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The almost-periodic solutions of the weakly coupled pendulum equations
Advances in Difference Equations volume 2018, Article number: 157 (2018)
Abstract
In this paper, it is proved that, for the networks of weakly coupled pendulum equations
there are many (positive Lebesgue measure) normally hyperbolic invariant tori which are infinite dimensional in both tangent and normal directions.
1 Introduction and main result
In the last several decades, models of (infinitely) many coupled oscillators have found diverse applications in various fields of science. Among a lot of examples are the collective dynamics of Josephson junctions [1, 2], lasers [3, 4], relativistic magnetrons [5], chemical reactions [6–9], circadian pacemakers [10, 11], intestinal electrical rhythms [12], a variety of biological processes [13–15], etc. The research of these systems has brought outstanding examples of different types of dynamical behavior that can be induced by the attendance of coupling. See [13, 14, 16–30] for more details. Meanwhile, the pendulum equation or analogous ones can be used to depict the synchronous electric motor models of a single machine infinite bus [31], Josephson junctions [32–34], super-conducting derive [35], shunted model of electrical rotator [36], and many other applications. Among those interesting models are the networks of weakly coupled pendulum equations
where \(\lambda_{n}, n\in\mathbb{Z}\) are constants and \(W_{n}\) are real analytic functions given in (1.6). While the existence of normally hyperbolic invariant tori which are infinite dimensional in both tangent and normal directions for (1.1) can be showed via KAM theory in this paper.
The classical KAM theory [37–39], founded by Kolmogorov, Arnold, and Moser in the last century, is a milestone of the evolution of Hamiltonian systems. It provided a new method for the research of Hamiltonian systems. The classical KAM theory established on a 2n-dimensional smoothly manifold affirms that most of the non-resonant tori of a non-degenerate integrable Hamiltonian system are not destroyed under perturbations, they only have a small deformation. The KAM theory has been developed into a very complete theory in the past nearly half a century.
In the 1980s, the distinguished KAM theory was triumphantly developed to infinitely dimensional Hamiltonian systems of short range so as to research a class of Hamiltonian networks of weakly coupled oscillators. To describe it more precisely, we consider from three aspects the infinitely dimensional Hamiltonian system
where P is of short range.
(i) Introducing the action-angle variables on \(u_{n},v_{n}\) for \(\forall n\in\mathbb{Z}\), then the Hamiltonian system (1.2) takes the form
where tangent frequencies \(\omega_{n}=\lambda_{n},n\in\mathbb{Z}\). Vittot and Bellissard [40], Fröhlich et al. [41] asserted that there is a full dimensional invariant torus with infinite dimension for (1.3). Pöschel [42] got also the above results for the Hamiltonian systems with a more general spatial structure for P. Thus, any solution starting from the torus is almost-periodic in time. For more results on the existence of almost-periodic solutions, see Bourgain [43] and Cong et al. [44].
(ii) Constructing the action-angle variables on \(u_{n},v_{n}\) for \(n\in\{ 1,2,\ldots,m\}\) and by abuse of notation to rearrange the subscript of \(u_{n},v_{n},\lambda_{n}\), we get
Then Hamiltonian (1.2) is of the form
Here tangent frequencies ω are m-dimensional and normal frequencies are infinite dimensional. Kuksin [45], Pöschel [46, 47], and Wayne [48] (in alphabetic order) concluded that, under the multiplicity of \(\lambda_{n}\) equals to 1 for \(\forall n\in\mathbb{Z}\) and some non-resonant conditions (Melnikov condition), most of elliptic type lower-dimensional invariant tori for (1.4) without being of short range will remain under small perturbations. While all of \(\lambda_{n}\) are the same, Yuan [49, 50] obtained similar KAM results for infinitely dimensional Hamiltonian system (1.4) of short range. On the other hand, the persistence problem of hyperbolic type lower-dimensional invariant tori with finite dimension was researched first by Moser [51]. Graff [52] then generalized Moser’s theory. Then, Zehnder in [53, 54] has brought a substitute proof of Graff’s conclusion by an implicit function technique. For more results on the evolutions in this direction, one can refer to [55–61]. By virtue of the KAM theory of this situation, we obtain that there are quasi-periodic solutions for the coupled pendulum equations.
(iii) For the case that both tangent frequencies ω and normal frequencies are infinite dimensional, according to our knowledge, not only of elliptic type but also of hyperbolic type, there has not been any KAM theorem to deal with this situation. However, the existence of almost-periodic solutions for the networks of weakly coupled pendulum equations (1.1) needs to be proved. That is the problem we are most concerned with in this paper.
