Reducibility for a class of almost periodic Hamiltonian systems which are degenerate

1.   Introduction and main results

Definition 1.1. If $f(t) = L(\omega_1t, \omega_2t, \cdots, \omega_rt)$ with $\theta_j = \omega_j t, j = 1, 2\cdots, r$, and $L(\theta_1, \theta_2, \cdots, \theta_r)$ is $2\pi$ periodic with respect to all $\theta_j$, we say a function $f$ is quasi-periodic with frequencies $\omega = (\omega_1, \omega_2, \cdots, \omega_r)$. Further, if $L(\theta)\; (\theta = (\theta_1, \theta_2, \cdots, \theta_r))$ is analytic on $D_\rho = \{\theta\in C^r| \ |Im\theta_j|\le\rho, j = 1, 2, \cdots, r\}$, we say $f(t)$ is analytic quasi-periodic on $D_\rho$. The norm of $f$ on $D_\rho$ is defined as $||f||_{\rho} = \sup\limits_{\theta\in D_\rho}|L(\theta)|.$

Definition 1.2. If $p_{ij}(t)(i, j = 1, 2\cdots n)$ are all analytic quasi-periodic on $D_\rho$, we say a matrix function $P(t) = (p_{ij}(t))_{1\le i, j \le n}$ is analytic quasi-periodic on $D_\rho$.

Define the norm of the matrix $P$ by $\parallel P\parallel_{\rho} = \max\limits_{1\le i\le n}\sum\limits_{j = 1}^n\parallel p_{ij} \parallel_{\rho}$. Obviously, $\parallel P_1P_2\parallel_{\rho}\le\parallel P_1\parallel_{\rho}\parallel P_2\parallel_{\rho}$. For simplicity, if the matrix $P$ is constant, denote $\parallel P\parallel = \parallel P\parallel_{\rho}$.

For almost periodic Hamiltonian systems, we use notations and definitions of finite spatial structure ^[1].

Definition 1.3. Assume $\tau$ is a family of subsets of $N$ and $N$ is a natural number set. If $\tau$ fulfills (i) $\cup_{\Lambda\in \tau}\Lambda = N;$ (ii) if $\Lambda_1, \Lambda_2\in \tau,$ then $\Lambda_1\cup\Lambda_2\in \tau$; (iii) $\phi\in \tau$, where $\phi$ is an empty set, we say ($\tau$, [$\cdot$]) is a finite spatial structure. Moreover, [$\cdot$] is called a weight function on $\tau$ if [$\phi$] = 0 and [$\Lambda_1\cup\Lambda_2$]$\le$[$\Lambda_1$]+[$\Lambda_2$].

Definition 1.4. Assume $k = (k_1, k_2, \cdots)\in Z^\infty$. Define the support of $k$ by supp$k = \{i\mid k_i\ne 0\}$. Denote $|k| = \sum\limits_{i = 1}^\infty|k_i|.$ The weight of its support is defined as $[k] = \inf_{suppk\subseteq\Lambda\in \tau}[\Lambda].$

Definition 1.5. If $P(t) = \sum\limits_{\Lambda\in \tau}P_\Lambda(t),$ where $P_\Lambda(t)$ is a quasi-periodic matrix with frequencies $\omega_\Lambda = \{\omega_i|i\in \Lambda\},$ we say $P(t)$ is an almost periodic matrix with weighted spatial structure ($\tau$, [$\cdot$]). In the context of integer modulus, frequencies $\omega$ of $Q(t)$ is the the biggest subset of $\bigcup\omega_\Lambda$.

Definition 1.6. Denote $P(t) = \sum\limits_{\Lambda\in \tau}P_\Lambda(t).$ When $m > 0,$ $\rho > 0,$ $|||P|||_{m, \rho} = \sum\limits_{\Lambda\in \tau}e^{m[\Lambda]}||P_\Lambda(t)||_\rho$ (see ^[1]) is defined as a weighted norm of $P(t)$. Clearly, for $m > 0,$ $\rho > 0,$ $||P(t)||_\rho\le |||P(t)|||_{0, \rho}\le |||P(t)|||_{m, \rho}.$

