde { \mathcal {E}}_{L} \leq \mathcal {L}_{+} \tilde { \mathcal {E}}_{0}= \mathcal {E}_{+} $
and
$\tilde { \mathcal {E}}_{L-1}> \mathcal {E}_{+}. $
Once we reach the Lth step, we stop the above iteration. Let
$$ \begin{align*} \Phi= \phi_{0}\circ\cdots\circ \phi_{L-1} : u= e^{ \tilde{W}_{0}} \cdots e^{ \tilde{W}_{L-1}} u_{L} \end{align*} $$
and
$x_{+}= u_{L}, B_{+}(t, \epsilon )= B(t, \epsilon ) + b_{L}(t,\epsilon ), P_{+}(t,\epsilon ) = \tilde {P}_{L}(t,\epsilon ), $
$$ \begin{align*} b_{L}(t, \epsilon)= \sum_{j=0}^{L-1} \mathcal{T}_{k}^{2} P_{j}(t, \epsilon)= \sum_{j=0}^{L-1} (a_{j}^{1}(t,\epsilon))_{|k|< K } J_{1} = \Delta_{L}(t,\epsilon) J_{1}. \end{align*} $$
Then, the orthogonal transformation
$x= \Phi x_{+}$
reduces the system (30) into
$$ \begin{align*} \frac{d x_{+}}{dt}= (A+ B_{+}(t, \epsilon)+ P_{+}(t, \epsilon) ) x_{+}, \end{align*} $$
satisfying
$$ \begin{align*} \|P_{+}\|_{r_{+}}\leq \tilde{ \mathcal{E}}_{L} \leq \mathcal{L}_{+} \tilde{ \mathcal{E}}_{0}\triangleq \mathcal{E}_{+} \end{align*} $$
and
$$ \begin{align*} \|\partial_{\epsilon}P_{+}\|_{r_{+}}\leq c \tilde{\mathcal{E}}_{L-1}^{{1}/{15}} \tilde{M}_{L-1} + c\tilde{\mathcal{E}}_{L-1}^{{16}/{15}}= \tilde{M}_{L}\triangleq M_{+}. \end{align*} $$
Moreover, since
$K< Q_{n+2},$
it follows that
$\mathcal {R}_{Q_{n+2}} B_{+}(t)=0. $
Below, we prove
$B_{+}(t, \epsilon )$
has the same form as
$A.$
Let
$ \Delta _{L}(t,\epsilon ) = \sum _{j=0}^{L-1} ( a^{1}_{j}(t,\epsilon ))_{|k| < K }. $
Due to
$ b_{L}(t,\epsilon ) = \Delta _{L}(t,\epsilon )J_{1},$
we have
$$ \begin{align*} B_{+}(t,\epsilon)&= B(t,\epsilon) +b_{L}(t,\epsilon) =\left( \begin{array}{@{}ccc@{\kern1pt}} 0 & \Xi(t,\epsilon) + \Delta_{L}(t,\epsilon) & 0\\ - (\Xi(t,\epsilon) + \Delta_{L}(t,\epsilon) ) & 0 &0 \\ 0 & 0 &0 \end{array} \right) \\ &= \left( \begin{array}{@{}ccc@{\kern1pt}} 0 & \Xi_{+}(t,\epsilon) & 0 \\ -\Xi_{+}(t,\epsilon) & 0 &0 \\ 0 & 0 &0 \end{array} \right)\!. \end{align*} $$
Accordingly, define
$$ \begin{align*} \Pi_{+}= \bigg\{\epsilon\in [0, \delta] ~ \bigg| ~ |\langle k, \omega \rangle \pm (\unicode{x3bb}+ [\Xi_{+}] ) | \geq \frac{ {\gamma}_{+}}{(|k|+1)^{ 3\tau} }\text{ for all } |k|< K_{+} \bigg\}, \end{align*} $$
where
$ 0 <{\gamma }_{+}\leq \gamma _{0} /2.$
Recall that
$$ \begin{align*} r_{+}= \tilde{r}_{0}(1-\eta). \end{align*} $$
By the definition of
$\sigma _{m},$
we know
$ \sigma _{m+1}= ({16}/{15}) \sigma _{m}$
if
$m\geq m_{0}.$
Then, by the selection of
$L\in \mathbb {Z}_{+}, $
that is,
$ \tilde {\mathcal {E}}_{L}\leq \mathcal {E}_{+}, $
$\tilde {\mathcal {E}}_{L-1}> \mathcal {E}_{+}=\mathcal {L}_{+} \tilde {\mathcal {E}}_{0},$
we have
$$ \begin{align*} \sum_{j=0}^{L-1} \sigma_{j}&= \sum_{j=0}^{m_{0}-1} \sigma_{j} + \sum_{j=m_{0}}^{ L-1} \sigma_{j} \\ &\quad < \frac{\eta}{4} + \frac{\tilde{c}}{8} \bigg( \frac{1}{Q_{n+1}^{ \mathcal{M}-\tau}}+ \frac{1}{\bar{Q}_{n+1}^{{2}/{\mathcal{A}}}} \bigg)\\ &\quad < \frac{\eta}{4}+ \frac{1}{4} \frac{\tilde{c}}{ Q_{n}^{{1}/{ 2\mathcal{A}^{4}}}} = \frac{\eta}{2}, \end{align*} $$
which implies
$\tilde {r}_{L}\geq r_{+}. $
Since
$\Phi =\phi _{0}\circ \cdots \circ \phi _{L-1}, $
where
$\phi _{i}: x_{i}=e^{\tilde {W}_{i}} x_{i+1}$
$(i=0, \ldots ,L-1),$
we obtain
$x_{0}= \Phi x_{L},$
that is,
$$ \begin{align*} x_{0}=e^{ \tilde{W}_{0}} x_{1} = e^{ \tilde{W}_{0}}e^{ \tilde{W}_{1}}\cdots e^{ \tilde{W}_{L-1}} x_{L}. \end{align*} $$
Then, the estimate (50) yields that