To describe it more accurately, by virtue of the technique of action-angle variables on \(x_{2j+1}, j\in\mathbb{Z}\) and the transformation \((x_{2j}-\pi,\dot{x}_{2j})=(1/\sqrt{2\lambda _{2j}}(u_{j}-v_{j}), \sqrt{\lambda_{2j}/2}(u_{j}+v_{j})),j\in\mathbb{Z}\), we transform the Hamiltonian of Eq. (1.1) into the form
where P is of short range. Obviously, both tangent frequencies \(\omega=(\omega_{j})_{j\in\mathbb{Z}}\) and normal frequencies \(\Omega =(\lambda_{2j})_{j\in\mathbb{Z}}\) are infinite dimensional belonging to case (iii). Meanwhile, \((u_{j},v_{j})_{j\in\mathbb{Z}}=(0,0)_{j\in \mathbb{Z}}\) is a hyperbolic equilibrium point on the normal direction. From this, we then mainly study the persistence of normally hyperbolic invariant tori which are infinite dimensional in both tangent and normal directions for this Hamiltonian system in this issue.
Before starting our theorem, we define the norm
in \({\mathbb{C}}^{\mathbb{Z}}\) and give the following assumptions:
(A1) \(\lambda_{n}, n\in\mathbb{Z}\) satisfy
(A2) \(W_{n}\) satisfies
with \(\alpha>0\) (arbitrarily small), and \(W=O( \vert x \vert ^{3})\) is real analytic in the strip domain \(\{x\in\mathbb{C}: \vert \operatorname{Im} x \vert <\delta_{0}\}\) for some constant \(\delta_{0}>0\).
The expression \(\frac{1}{2}y^{2}+\lambda^{2}(1-\cos x)=h\) with \(\frac {\lambda^{2}}{2}\leq h\leq\lambda^{2}\) denotes a simple closed curve Γ which encloses \((0, 0)\) in the \((x, y)\)-plane. Let \(\rho=\rho(h)\) be the area enclosed by \(\Gamma(h)\), i.e.,
Then we can see that \(\rho'(h)>0,\rho''(h)\neq0\) for any \(h\in [\frac{\lambda^{2}}{2},\lambda^{2}]\).
Equation (1.1) can be regarded as a perturbation of the following system:
For any \(\eta=(h_{j})_{j\in\mathbb{Z}}, h_{j}\in[\frac{\lambda ^{2}}{2},\lambda^{2}]\), then, by the fact that \(\frac{1}{2}y^{2}+\lambda ^{2}(1-\cos x)=h_{j},j\in\mathbb{Z}\) is a first integral of (1.7a), \(\prod_{j\in\mathbb{Z}}\Gamma(h_{j})\) is an invariant torus with the frequencies \(\omega(\eta)=(H'_{0}(\rho(h_{j})))_{j\in \mathbb{Z}}\) for (1.7a), where \(H_{0}\) is the inverse of \(\rho =\rho(h)\). Observe that \((\pi,0)\) is an equilibrium of (1.7b). Thus,
is an invariant torus with the frequencies \(\omega(\eta)\) for (1.7a)–(1.7b). Therefore, any solution of (1.7a)–(1.7b) starting from \(\mathcal{T}(\eta)\) is a trivial breather for (1.7a)–(1.7b). Our goal is to show that the torus \(\mathcal{T}(\eta)\) remains under the small perturbation. Here is our main result which expresses that there does persist a large Cantor sub-family of rotational \(\mathbb{Z}\)-tori which are only slightly deformed, thus the solutions starting from the persisted tori are almost-periodic breathers of (1.1).