If the quasi-periodic equation

by a non-sigular mapping $x = \psi(t)y$, where $\psi(t)^{-1}$ and $\psi(t)$ are bounded and quasi-periodic, (1.1) can become

with the matrix $C$ is constant, we call (1.1) is reducible. When the matrix $B(t)$ is periodic, the famous Floquent theorem tells us by a (double-)periodic transformation, $\dot{x} = B(t)x$ is reducible. However, for the quasi-periodic situation it is not true. Under some "full spectrum" conditions, the authors ^[2] obtained the quasi-periodic system (1.1) is reducible. For linear systems, the authors in ^[3] studied the quasi-periodic system

where $A$ is an $n\times n$ constant matrix with different eigenvalues $\lambda_1, \lambda_2, \cdots, \lambda_{n}.$ If non-degeneracy conditions

and non-resonance conditions $\mid\langle k, \omega\rangle\sqrt{-1}+ \lambda_i-\lambda_j\mid\ge\frac{\alpha_0}{\mid k\mid^{\tau}}$ are satisfied, where $\forall k\in Z^r\setminus\{0\}, \; \forall i, j = 1, 2, \cdots, n,$ $\alpha_0 > 0$ is a small constant, $\tau > r-1$, $\bar \lambda_i(\varepsilon)(i = 1, 2, \cdots, n)$ are eigenvalues of $A+\varepsilon\bar Q$ and $\bar Q$ is the average of $Q(t)$, for $\varepsilon\in E$ with the nonempty Cantor subset $E$, (1.2) is reducible.

In ^[4], $\bar \lambda_i(\varepsilon)-\bar \lambda_j(\varepsilon)$ are called degenerate if non-degeneracy conditions (1.3) do not hold. The authors ^[4] considered this degenerate case. They proved a similar result under weaker non-degeneracy conditions.

Previously, the reducibility for analytic quasi-periodic systems were mainly considered. The finitely smooth case was considered in ^[5].

In KAM theorems, non-degeneracy conditions are always necessary. But when the hamiltonian system is two degrees of freedom, a special result ^[6] is obtained. Without any non-degeneracy condition, the authors ^[7] obtained the reducible result for the linear two-dimensional quasi-periodic system depending on a small parameter analytically. For the case that depends on the small parameter smoothly, there is a similar result ^[8]. Without any non-degeneracy condition, the authors ^[9] obtained the reducible result for the nonlinear two-dimensional quasi-periodic system. Recently, for the two dimensional almost periodic system, we also obtain a similar result in ^[10].

For nonlinear quasi-periodic systems, the authors ^[11] studied the following system

where the matrix $A$ is constant and $h$ is $O(x^2).$ If non-degeneracy conditions and non-resonance conditions are satisfies, using an analogous way as ^[3], the system (1.4) is reducible. When the system (1.4) becomes the hamiltonian system with multiple eigenvalues, we obtain an analogous result in ^[12].

In ^[13], under non-resonance and non-degeneracy conditions, Xu further considered the reducibility for the almost periodic system.

Motivated by ^[1,4,13], here we consider the reducibility for the higher dimensional Hamiltonian almost periodic system under weaker non-degeneracy conditions, which is called degenerate in ^[4].

Here non-resonance conditions are presented by so called approximation function. If $\Delta:\; [1, +\infty)\rightarrow [1, +\infty)$, $\Delta(1) = 1,$

and

we say an increasing function $\Delta(t)$ is an approximation function ^[1]. obviously, when $\Delta(t)$ is an approximation function, so is $\Delta^4(t)$.

Let

There exists a sequence $\bar \varrho_1\ge \bar\varrho_2\ge \cdots\ge 0$, such that $\sum\limits_{v = 0}^{\infty}\bar \varrho_v = \varrho$ and $\psi(\varrho) = \frac12\prod\limits_{v = 0}^{\infty}(\Gamma(\bar\varrho_v))^{(\frac32)^{-(v+1)}}.$ For the details, see ^[1].

Suppose $\omega = (\omega_1, \omega_2, \cdots)$ is frequencies of $Q(t)$, $\lambda_1, \lambda_2, \cdots, \lambda_{2n}$ are the different eigenvalues of $A$, $\Delta(t)$ is an approximation function that fulfills

For Theorem 1.1 of this paper, non-resonance conditions are

Since ^[1], when we choose $\Delta(t)$ which satisfies (1.5) and $[\Lambda] = 1+\sum\limits_{i\in \Lambda}\log^r(1+|i|)\; (r > 2)$, there exists $\omega = (\omega_1, \omega_2, \cdots)$ ^[1] which fulfills non-resonance conditions (1.6). The following theorem is the main result of this paper.