$$ \begin{align*} \|\Phi-\mathrm{id}\|_{r_{+}} &\leq \bigg\| \prod_{i=0}^{L-1}e^{ \tilde{W}_{i}} -\mathrm{Id} \bigg\| \leq\|e^{\sum_{i=0}^{L-1}\tilde{ W}_{i}}-\mathrm{Id} \|\\ &\leq 2 \sum_{i=0}^{L-1} \|\tilde{W}_{i}\|\leq \frac{\tilde{\mathcal{E}}_{0}^{{14}/{15}}}{4}. \end{align*} $$
In the differentiable case,
$$ \begin{align*} \|\partial_{\epsilon}\Phi\|_{{r_{+}}}\leq 2 e \mathcal{L}^{- {1}/{240}}C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau} ( 2C_{3}(\tau) \gamma ^{-( \mathcal{A}\tau+2)} Q_{n+1}^{18\tau}\mathcal{E} + M ). \end{align*} $$
In addition, it follows from equation (47) that the estimate (35) holds. Thus, we get all the estimates (35)–(34).
5.3.2 Setting of parameters and iteration
In this section, we choose some appropriate parameter sequences to ensure that the KAM step can iterate infinitely many times.
Let
$0<r_{*}<r, \tau>2, \gamma >0, $
and
$ \mathcal {A}, \mathcal {M}$
are defined as in equation (31). Since
${{r}/{{r_{*}}}> 1, }$
there exists
$\tilde {c}_{0} = \tfrac 12 (({r}/{{r_{*}}}) +1 )$
such that
$ r> \tilde {c}_{0}r_{*}. $
Let
$$ \begin{align} \tilde{c}= \frac{1}{12} \bigg(1- \frac{1}{\tilde{c}_{0}}\bigg)< 1,\quad T_{1}= \bigg(\frac{4^{5}\cdot 10^{\tau} J(\tau) \ln2 }{\tilde{c} r_{*} } \bigg)^{\mathcal{A}}, \end{align} $$
$$ \begin{align*} T= \max \bigg \{T_{0}\bigg( \frac{\gamma}{2}\bigg)^{-({\mathcal{A}}/{2})}, T_{1}\gamma^{-\mathcal{A}}, 4^{\mathcal{A}^{4}} \bigg \}. \end{align*} $$
These constants
$ \varepsilon _{0}(\tau , r_{*}, \tilde {c} ), \varepsilon _{1}(\tau , r_{*}, \tilde {c}), J(\tau ) , T_{0}(\tau , r_{*}, \tilde {c})$
are defined as in Lemma 15.
Due to the choice of Lemma 10, there exists some
$n_{0}\in \mathbb {Z}_{+} $
such that
$Q_{n_{0}}\leq T^{\mathcal {A}^{4}},$
whereas
$\bar {Q}_{n_{0}}\geq T.$
Since
$Q_{n_{0}}\leq T^{\mathcal {A}^{4}},$
we can take
$\mathcal {E}$
sufficiently small, depending on
$r, r_{*}, \gamma , \gamma _{0}, \tau $
, and
$M,$
but not on
$\alpha ,$
such that
$$ \begin{align} \mathcal{E}< \min\bigg\{\varepsilon_{0} \bigg(\frac{\gamma}{2} \bigg)^{J}, \frac{\varepsilon_{1} \gamma}{2}, \frac{1}{T^{120 \tau \mathcal{A}^{4}}} \bigg\} \leq \min \bigg\{\varepsilon_{0} \bigg(\frac{\gamma}{2}\bigg)^{J}, \frac{\varepsilon_{1} \gamma}{2}, \frac{1}{Q_{n_{0}}^{120\tau}}\bigg\}. \end{align} $$
For any given
$\mathcal {E}> 0$
satisfying equation (53), we define some sequences inductively:
$$ \begin{align*} \mathcal{E}_{0}= \mathcal{E},\quad M_{0}= M,\quad r_{0}=r,\quad \gamma_{0}=\gamma, \end{align*} $$
$$ \begin{align*} \gamma_{n}= \gamma\bigg(1-\frac{1}{2^{n}}\bigg),\quad \eta_{n}= \tilde{c} Q_{n_{0}+n }^{ - ({1}/{2 \mathcal{A}^{4}})},\quad r_{n+1}= r_{n}(1- \eta_{n})^{2}, \end{align*} $$
$$ \begin{align*} K_{n}= \bigg[\frac{\gamma_{n}}{4\cdot 10^{\tau}}\max \bigg\{\frac{\bar{Q}_{n_{0}+n}}{ Q_{n_{0}+n }^{\tau} }, \bar{Q}_{n_{0}+n}^{ {3}/{\mathcal{A}}} \bigg\}\bigg], \end{align*} $$
$$ \begin{align} \mathcal{L}_{n+1}= e^{ -c_{0} \gamma_{n} r_{n+1} (({\bar{Q}_{n_{0}+n+1}}/{Q_{n_{0} +n+1}^{\mathcal{M}} }) + {\bar{Q}_{n_{0}+n+1}^{{1}/{\mathcal{M}^{1/4}}}} )},\quad \mathcal{E}_{n+1}= \mathcal{L}_{n+1} \mathcal{E}_{n}, \end{align} $$
$$ \begin{align*} M_{n}= \tilde{M}_{L_{n}}\quad \mbox{with}\quad \tilde{M}_{k}=c \tilde{\mathcal{E}}_{k-1}^{{1}/{15}} \tilde{M}_{k-1}+ c\tilde{\mathcal{E}}_{k-1}^{{16}/{15}}\quad(k=1, \ldots, L_{n}), \end{align*} $$
$$ \begin{align*} \tilde{M}_{0}=M,\quad \tilde{\mathcal{E}}_{k}= \tilde{\mathcal{E}}_{0}^{({16}/{15})^{k}},\quad \tilde{\mathcal{E}}_{0}= \mathcal{E}, \end{align*} $$
where
$L_{n}$
is the number of steps in the KAM iteration.