Theorem 1.1
Suppose that Eq. (1.1) satisfies assumptions (A1), (A2). Then, for the set \(\Omega=[\frac{\lambda^{2}}{2},\lambda ^{2}]^{\mathbb{Z}}\), there is a positive constant \(\epsilon^{*}\) sufficiently small such that, when \(0<\epsilon<\epsilon^{*}\), there are a set \(\mathcal{S}\subset\Omega\) with \(\operatorname{Prob}(\mathcal{S})\) arbitrarily close to one (depending on ϵ), a family of \(\mathbb{Z}\)-tori
over \(\mathcal{S}\), and an analytic embedding
which is a higher order perturbation of the inclusion map \(\Phi _{0}:\bigcup_{\eta\in\Omega}\mathcal{T}(\eta)\hookrightarrow \mathbb{R}^{\mathbb{Z}}\times\mathbb{T}^{\mathbb{Z}}\times\mathbb {R}^{\mathbb{Z}}\times\mathbb{R}^{\mathbb{Z}}\) restricted to \(\mathcal{T}[\mathcal{S}]\), such that the restriction Φ to each \(\mathcal{T}(\eta)\) in the family is an embedding of a rotational \(\mathbb{Z}\)-torus for (1.1). Moreover, any solution of (1.1) starting from \(\Phi(\mathcal{T}(\eta ))\) is an almost-periodic breather of frequencies \(\omega^{*}\) with \(\vert \omega^{*}-\omega \vert _{\infty}=O(\epsilon^{1/6})\).
This paper is organized as follows. In Sect. 2, Eq. (1.1) is, by the technique of action-angle variables, reduced to a normal form to which a KAM theorem is applicable. In Sect. 3, a KAM theorem and its iterative lemma are given, and the proof for the iterative lemma is finished. Theorem 1.1 and the KAM theorem are proven in Sect. 4.
2 Reduced to normal form
In this section, we will find a series of changes in variables to transform Eq. (1.1) into a normal form.
Let \(\dot{x}_{n}=y_{n}\). Then (1.1) is a Hamiltonian system with its Hamiltonian
We now carry out the standard reduction to action-angle variables. To construct the map \((x, y) \mapsto(\theta, \rho)\), where ρ and θ are action and angle variables, respectively, we let \(H_{0}(\rho)\) be the value of the function \(\frac{1}{2}y^{2}+\lambda^{2}(1-\cos x)\) on the closed curve which encloses area ρ in the \((x, y)\)-plane, i.e., we define \(H_{0}(\rho)\) implicitly by
We now define a generating function \(S(x, \rho)\) as follows:
where \(\Gamma^{*}\) is a part of the closed curve \(\frac{1}{2}y^{2}+\lambda ^{2}(1-\cos x)=H_{0}(\rho)\) connecting the y-axis with point \((x, y)\), oriented clockwise. We define the map \(\psi: (\theta, \rho) \mapsto (x, y)\) via
Then
Thus,
Let
Then
This implies that Ψ is symplectic. Thus, Hamiltonian H is transformed into
By assumption (A2), there exists the inverse \(H_{0}^{-1}\) of \(H_{0}\). Let \([\mu,\nu]=H_{0}^{-1}([\frac{\lambda^{2}}{2},\lambda^{2}])\). For any \(\xi=(\xi_{j})_{j\in\mathbb{Z}}\in[\mu,\nu]^{\mathbb {Z}}\), let \(\rho=I+\xi\), where \(I=(I_{j})_{j\in\mathbb{Z}}\). Expand \(H_{0}(\xi_{j}+I_{j})\) in \(\xi_{j}\) by Taylor’s formula:
Let \(\omega=(H'_{0}(\xi_{j}))_{j\in\mathbb{Z}}, \Pi=[\mu,\nu ]^{\mathbb{Z}}\), and from transformation (2.4), we can denote
Then \(\xi\in\Pi\) and (2.5) can be written as
where the constant \(\sum_{j\in\mathbb{Z}}H_{0}(\xi_{j})\) is omitted since it does not affect the dynamics.
Now we need to introduce the domain of the definition for Hamiltonian H. Set
here \(\rho^{0}_{j}=\frac{1}{4}\mu e^{- \vert j \vert ^{1+\alpha}}\), \(\varrho ^{0}_{j}=\frac{1}{4}\sqrt{\rho^{0}_{j}}\). \(f_{j}(I,\theta,u,v), j\in\mathbb{Z}\) are real analytic on the domain \(\mathcal{D}\) and satisfy
for some \(K>0\).
Letting
Then Hamiltonian (2.6) is of the form
where the Hamiltonian H satisfies the following conditions:
(B1) H is real analytic in \(\mathcal{D}\).
(B2) (Non-degenerate) There are constants \(\delta_{b}>\delta_{a}> 0\) such that on some complex neighborhood of Π
(B3) The inequality
holds on the domain \(\mathcal{D}\), where \(K_{1}\) is a positive constant.