Theorem 1.1. Consider the linear Hamiltonian system

where $A = \mathit{\mbox{diag}}(\lambda_1, \lambda_2, \cdots, \lambda_{2n})$ is a $2n\times2n$ constant Hamiltonian matrix with $\lambda_i\ne \lambda_j,$ $i\ne j,$ $1\le i, j\le 2n,$ and $\lambda_{p+n} = -\lambda_p,$ $p = 1, 2, \cdots, n.$ Suppose a small parameter $\varepsilon\in(0, \varepsilon_0)$, $Q(t, \varepsilon) = \sum\limits_{\Lambda\in \tau}Q_\Lambda(t, \varepsilon)$ is Hamiltonian analytic almost periodic in t with frequencies $\omega = (\omega_1, \omega_2, \cdots)$ on $D_\rho$ and analytic in $\varepsilon$.

Assume

$(A_1)$ (non-resonance conditions) The frequencies $\omega = (\omega_1, \omega_2, \cdots)$ satisfies

for $\forall k\in Z^\infty\backslash\{0\},$ $1\le i, j\le 2n$, where $\alpha > 0$ is a small constant.

$(A_2)$ Let $\bar q_{ii}$ be the average of $q_{ii}(t)$ and $\bar R_0 = \mathit{\mbox{diag}}(\bar q_{11}, \bar q_{22}, \cdots, \bar q_{2n, 2n})$. Assume when $j\ne i$, $\varepsilon(\bar q_{jj}-\bar q_{ii})$ satisfies one of the following forms:

where $\mu_i\ne 0,$ $i = 1, 2, \cdots, p,$ $1\le l_1 < l_2 < \cdots < l_p,$ and $o(\varepsilon^{l})$ is of order smaller than $\varepsilon^l$ as $\varepsilon\rightarrow0.$

$(A_3)$ There exists $m > 0$ satisfying $|||Q(t, \varepsilon)|||_{m, \rho} < +\infty.$ Then for $\varepsilon\in \tilde E,$ there exists an analytic symplectic almost periodic mapping $x = \phi(t, \varepsilon)y$, where $\phi(t, \varepsilon)$ and $Q(t, \varepsilon)$ have the same spatial structure and frequencies, such that (1.7) becomes the Hamiltonian system

where $\tilde E\subset(0, \varepsilon_0)$ is a non-empty Cantor subset of positive Lebesgue measure satisfying $\mathit{\mbox{meas}}((0, \varepsilon_0)\setminus \tilde E) = o(\varepsilon_0)$ when $\varepsilon_0\rightarrow 0,$ and a constant matrix $A_\infty$ is Hamiltonian.

Remark 1: We understand the smoothness of the function in $\varepsilon$ for Cantor set $\tilde E$ in the sense of Whitney ^[14].

Remark 2: Generally, $Q$ depends on $\varepsilon$. Sometimes this dependence is not shown explicitly for simplicity.

Remark 3: If $\alpha$ is small enough and $\forall \lambda = (\lambda_1, \lambda_2, \cdots, \lambda_{2n})$ is given, by ^[1], there exists $\omega\in R^\infty$ satisfying (1.8).

Remark 4: Now the Hamiltonian system is

where

Since it is the Hamiltonian system, there exists a symmetric matrix $S(t, \varepsilon)$ such that $JS(t, \varepsilon) = A+\varepsilon Q(t, \varepsilon).$

Remark 5: In ^[4], the degenerate case is also the condition $(A_2).$ However, it is the quasi-periodic case for ^[4] and it is the almost periodic case for this paper.

2.   The lemmas

To prove Theorem 1.1, in this section we formulate some lemmas which will be used in the next section. Below $c > 0$ indacate a constant.