By the choice of the above parameters, we can verify that for all
$n\in \mathbb {Z}_{+}$
,
$r_{n}, B_{n}, \bar {Q}_{n_{0}+n}, P_{n} $
satisfy that
$$ \begin{align*} r_{n}\geq r_{*},\quad \bar{Q}_{n_{0}+n} \geq T_{0} \gamma_{n} ^{-\mathcal{A}/2}, \end{align*} $$
$$ \begin{align*} \|B_{n}\|_{r_{n}, U_{\delta} }\bigg( \frac{\bar{Q}_{n_{0}+n -1}}{Q_{n_{0}+n -1}^{\mathcal{A}^{4}}}+\bar{Q}_{n_{0}+n -1}^{{1}/{\mathcal{A}}}\bigg) \leq \varepsilon_{1} \gamma_{n} \bigg( \frac{\bar{Q}_{n_{0}+n -1}}{Q_{n_{0}+n -1}^{\mathcal{M}}}+\bar{Q}_{n_{0}+n -1}^{ {1}/{\mathcal{M}^{1/4}}}\bigg), \end{align*} $$
$$ \begin{align*} \| P_{n}\|_{r_{n}, U_{\delta}} \leq \mathcal{E}_{n} \leq \varepsilon_{0} \gamma_{n}^{J} \mathcal{L}_{n}. \end{align*} $$
For detailed proofs, we refer to [Reference Zhang, Xu and Xu40].
After setting the parameters, we will iterate KAM iteration infinitely and prove its convergence. For the first step, let
$B_{0}=0, P_{0}=P.$
By our selection
$ \bar {Q}_{n_{0}} \geq T \geq T_{0} \gamma ^{- ({\mathcal {A}}/{2})}$
and equation (53), we have
$$ \begin{align*} \|P_{0}\|_{r_{0}, U_{\delta}}\leq \mathcal{E}_{0}\leq \min \bigg\{\varepsilon_{0} \bigg(\frac{\gamma}{2}\bigg)^{J}, \frac{1}{Q_{n_{0}}^{120\tau}} \bigg\}. \end{align*} $$
In the differentiable case, suppose that
$$ \begin{align*} \|\partial_{\epsilon}P_{0}\|_{r_{0}, U_{\delta}}\leq M_{0}. \end{align*} $$
Meanwhile, we can check equations (36) and (37) hold. Thus, we apply Lemma 15 and get the transformation
$\Phi _{0}: D(r_{1}) \times U_{\delta }\rightarrow D(r_{0}) \times U_{\delta } $
such that for all
$ \epsilon \in U_{\delta },$
the system (9) is transformed into
$$ \begin{align*} \dot{x}_{1}=( A + B_{1}(t, \epsilon)+ P_{1}(t, \epsilon)) x_{1}, \end{align*} $$
satisfying
$$ \begin{align*} \|B_{1}\|_{r_{1}, U_{\delta}}\leq 2\mathcal{E}_{0}\leq \varepsilon_{1} \gamma \end{align*} $$
and
$$ \begin{align*} \| P_{1} \|_{ r_{1}, U_{\delta} }\leq \mathcal{E}_{0} \mathcal{L}\leq \varepsilon_{0} \gamma^{J}\mathcal{L}. \end{align*} $$
In the differentiable case,
$$ \begin{align*} \|\partial_{\epsilon}P_{1} \|_{ r_{1}, U_{\delta} } \leq M_{1}, \end{align*} $$
where
$$ \begin{align*} M_{1}= \tilde{M}_{L_{1}}\quad\mbox{with}\quad \tilde{M}_{k}=c \tilde{\mathcal{E}}_{k-1}^{{1}/{15}} \tilde{M}_{k-1}+ c\tilde{\mathcal{E}}_{k-1}^{{16}/{15}}(k=1, \ldots, L_{1}), \end{align*} $$
$$ \begin{align*} \tilde{M}_{0}=M,\quad\tilde{\mathcal{E}}_{k}= \tilde{\mathcal{E}}_{0}^{({16}/{15})^{k}},\quad \tilde{\mathcal{E}}_{0}= \mathcal{E}, \end{align*} $$
where
$L_{1}$
is the number of steps in the KAM iteration. Note that the above estimates mean that all the conditions of Lemma 15 for the next step are established.