3 KAM theorem and its iterative lemma
3.1 Statement of KAM theorem
Let \(\hat{\mathbb{T}}^{\mathbb{Z}}=\mathbb{C}^{\mathbb{Z}}/(2\pi \mathbb{Z})^{\mathbb{Z}}\). Define the phase space
We now consider a small perturbation
of an infinite dimensional Hamiltonian in the parameter-dependent normal form
on the phase space \(\mathcal{P}\) with the symplectic structure
The Hamiltonian equations of motion of \(H_{0}\) are as follows:
here \(\Lambda=\operatorname{diag}(\lambda_{2j})_{j\in\mathbb{Z}}\). Hence, for each \(\xi\in\Pi\), there is an infinite dimensional invariant torus: \(\mathcal{T}^{\mathbb{Z}}_{0}=\mathbb{T}^{\mathbb {Z}}\times\{0\}\times\{0\}\times\{0\}\) for \(H_{0}\).
Our aim in this issue is to prove the persistence of the torus \(\mathcal{T}^{\mathbb{Z}}_{0}\) under the small perturbation ϵP for “most” \(\xi\in\Pi\) via a KAM method similar to that in [41].
Theorem 3.1
Suppose that Hamiltonian (2.9) satisfies conditions (B1)–(B3). Then there exists a small constant \(\epsilon^{*}\) such that, if \(0<\epsilon<\epsilon^{*}\), then there are a set \(\Pi_{\infty}\subset\Pi\) with \(\operatorname{Prob}(\Pi_{\infty})\) arbitrarily close to one (depending on ϵ), an analytic torus embedding \(\mathcal{C}^{\infty}: \mathbb{T}^{\mathbb {Z}}\times\Pi_{\infty}\rightarrow\mathcal{P}\), and a map \(\omega ^{\infty}: \Pi_{\infty}\rightarrow\mathbb{R}^{\mathbb{Z}}\) such that, for each \(\xi\in\Pi_{\infty}\), the map \(\mathcal{C}^{\infty }\) restricted to \(\mathbb{T}^{\mathbb{Z}}\times\{\xi\}\) is an analytic embedding of rotational torus with frequencies \(\omega^{\infty}\) satisfying \(\vert \omega^{\infty}-\omega \vert _{\infty}<\epsilon^{1/6}\) for the Hamiltonian H defined by (2.9).
3.2 Iterative constants and iterative domains
In what follows, we denote by \(C,C_{1}, C_{2},\ldots \) positive constants which arrive in estimates, and by \(K,K_{1},K_{2},\ldots \) positive constants which arrive in lemmas and theorems. Both of them are independent of ϵ and the number m of the iteration, and may be different in different parts of the text. Let \(C(m)\) be the function of m of the form \(C_{1}m^{C_{2}m}\) or \(C_{1}m^{C_{2}m^{2}}\) or \(C_{1}m^{C_{2}m^{4}}\).
As usual, the KAM theorem is proved by the Newton-type iteration procedure which involves an infinite sequence of coordinate changes. In order to make our iteration procedure run, we need the following iterative constants and iterative domains.
1. \(\epsilon_{m}=\epsilon^{(\frac{5}{4})^{m}}, \epsilon_{m}\) bounds the size of the interaction after m iterations.
2. \(\delta_{m+1}=\delta_{m}-b_{m}=\delta_{m}-\delta_{0}/[64(m+1)^{2}]\), \(\delta_{m}\) measures the size of the analyticity domain in the angular variables after m iterations, and \(b_{m}\) is the amount by which the domain shrinks in the \((m+1)\)th step.
3. \(w_{m}=(\epsilon_{m})^{2\gamma}\), \(w_{m}\) measures the size of the analyticity domain in the frequency space. γ is a small positive constant.
4. \(L_{m}=\{2(1+\beta) \vert \ln\epsilon_{m} \vert /3\}^{1/1+\alpha}\); \(L_{m}\) determines the size of the region we must consider at the mth iterative step. Here β is a small positive constant, α is the constant in (1.6).
5. \(M_{m+1}=3 \vert \ln\epsilon_{m} \vert /(2b_{m})\), \(M_{m}\) determines the number of Fourier coefficients we must consider at the mth step of the iteration, \(b_{m}\) is defined in (2).
6.
\(\rho^{m}\) measures the size of the analyticity domain for the action variables.
7.
\(\varrho ^{m}\) measures the size of the analyticity domain for variables \(u,v\).
8.
[Here, \(n(j)\) is defined by \(L_{n(j)}< \vert j \vert \leq L_{n(j)+1}\)].