Lemma 2.1. Suppose $D(t)$ and $G(t)$ are almost periodic matrices with the same spatial structure and frequencies. If $|||D(t)|||_{m, \rho}, |||G(t)|||_{m, \rho} < +\infty,$ then $DG$ is also an almost periodic matrix with the same spatial structure and frequencies as $D$ and $G$. Furthermore, $||| DG|||_{m, \rho}\le||| D|||_{m, \rho}||| G|||_{m, \rho}$.

The proof can be seen in ^[13].

To solve the transformation equation, we give the following lemma.

Lemma 2.2. Consider the equation

where $A = \mathit{\mbox{diag}}(\lambda_1, \lambda_2, \cdots, \lambda_{2n})$, $|\lambda_l-\lambda_m|\ge \mu$ with a constant $\mu > 0,$ $l\ne m$, $1\le l, m\le 2n,$ and $\lambda_{i+n} = -\lambda_i,$ $1\le i\le n.$ Suppose $Q(t) = (q_{ij}(t))_{1\le i, j\le 2n} = \sum\limits_{\Lambda\in \tau} Q_\Lambda(t)$ is analytic Hamiltonian almost periodic in t with frequencies $\omega = (\omega_1, \omega_2, \cdots)$ on $D_\rho$ with finite spatial structure ($\tau$, [$\cdot$]). Suppose $\bar q_{ii} = 0,$ $i = 1, 2, \cdots, 2n,$ where $\bar q_{ii}$ is the average of $q_{ii}(t)$ in $t$. Assume

for $\forall k\in Z^\infty\backslash\{0\},$ $1\le i, j\le 2n$. Then there exists a unique analytic Hamiltonian almost periodic solution $P(t)$ with the same frequencies and spatial structure as $Q(t)$, and $||| P|||_{m-\bar m, \rho-\bar \rho}\le\frac{c\Gamma(\bar m)\Gamma(\bar \rho)}{\alpha }||| Q|||_{m, \rho}$, $|||\varepsilon\frac{\partial P}{\partial\varepsilon}|||_{m-\bar m, \rho-\bar \rho}\le\frac{c\Gamma^2(\frac{\bar m}{2})\Gamma^2(\frac{\bar \rho}{2})}{\alpha^2 } (||| Q|||_{m, \rho}+ |||\varepsilon\frac{\partial Q}{\partial\varepsilon}|||_{m, \rho}),$ where $\Gamma(\varrho) = \sup\limits_{t\ge 0}(\Delta^3(t)e^{-\varrho t})$, $0 < \bar m < m$, $0 < \bar \rho < \rho$.

Proof. Now we need solve the equation

Let

Comparing the coefficients of (2.3), it follows $p^{ii}_{\Lambda0} = 0;$ or else,

Then we have

So

Denote $P = \sum\limits_{\Lambda\in \tau}P_\Lambda.$ Since (2.4), it follows

Let us estimate $\parallel\varepsilon\frac{\partial P}{\partial\varepsilon}\parallel_{m-\bar m, \rho-\bar\rho}.$ Moreover, $\frac{dp^{ii}_{\Lambda0}(\varepsilon)}{d\varepsilon} = 0,$ and

Then it follows

So

Then

Moreover, by $A$ and $Q$ are Hamiltonian, it follows $Q = JQ_J$ and $A = JA_J$, where $Q_J$ and $A_J$ are symmetric. Denote $P_J = J^{-1}P$. Below we prove $P_J$ is symmetric. (2.1) becomes

(2.5) becomes

By the solution of (2.5) is unique, it follows $(P_J) = (P_J)^T.$ So $P$ is Hamiltonian.

The following lemma is used for the estimate of the measure.

Lemma 2.3. Assume

where $\chi\in R$. Let $\tilde\alpha\le \frac\alpha2,$ $\sigma\ne 0,$ and

where $q\in Z^+$ and $\Delta(t)$ is an approximation function that fulfills (1.5). Suppose $g(\varepsilon)$ fulfills $|g'(\varepsilon)|\le c$ for $\varepsilon\in(0, \varepsilon_0)$, and $g(\varepsilon) \to 0$ when $\varepsilon \to 0$. If $\varepsilon_0$ is small enough, it follows

Proof. Denote $\varphi(\varepsilon) = \langle k, \omega\rangle-(\chi +\sigma\varepsilon^q+\varepsilon^q g(\varepsilon))$ and fix $k\ne 0$. Let