Inductively, there exists an orthogonal transformation
$\Phi _{n-1}: x_{n-1} =e^{ \tilde {W}_{0}^{n-1}} \cdots e^{ \tilde {W}_{L_{n-1}-1}^{n-1}} x_{n} $
such that for any
$\epsilon \in U_{\delta },$
it reduces the system of the previous step into a desired form and satisfies that
$$ \begin{align*} \|\Phi_{n-1}-\mathrm{id}\|_{r_{n}, U_{\delta}} \leq \frac{\mathcal{E}_{n}^{{14}/{15}}}{4}. \end{align*} $$
In the differentiable case,
$$ \begin{align*} \|\partial_{\epsilon}\Phi_{n-1}\|_{r_{n}, U_{\delta}}&\leq \sum_{j=0}^{L_{n-1}-1} \|e^{\tilde{W}_{0}^{n-1}} \cdots e^{\tilde{W}_{j}^{n-1}} \partial_{\epsilon} \tilde{W}_{j}^{n-1}\cdots e^{ \tilde{W}_{L_{n-1}-1}^{n-1}}\|_{r_{n}, U_{\delta}} \\ &\leq 2 e \mathcal{L}_{n-1}^{- {1}/{240}}C_{3}(\tau) \gamma_{n-1}^{-( \mathcal{A}\tau+2)}\kern-0.5pt Q_{n_{0}+n}^{18\tau} ( 2C_{3}(\tau) \gamma_{n-1}^{-( \mathcal{A}\tau+2)}\kern-0.5pt Q_{n_{0}+n}^{18\tau}\mathcal{E}_{n-1} \kern1.3pt{+}\kern1.3pt M_{n-1}\kern-0.4pt), \end{align*} $$
where the last inequality is due to the estimate (48).
$$ \begin{align*} \|\partial_{\epsilon}\Phi_{n-1}\|_{r_{n}, U_{\delta}} \kern1.4pt{\leq}\kern1.4pt 2 e \mathcal{L}_{n-1}^{- {1}/{240}}C_{3}(\tau) \gamma_{n-1}^{-( \mathcal{A}\tau+2)} Q_{n_{0}+n}^{18\tau} ( 2C_{3}(\tau) \gamma_{n-1}^{-( \mathcal{A}\tau+2)} Q_{n_{0}+n}^{18\tau}\mathcal{E}_{n-1} \kern1.4pt{+}\kern1.4pt M_{n-1}\kern-1pt). \end{align*} $$
Let
$\Phi ^{n}= \Phi _{0} \cdots \Phi _{n-1}.$
Then, for any
$\epsilon \in U_{\delta },$
the orthogonal transformation
$x =\Phi ^{n} x_{n} $
reduces the system (9) into the following form:
$$ \begin{align*} \frac{ d x_{n}}{ dt} = ( A + B_{n}(t,\epsilon) + P_{n}(t, \epsilon)) x_{n}, x_{n}\in \mathbb{R}^{3}, \end{align*} $$
where
$B_{n}(t,\epsilon ), P_{n}(t,\epsilon )\in C^{a, m}(\mathbb {T}_{r_{n}}^{2} \times U_{\delta })(m=0,1),$
$$ \begin{align*} B_{n}(t, \epsilon)=\left( \begin{array}{@{}ccc@{\kern1pt}} 0 & \Xi^{n}(t, \epsilon) & 0\\ - \Xi^{n}(t, \epsilon) & 0 & 0\\ 0 & 0 & 0 \end{array} \right)\!, \end{align*} $$
and
$$ \begin{align} \| B_{n}- B_{n-1}\|_{r_{n},U_{\delta}}\leq 2 \mathcal{E}_{n-1},\quad \|P_{n}\|_{r_{n}, U_{\delta}}\leq \mathcal{E}_{n},\quad \|[P_{n}]\|_{r_{n}, U_{\delta}}\leq \mathcal{E}_{n}. \end{align} $$
Noting that
$P_{0}(t,0)= P(t,0)\equiv 0,$
we have
$P_{n}(t,0)\equiv 0$
and
$B_{n}(t,0)\equiv 0.$
In the differentiable case,
$$ \begin{align} \|\partial_{\epsilon}P_{n}\|_{r_{n},\quad U_{\delta}}\leq M_{n},\quad \|\partial_{\epsilon}[P_{n}]\|_{r_{n},\quad U_{\delta}}\leq M_{n}. \end{align} $$
Define the non-resonant set of the nth step
$$ \begin{align*} \Pi_{n}= \{\epsilon\in [0, \delta] :| \langle k, \omega\rangle \pm(\unicode{x3bb}+[\Xi^{n}])|\geq \frac{ \gamma_{n}}{(|k|+1)^{3\tau}},\ |k| < K_{n} \}, \end{align*} $$
which satisfies the properties
$$ \begin{align} \Pi_{n}\subset \Pi_{n-1} \subset \cdots \subset \Pi_{0}\quad \text{for all } n\in \mathbb{Z}_{+}. \end{align} $$
So, for any
$\epsilon \in \Pi _{n}, $
the above extended transformation is a true one and the new quasi-periodic system comes exactly from the original one under the orthogonal transformation.