9.
10. \(\{\Pi_{m}\}^{\infty}_{m=0}\): be a sequence of compact subsets of \(\mathbb{R}^{\mathbb{Z}}_{+}\) with
here \(\Pi_{0}=\Pi\).
11. \(\mathcal{D}^{l}_{m}=\{(I,\theta,u,v)\in\mathbb{C}^{\mathbb {Z}}\times\mathbb{C}^{\mathbb{Z}}\times\mathbb{C}^{\mathbb {Z}}\times\mathbb{C}^{\mathbb{Z}}: \vert I_{j} \vert <\frac{\rho^{m}_{j}}{2^{l}}, \vert \operatorname{Im} \theta_{j} \vert <\delta _{m}-(1-\frac{1}{2^{l}})b_{m}, \vert u_{j} \vert <\frac{\varrho^{m}_{j}}{2^{l}}, \vert v_{j} \vert <\frac {\varrho^{m}_{j}}{2^{l}},j\in\mathbb{Z}\},l=0,1,2; \text{ and denote } \mathcal{D}_{m}=\mathcal{D}^{0}_{m}\).
12. \(\mathcal{O}_{m}=\{\xi\in\mathbb{C}^{\mathbb{Z}}: \vert \xi_{j}-\xi '_{j} \vert < w_{m},j\in\mathbb{Z}, \text{for some } \xi'\in\Pi_{m}\} \).
3.3 Iterative lemma
The proof of Theorem 3.1 uses the KAM method with a novel addition:
We introduce a sequence of length scales, \(L_{m}\nearrow\infty\), and at the mth stage of our iterative procedure, we consider only sites \(j: \vert j \vert \leq L_{m}\).
As a standard way of proving the theorem, we must give the iterative lemma.
Lemma 3.2
Consider a family of Hamiltonians \(\mathcal{H}_{l}\ (0\leq l\leq m)\):
where \((I,\theta,u,v,\xi)\in\mathcal{D}_{l}, \Lambda^{l} u\cdot v=\sum_{j\in\mathbb{Z}}\Lambda^{l}_{j}u_{j}v_{j}\). Write \(P_{l}:=P'_{2l}+P'_{3l}\), where \(P'_{3l}=P_{l}-P'_{2l}\) and
in the usual multi-index notation, where
Assume that, for \(0 \leq l \leq m\), the following conditions hold true:
\((l.1)\) \(\mathcal{H}_{l}\) is real analytic in the domain \(\mathcal {D}_{l} \times \mathcal{O}_{l}\), \(\mathcal{H}_{0}=H\);
\((l.2)\) \(P_{l}=P^{l}+\epsilon\sum_{L_{l}\leq \vert j \vert < L_{l+1}}f_{j}\) and \(P^{l}\) depends only on \((I_{j},\theta_{j},u_{j},v_{j},\xi_{j})\) with \(\vert j \vert \leq L_{l}\), \(P^{0}=\epsilon\sum_{ \vert j \vert < L_{0}}f_{j}(I,\theta,u,v,\xi)\), and \(\vert P^{l} \vert ^{\mathcal{D}_{l}\times\mathcal{O}_{l}}\leq C(l)\epsilon_{l}\);
\((l.3)\) \(\omega^{l}_{j}=\omega^{l-1}_{j}+R^{l-1}_{j00}(0,\xi),l\geq 1,\omega^{0}=\omega\), and \(R^{l-1}_{j00}(0,\xi)=0\) with \(\vert j \vert > L_{l}\), and \(\vert \omega^{l}_{j}-\omega_{j} \vert ^{\mathcal{O}_{l-1}}\leq\hat {\omega}_{j}, \vert \partial_{\xi_{i}}(\omega^{l}_{j}-\omega_{j}) \vert ^{\mathcal {O}_{l}}\leq\eta^{l}_{ij}\);
\((l.4)\) \(\Lambda^{l}_{j}=\Lambda^{l-1}_{j}+R^{l-1}_{0jj}(0,\xi),l\geq 1,\Lambda^{0}=\Lambda\), and \(R^{l-1}_{0jj}(0,\xi)=0\) with \(\vert j \vert > L_{l}\), \(\vert \Lambda^{l}_{j}-\Lambda_{j} \vert ^{\mathcal{O}_{l}}\leq\hat{\omega}_{j}\);
\((l.5)\) \(\operatorname{Prob}(\Pi_{l})\geq1-\sum^{l}_{j=0}(\epsilon_{j})^{\kappa}\) for some \(\kappa>0\);
\((l.6)\) Writing \(\mathcal{C}^{l}=\mathcal{C}_{1}\circ\cdots\circ \mathcal{C}_{l}=[I+\Phi^{l},\theta+\Psi^{l},u+\phi^{l},v+\psi^{l}]\), we have
and \(\mathcal{C}^{l}=\textit{ identity at sites}\), j, with \(\vert j \vert > L_{l}\).