We first consider the case $\varepsilon^q\le \frac{\alpha}{4|\sigma| \Delta(|k|)\Delta([k])}.$ If $\varepsilon^q\le \frac{\alpha}{4|\sigma| \Delta(|k|)\Delta([k])},$ it follows $|\sigma\varepsilon^q+\varepsilon^q g(\varepsilon)|\le \frac{\alpha}{2\Delta(|k|)\Delta([k])}.$ So

Thus, we only consider the case $\varepsilon_0^q\ge \varepsilon^q\ge \frac{\alpha}{4|\sigma|\Delta(|k|)\Delta([k])}.$ So

For $\varepsilon_0$ sufficiently small, we get

By (2.6) and (2.7), we have

Then since (1.5), it follows

The following lemma is used for the convergance of KAM iteration.

Lemma 2.4. (^[11]) A sequence $\{\eta_v\}$ satisfies

where $\eta_v$ are all positive real numbers, $1 < z < 2$ and $\bar{\gamma} > 0$. It follows that

This Lemma is used for the convergence of KAM iteration.

3.   Proof of Theorem 1.1

KAM-step. At $v$-th step, consider the Hamiltonian system

where $A_0 = A,$ $Q_0 = Q,$ $A_v = \mbox{diag}(\lambda_1^v, \lambda_2^v, \cdots, \lambda_{2n}^v)$, $|\lambda_i^v-\lambda_j^v|\ge \mu > 0,$ $i\ne j$, $1\le i, j\le 2n,$ $\lambda_{d+n} = -\lambda_d,$ $1\le d\le n,$ and $Q_v$ is almost periodic on $D_{\rho_v}.$

Let $Q_v = (q_{ij}^v)_{1\le i, j\le 2n}$ $R_v = \mbox{diag}(q_{11}^v, q_{22}^v, \cdots, q_{2n, 2n}^v),$ $R_0 = \mbox{diag}(q_{11}, q_{22}, \cdots, q_{2n, 2n}).$ Denote the average of $R_v$ by $\bar R_v = \mbox{diag}(\bar q_{11}^v, \bar q_{22}^v, \cdots, \bar q_{2n, 2n}^v).$ Hamiltonian system (3.1) becomes

where $A_{v+1} = A_v+\varepsilon^{2^{v}}\bar R_v = \mbox{diag}(\lambda_1^{v+1}, \lambda_2^{v+1}, \cdots, \lambda_{2n}^{v+1})$ and $\widetilde Q_v = Q_v-\bar R_v.$

We now make the symplectic mapping $x_v = e^{\varepsilon^{2^{v}}P_v(t)}x_{v+1}$ to (2.2) to obtain

Expand $e^{\varepsilon^{2^{v}}P_v}$ and $e^{-\varepsilon^{2^{v}}P_v}$ into $e^{\varepsilon^{2^{v}}P_v} = I+\varepsilon^{2^{v}}P_v+B_v$ and $e^{-\varepsilon^{2^{v}}P_v} = I-\varepsilon^{2^{v}}P_v+\widetilde{B}_v$, where $B_v = \frac{(\varepsilon^{2^{v}}P_v)^2}{2!}+\frac{(\varepsilon^{2^{v}}P_v)^3}{3!}+\cdots$ and $\widetilde{B}_v = \frac{(\varepsilon^{2^{v}}P_v)^2}{2!}-\frac{(\varepsilon^{2^{v}}P_v)^3}{3!}+\cdots.$ (3.3) becomes

where

We want to have that

that is,

Clearly, $A_{v+1}$ and $\widetilde{Q}_v$ are Hamiltonian. Since Lemma 2.2, if

for $\forall k\in Z^\infty\backslash\{0\}$, where $1\le i, j\le 2n,$ $\alpha_{v+1} = \frac{\alpha}{(v+1)^2},$ for the Eq (3.5), we find an unique analytic almost periodic Hamiltonian solution $P_v(t)$ with frequencies $\omega$, and

where $\Gamma(\rho) = \sup\limits_{t\ge 0}(\Delta^3(t)e^{-\rho t})$, $0 < \bar m_v < m_v$, $0 < \bar \rho_v < \rho_v$.