5.3.3 Convergence of the KAM scheme
In this section, we will prove the convergence of infinite iterations, the properties (57) of the non-resonant sets
$ \Pi _{n}$
and
$\Pi _{\infty } \subset \bigcap _{n\geq 1 } \Pi _{n},$
where the definition of
$\Pi _{\infty }$
is given in conclusion (3) of Theorem 6.
First, consider the convergence of
$\Phi ^{n}.$
By Lemma 15, we observe that
$$ \begin{align*} \|\Phi_{n}-\mathrm{id}\|_{r_{n+1}, U_{\delta}}\leq \frac{\mathcal{E}_{n}^{14/15}}{4} \end{align*} $$
and
$$ \begin{align*} \|(D\Phi_{n}-\mathrm{Id})\|_{r_{n+1}, U_{\delta}} \leq\frac{\mathcal{E}_{n}^{{14}/{15}} }{16}, \end{align*} $$
where D denotes the Jacobian with respect to
$x.$
By induction, we have
$D\Phi ^{n}=D\Phi _{0} \cdots D\Phi _{n-1},$
with the Jacobians evaluated at different points, and
$$ \begin{align*} \|D\Phi^{n}\|=\|D\Phi_{0}\cdots D\Phi_{n-1}\|\leq \prod_{k\geq 0}\bigg(1+ \frac{\mathcal{E}_{k}^{{14}/{15}} }{16}\bigg) < \infty, \end{align*} $$
which is uniformly bounded on
$D_{n}\times U_{\delta }.$
Then,
$$ \begin{align*} \|\Phi^{n+1}-\Phi^{n}\|_{r_{n+1},U_{\delta}}&= \|\Phi^{n}\circ \Phi_{n}-\Phi^{n}\|_{_{r_{n+1},U_{\delta}}}\\ &\leq \|D\Phi^{n}\|_{r_{n}, U_{\delta}}\|\Phi_{n}-\mathrm{id}\|_{r_{n+1},U_{\delta}}\\ &\leq \frac{\mathcal{E}_{n}^{{14}/{15}}}{2} \end{align*} $$
and
$$ \begin{align*} \|\Phi^{n+1}- \mathrm{id}\|_{r_{n+1}, U_{\delta}}\leq \sum_{k\geq 0} \frac{\mathcal{E}_{k}^{{14}/{15}}}{2} \leq \mathcal{E}_{0}^{{14}/{15}}. \end{align*} $$
In the differentiable case, we consider
$\partial _{\epsilon }\Phi ^{n}.$
By the definition, we have
$\Phi ^{n+1}= \Phi ^{n} \cdot e^{\tilde {W}_{n}}.$
Taking the derivative on both sides, we get
$\partial _{\epsilon }\Phi ^{n+1}-\partial _{\epsilon }\Phi ^{n}=\partial _{\epsilon } \Phi ^{n} (e^{\tilde {W}_{n}}-I) + \Phi ^{n}e^{\tilde {W}_{n}} \partial _{\epsilon }\tilde {W}_{n}. $
Thus, for any
$\epsilon \in [0,\delta ]$
, we have
$$ \begin{align*} &\|\partial_{\epsilon}\Phi^{n+1}-\partial_{\epsilon}\Phi^{n}\|_{r_{n+1}, U_{\delta}} \\ &\quad \leq c \mathcal{L}_{n}^{- {1}/{240}}C_{3}(\tau) \gamma_{n}^{-( \mathcal{A}\tau+2)} Q_{n_{0}+n+1}^{18\tau} ( C_{3}(\tau) \gamma_{n}^{-( \mathcal{A}\tau+2)} Q_{n_{0}+n+1}^{18\tau}\mathcal{E}_{n} + M_{n}). \end{align*} $$
Let
$ \Phi ^{*}= \lim \nolimits _{n\rightarrow \infty } \Phi ^{n}.$
Then,
$$ \begin{align*} \|\Phi^{*}- \mathrm{id}\|_{r_{*}, U_{\delta}} \leq \mathcal{E}_{0}^{{14}/{15}} \end{align*} $$
and
$$ \begin{align*} \|\partial_{\epsilon}\Phi^{*}\|_{r_{*}, U_{\delta}}\leq c \mathcal{L}_{0}^{- {1}/{240}}C_{3}(\tau) \gamma_{0}^{-( \mathcal{A}\tau+2)} Q_{n_{0}+1}^{18\tau} ( C_{3}(\tau) \gamma_{0}^{-( \mathcal{A}\tau+2)} Q_{n_{0}+1}^{18\tau}\mathcal{E}_{0} + M_{0} ). \end{align*} $$
Second, consider the convergence of
$B_{n}.$
Since
$B_{n}=\sum _{j=0}^{L_{n}-1} \mathcal {T}_{K}^{1} P_{j}(t,\epsilon )$
and equation (55), we have
$$ \begin{align} \|B_{n}\|\leq 2 \sum_{j=0}^{n-1} \mathcal{E}_{n}\leq 4 \mathcal{E}_{0}. \end{align} $$
Moreover, due to
$P_{n}(t,0)=0,$
thus,
$B_{n}(t,0)=0 \text { for all } n\geq 0.$
In the differentiable case, by equation (56),
$$ \begin{align} \|\partial_{\epsilon} B_{n}(t,\epsilon)\|\leq 2\sum_{j=0}^{n-1} M_{j}\leq 4 M_{0}. \end{align} $$
Let
$B_{*}= \lim \nolimits _{n\rightarrow \infty } B_{n}.$
Then, equations (58) and (59) yield that
$B_{*}(t,0)=0$
,
$\|B_{*}(t,\epsilon )\|\leq 4\mathcal {E}_{0} $
, and
$ \| \partial _{\epsilon }B_{*}(t,\epsilon )\| \leq 4M_{0},$
when
$m=1.$
Third, we will prove conclusion (3) of Theorem 6. According to
$\|[\Xi ^{n+1}_{i}]- [\Xi ^{n}_{i}]\|\leq 2 \mathcal {E}_{n},$
if
$\mathcal {E}_{n} \leq {\gamma }/{ K_{n}^{3\tau }2^{n+3}}$
, then
$\Pi _{n+1}\subset \Pi _{n}. $
In fact, since
$\bar {Q}_{n_{0}}\geq T_{1}\gamma ^{-\mathcal {A}}$
and equation (52), one can check
$\mathcal {L}_{1} \leq {1}/{2^{J(\tau )}}, $
which implies that
$\mathcal {L}_{n} \leq {1}/{2^{J(\tau )}} \text { for all } n\geq 1.$
In addition, by the definition of
$\mathcal {L}_{n},$
$$ \begin{align*} \mathcal{L}_{n}= e^{ -c_{0} \gamma_{n-1} r_{n} (({\bar{Q}_{n_{0}+n}}/{Q_{n_{0}+n}^{\mathcal{M}} }) + {\bar{Q}_{n_{0}+n}^{{1}/{\mathcal{M}^{1/4}}}} )} \leq \frac{1} {(c_{0} \gamma_{n-1} r_{n})^{9 \tau} (({\bar{Q}_{n_{0}+n}}/{Q_{n_{0}+n}^{\mathcal{M}} }) + {\bar{Q}_{n_{0}+n}^{{1}/{\mathcal{M}^{1/4}}}} )^{9\tau}}, \end{align*} $$
and by the definition of
$K_{n}$
,
$K_{n}= [({\gamma _{n}}/({4\cdot 10^{\tau }})) \bar {Q}_{n_{0}+n}^{ {3}/{\mathcal {A}}} ], $
if
$\bar {Q}_{n_{0}+n}\leq Q_{n_{0}+n}^{\mathcal {A}};$
$K_{n}= [({\gamma _{n}}/({4\cdot 10^{\tau }}))({\bar {Q}_{n_{0}+n}}/{ Q_{n_{0}+n }^{\tau }})]$
if
$\bar {Q}_{n_{0}+n} \geq Q_{n_{0}+n}^{\mathcal {A}}.$
In the following, we only consider the case of
$\bar {Q}_{n_{0}+n}\leq Q_{n_{0}+n}^{\mathcal {A}}.$
Altogether, the definitions of
$\mathcal {E}_{n},$
the above estimates,
${\mathcal {E}_{0}\leq \varepsilon _{0} ({\gamma }/{2})^{J}}$
and
$J(\tau )= [120\tau \mathcal {A}^{5}]$
, yield that
$$ \begin{align*} \mathcal{E}_{n}&= \mathcal{L}_{n}\mathcal{L}_{n-1}\cdots \mathcal{L}_{1} \mathcal{E}_{0} \leq \mathcal{L}_{n} \frac{1}{ 2^{(n-1) J }} \mathcal{E}_{0} \leq \mathcal{L}_{n} \frac{\gamma^{J}}{ 2^{n J }} \varepsilon_{0}\\ &\leq \frac{1} {(c_{0} \gamma_{n-1} r_{n})^{9\tau} (({\bar{Q}_{n_{0}+n}}/{Q_{n_{0}+n}^{\mathcal{M}} }) + {\bar{Q}_{n_{0}+n}^{{1}/{\mathcal{M}^{1/4}}}})^{9\tau}} \frac{\gamma^{J}}{ 2^{n J }} \varepsilon_{0} \\ &\leq \frac{1}{K_{n}^{3\tau}} \frac{\gamma}{ 2^{n+3}} \end{align*} $$
if
$\varepsilon _{0} \leq (c_{0}r_{*})^{9\tau }. $
Let
$\Pi _{*}= \bigcap _{n\geq 1 } \Pi _{n}.$
Because
$\lim \nolimits _{n\rightarrow \infty } P_{n}=0, $
then for any
$\epsilon \in \Pi _{*},$
the transformation
$x= \Phi ^{*} y $
reduces system (4) into
$$ \begin{align*} \frac{d y}{dt} = (A + B_{*}(t,\epsilon)) y, \end{align*} $$
where
$A= \unicode{x3bb} J_{1},$
$B_{*}(t,\epsilon )$
has the form
$$ \begin{align} B_{*}(t,\epsilon) = \left( \begin{array}{@{}ccc@{\kern1pt}} 0 & \Xi^{*}(t,\epsilon) & 0 \\ -\Xi^{*}(t,\epsilon) & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\!. \end{align} $$
At last, we prove
$\Pi _{\infty } \subset \Pi _{*},$
which implies conclusion (3) in Theorem 6. To prove
$\Pi _{\infty } \subset \Pi _{*},$
it is enough to verify that
$\Pi _{\infty } \subset \Pi _{n} \text { for all } n\geq 1.$
Suppose
$\epsilon \in \Pi _{\infty },$
then for
$|k|< K_{n},$
we have
$$ \begin{align*} |\langle k, \omega\rangle \pm (\unicode{x3bb}+[\Xi^{n}]) |&\geq \frac{\gamma}{|k|^{3\tau}}- |[\Xi^{n}]- [\Xi^{*}]|\\ &\geq \frac{\gamma}{|k|^{3\tau}}- 4\mathcal{E}_{n} \geq \frac{\gamma_{n}}{|k|^{3\tau}}, \end{align*} $$
it follows that
$\epsilon \in \Pi _{n}.$
Thus, the proof of Theorem 6 is completed.