Now (3.4) is changed to the Hamiltonian system

where $\varepsilon^{2^{v+1}}Q_{v+1} = Q_v^{(1)}.$

Since $\widetilde{Q}_v-\dot{P}_v = P_vA_{v+1}-A_{v+1}P_v,$ it follows

Under the symplectic mapping $x_v = e^{\varepsilon^{2^{v}}P_v}x_{v+1},$ Hamiltonian system (3.1) becomes Hamiltonian system (3.8).

KAM iteration. Let us prove the convergence of KAM iteration when $v\rightarrow \infty.$ At $v$-th step, define $\alpha_{v+1} = \frac{\alpha}{(v+1)^2}$, $\rho_{0} = \tilde\rho,$ $m_{0} = \tilde m,$ $\bar m_v\searrow0,$ $\bar \rho_v\searrow0,$ $\sum\limits_{v = 0}^{\infty}\bar m_v = \frac12\tilde m,$ $\sum\limits_{v = 0}^{\infty}\bar \rho_v = \frac12\tilde\rho,$ $m_{v+1} = m_{v}-\bar m_{v},$ $\rho_{v+1} = \rho_{v}-\bar \rho_{v}.$ Let $|||\cdot|||_v = |||\cdot|||_{m_v, \rho_v}.$ Since Lemma A.1 of ^[1], there exists a constant $H > 0,$ which satisfies

By (3.7) and (3.9), if $|||\varepsilon^{2^{v}} P_v|||_v\le\frac12,$ it follows

Denote $\eta_v = ||| Q_v|||_{v}+ |||\varepsilon\frac{\partial Q_v}{\partial\varepsilon}|||_{v}.$ Since (3.10) and (3.11), there exists $1 < z < 2$ satisfying $\eta_{v+1}\le (\bar{\gamma}z^v)^{\bar{\gamma}z^v}\eta_{v}^2.$ Since Lemma 2.4, it follows $\eta_v\le M^{2^{v}}$ with a constant $M > 0$. Thus,

If $0 < M\varepsilon < 1,$ by (3.12), it follows that

By (2.2), it follows that

When $0 < M\varepsilon < 1,$ $A_v$ is convergent as $v\rightarrow \infty$. Denote

By (3.7), (3.10) and (3.12), we have

Thus, there exists a symplectic mapping $x = \phi(t, \varepsilon)y$, such that Hamiltonian system (1.7) becomes Hamiltonian system (1.9).

Estimate of measure. Below let us prove if $\varepsilon_0$ is sufficiently small, for most $\varepsilon\in(0, \varepsilon_0)$, non-resonance conditions

hold, where $0\ne k\in Z^\infty$, $v = 0, 1, 2\cdots,$ and $i, j = 1, 2, \cdots, 2n$. Denote

If $i = j,$ since (1.8), (3.14) holds. So we merely need prove for most $\varepsilon\in(0, \varepsilon_0)$,

Without generality, since ($A_2$), we assume

There exists an integer $N\ge 0$ satisfy

So after $N+2$ KAM steps, by (3.12), we have

where $|f'(\varepsilon)|\le c$, $f(\varepsilon)\rightarrow0$ when $\varepsilon\rightarrow0$, and the integer $t\ge N+2.$ Moreover, for previous $N+1$ KAM steps, by (3.15), we have

where $s = 1, 2, \cdots, N+1,$ $0 < q\le l_1$ is an integer, and $\sigma\ne 0$ is a constant. Now substitute (3.18) ($s = N+1$) into (3.17), by (3.16), we have

where $|K'(\varepsilon)|\le c$, $K(\varepsilon)\rightarrow0$ as $\varepsilon\rightarrow0$. By (3.18), (3.19), ($A_1$), and Lemma 2.3, it follows

Denote $\tilde E = \cap_{v = 0}^\infty E_{v+1}.$ By (3.20), it follows that

So when $\varepsilon_0$ is sufficiently small, non-resonance conditions (3.14) hold for most $\varepsilon\in(0, \varepsilon_0)$.

We prove Theorem 1.1 completely.

Acknowledgments

The authors were supported by the Natural Science Foundations for Colleges and Universities in Jiangsu Province grant 18KJB110029.

Conflict of interest

The authors declare that they do not have any conflicts of interest regarding this paper.