Acknowledgements
We would like to thank the referees for their careful reading and valuable suggestions, which have helped improve the presentation of this paper. The work was supported by the National Natural Science Foundation of China (Grant Nos. 11001048, 11871146, 12371161) and the Natural Science Foundation of Jiangsu Province, China (No. BK20201262).
A Appendix. The properties of
$so(3, \mathbb {R} ).$
In this section, we give some basic properties of
$so(3, \mathbb {R} ).$
For detailed proofs, we refer to [Reference Zhang, Xu and Xu40]. First, denote
$$ \begin{align} J_{1}=\left( \begin{array}{@{}ccc@{\kern1pt}} 0 & 1 & 0\\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)\!,\quad J_{2}=\left( \begin{array}{@{\kern1pt}ccc@{}} 0 & 0 & 1\\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{array} \right)\!,\quad J_{3}=\left( \begin{array}{@{}ccc@{\kern1pt}} 0 & 0 & 0\\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{array} \right)\!. \end{align} $$
Then,
$(J_{1}, J_{2}, J_{3})$
form a basis of
$so(3, \mathbb {R}),$
that is,
$\text { for all } A\in so(3, \mathbb {R}), $
there exist
$a_{1}, a_{2} $
and
$a_{3} \in \mathbb {R}$
such that
$$ \begin{align*} A= a_{1} J_{1}+ a_{2} J_{2}+ a_{3} J_{3}. \end{align*} $$
Lemma A.1. Let
$A= a_{1} J_{1}+ a_{2} J_{2}+ a_{3} J_{3}.$
Then, the matrix A has three eigenvalues
$0, \pm ia, $
where
$a= \sqrt {a_{1}^{2}+ a_{2}^2 +a_{3}^{2}}.$
If
$x_{3}\in \mathbb {R}^{3}$
is a unit eigenvector corresponding to eigenvalue
$zero, $
then there exist orthogonal unit vectors
$x_{1}, x_{2}\in \mathbb {R}^{3}, $
such that
${W=(x_{1}, x_{2}, x_{3}) }$
is an orthogonal matrix and
$$ \begin{align*} W^{-1} A W= aJ_{1}. \end{align*} $$
Lemma A.2. Let
$W=(x_{1}, x_{2}, x_{3})$
be an orthogonal matrix and satisfy that
$|W|=1, $
where
$ x_{i}\in \mathbb {R}^{3}, i=1,2,3.$
If
$\tilde {x}_{3} =x_{3}+ \hat {x}_{3} $
is a unit vector, where
$ \hat {x}_{3}$
is a sufficiently small vector, there exist two orthogonal unit vectors
$\tilde {x}_{1}, \tilde {x}_{2}\in \mathbb {R}^{3}, $
such that
$\tilde {W}= (\tilde {x}_{1}, \tilde {x}_{2}, \tilde {x}_{3} ) $
is an orthogonal matrix,
$|\tilde {W}|=1,$
and
$$ \begin{align*} \|\tilde{W}-W \|\leq c |\hat{x}_{3}|, \end{align*} $$
where c is a constant, which is independent of the vector
$ \hat {x}_{3}. $
Proof. Let
$ \bar {x}_{1}= x_{1} + \delta $
be perpendicular to the vector
$\tilde {x}_{3},$
where
$\delta $
is determined later. Then, by
$$ \begin{align*} \begin{cases} \bar{x}_{1}=x_{1}+ \delta,\\ \tilde{x}_{3}=x_{3}+\hat{x}_{3},\\ \bar{x}_{1}\cdot \tilde{x}_{3} =0, \end{cases} \end{align*} $$
we have
$$ \begin{align*} x_{1}\cdot x_{3} + x_{1}\cdot \hat{x}_{3}+ \delta \cdot \tilde{x}_{3}=0. \end{align*} $$
Additionally, because
$W=(x_{1}, x_{2}, x_{3})$
is an orthogonal matrix,
$ x_{1}\cdot x_{3}=0,$
it follows that
$$ \begin{align*} \delta \cdot \tilde{x}_{3}=-x_{1}\cdot \hat{x}_{3}. \end{align*} $$
Let
$\delta = a \tilde {x}_{3}. $
Substituting it into the above formula gives
$a= -x_{1}\cdot \hat {x}_{3},$
and
$$ \begin{align} \delta= (-x_{1}\cdot \hat{x}_{3})\tilde{x}_{3}. \end{align} $$
After unitizing
$\bar {x}_{1}, $
we get a unit vector
$\tilde {x}_{1}= { \bar {x}_{1}}/{ |\bar {x}_{1}|}. $
Then,
$$ \begin{align} \hat{x}_{1}= \tilde{x}_{1}- x_{1}= \frac{\delta}{|\bar{x}_{1}| } + x_{1} \frac{1- |\bar{x}_{1}|}{|\bar{x}_{1}|}. \end{align} $$
When
$\hat {x}_{3}$
is sufficiently small, by equation (A.2), we have
$$ \begin{align*} |\delta|\leq |x_{1}|\cdot | \hat{x}_{3}|\cdot| \tilde{x}_{3}|=|\hat{x}_{3}|, \end{align*} $$
without loss of generality, suppose
$|\delta | \leq 1/2, $
then
$$ \begin{align} |\bar{x}_{1}|= |x_{1}+\delta| \geq |x_{1} | -|\delta| \geq \tfrac{1}{2}. \end{align} $$
Then, because
$$ \begin{align} |\bar{x}_{1}|^{2}= |x_{1}+\delta|^{2}= 1+ 2\langle \delta, x_{1}\rangle + |\delta|^{2}, \end{align} $$
if
$\hat {x}_{3} $
is sufficiently small, by equation (A.5), we obtain
$$ \begin{align*} |\bar{x}_{1}|= 1+|\delta| + O(|\delta|^{2}), \end{align*} $$
so
$$ \begin{align*} |1- |\bar{x}_{1}| |\leq c |\hat{x}_{3}|. \end{align*} $$
Altogether, by equations (A.3), (A.4), and (A.5), we have
$$ \begin{align*} |\hat{x}_{1}|\leq 2 |\delta| + 2c |\hat{x}_{3} |\leq c |\hat{x}_{3} |. \end{align*} $$
At last, according to the right-hand rule of vector product and
$|\tilde {W}|= |( \tilde {x}_{1}, \tilde {x}_{2}, \tilde {x}_{3})| =1, $
we have
$$ \begin{align*} \tilde{x}_{2}&= \tilde{x}_{3} \times \tilde{x}_{1}\\ &= x_{3} \times x_{1} + x_{3} \times \hat{x}_{1} + \hat{x}_{3}\times \tilde{x}_{1}. \end{align*} $$
Due to
$|(x_{1}, x_{2}, x_{3})|=1, $
it follows that
$x_{3}\times x_{1}= x_{2}, $
and
$$ \begin{align*} \tilde{x}_{2}= x_{2}+ \hat{x}_{3}\times \tilde{x}_{1} +x_{3}\times \hat{x}_{1}. \end{align*} $$
Therefore,
$$ \begin{align*} \hat{x}_{2}= \tilde{x}_{2}-x_{2}= \hat{x}_{3}\times \tilde{x}_{1} +x_{3}\times \hat{x}_{1} \end{align*} $$
and
$$ \begin{align*} |\hat{x}_{2}| \leq |\hat{x}_{3}|\cdot |\tilde{x}_{1}| + |x_{3}|\cdot |\hat{x}_{1}|\leq (1+c) |\hat{x}_{3}|. \end{align*} $$
Furthermore,
$$ \begin{align*} \|\tilde{W}- W \|= \|(\hat{x}_{1}, \hat{x}_{2}, \hat{x}_{3})\|\leq c |\hat{x}_{3}|, \end{align*} $$
where c is a constant, which is independent of
$\hat {x}_{3}$
.
Lemma A.3. Consider the differential equation
$\dot {x}= A(t) x, x \in \mathbb {R}^{3},$
where A is a skew symmetric matrix. Let
$ x=e^{W(t)}y,$
where
$W(t)$
is a skew symmetric matrix, which transforms the above differential equation into
$ \dot {y}= B(t) y, $
then
$B(t)$
is still a skew symmetric matrix.
Proof. By the transformation
$ x=e^{W(t)}y,$
the differential equation
$\dot {x}= A(t) x$
is transformed into
$$ \begin{align} \dot{y} = \bigg(e^{-W} A e^{ W}- e^{-W} \frac{d}{d t} e^{W} \bigg) y. \end{align} $$
Since W is a skew symmetric matrix,
$$ \begin{align*} e^{W}\cdot ( e^{W})^{T}= e^{W} \cdot e^{W^{T}}= e^{W+ W^{T}} = e^{W- W} =I, \end{align*} $$
$ e^{W} $
is an orthogonal matrix, which yields that
$e^{ -W} A e^{W}$
is a skew symmetric matrix. Differentiating the equation
$ (e^{W })^{T} e^{W}=I $
with respect to
$t, $
we have
$$ \begin{align*} \bigg(\frac{d}{d t} e^{W}\bigg)^{T} e^{W}+ (e^{W})^{T} \bigg(\frac{d}{d t} e^{W}\bigg)= 0. \end{align*} $$
Therefore,
$$ \begin{align*} \bigg(e^{- W} \frac{d}{d t} e^{W}\bigg)^{T} &= \bigg(\frac{d}{d t} e^{W}\bigg)^{T} (e^{ -W})^{T} = \bigg(\frac{d}{d t} e^{W}\bigg)^{T} e^{ W}\\ & = - (e^{W})^{T} \frac{d}{d t} e^{W}= - e^{-W} \frac{d}{d t} e^{W}, \end{align*} $$
which implies that
$e^{-W} ({d}/{d t}) e^{W} $
is a skew symmetric matrix. Altogether, the system (A.6) is still a skew symmetric system.