Then there is a positive constant \(\epsilon^{*}\) small enough such that, if \(0 <\epsilon<\epsilon^{*}\), there is a set \(\Pi_{m+1}\subset\Pi _{m}\) with \(\operatorname{Prob}(\Pi_{m}\setminus\Pi_{m+1})\leq(\epsilon _{m})^{\kappa}\), and a change of variables \(\mathcal{C}_{m+1}:\mathcal {D}_{m+1}\times\mathcal{O}_{m+1}\rightarrow\mathcal{D}_{m}\times \mathcal{O}_{m}\) is real analytic in \(\mathcal{D}_{m+1}\times\mathcal {O}_{m+1}\). Furthermore, the new Hamiltonian \(\mathcal{H}_{m+1} = \mathcal{H}_{m}\circ\mathcal{C}_{m+1}=\mathcal{H}_{0}\circ\mathcal {C}^{m+1}\) is of the form
and satisfies all the above conditions \((l.1)\)–\((l.6)\) with l being replaced by \(m + 1\).
3.4 Derivation of homological equations
Step 1. Splitting the perturbation. Let us consider the Hamiltonian \(\mathcal{H}_{m}\). Following Kuksin [45] and Yuan [49], we split the perturbation \(P_{m}\) into an “essential” part \(P'_{2m}\) (i.e., \(l = m\) in (3.5)) which is linear in I, quadratic in \(u,v\), and an unessential part \(P'_{3m}\).
Lemma 3.3
If \(0<\epsilon<\epsilon^{*}\ll1\), then the following estimates hold true:
-
(a)
$$ \begin{aligned} &\bigl\vert R^{m}_{j00}(0,\xi) \bigr\vert ^{\mathcal{O}_{m}}\leq(\epsilon _{m})^{\frac{2}{9}},\qquad \bigl\vert R^{m}_{0jj}(0,\xi) \bigr\vert ^{\mathcal{O}_{m}}\leq( \epsilon _{m})^{\frac{2}{9}},\quad \vert j \vert \leq L_{m+1}, \\ &\bigl\vert \partial_{\xi_{i}} R^{m}_{j00}(0,\xi ) \bigr\vert ^{\mathcal{O}_{m+1}}\leq(\epsilon_{m})^{\frac{1}{6}}, \quad \vert i \vert , \vert j \vert \leq L_{m+1},\\ &\partial_{\xi_{i}} R^{m}_{j00}(0,\xi)=0,\quad \textit{otherwise}; \\ &R^{m}_{j00}(0,\xi)=R^{m}_{0jj}(0, \xi)=0,\quad \vert j \vert >L_{m+1}. \end{aligned} $$(3.8)
-
(b)
\(\vert P'_{3m} \vert ^{\mathcal{D}_{m+1}\times\mathcal{O}_{m}}\leq C(m+1)\epsilon_{m+1}\);
-
(c)
the functions \(P'_{2m}\) and \(P'_{3m}\) are real analytic and depend only on \((I_{j},\theta_{j},u_{j},v_{j},\xi_{j})\) with \(\vert j \vert \leq L_{m+1}\).
Proof
For (a), we consider \(R^{m}_{j00}(\theta,\xi)\). Since \(\vert P_{m} \vert ^{\mathcal{D}_{m}\times\mathcal{O}_{m}}\leq C(m)\epsilon_{m}\), by Assumption \((l.2)\), we get that \(\vert P'_{2m} \vert ^{\mathcal{D}_{m}\times \mathcal{O}_{m}}\leq C(m)\epsilon_{m}\). Therefore,
For any k with \(\vert k \vert \leq L_{m+1}\), let \(I^{*}(m)\) satisfy
Then \(\vert \sum_{ \vert j \vert \leq L_{m+1}}R^{m}_{j00}(\theta,\xi )I^{*}(m)_{j} \vert ^{\mathcal{D}_{m}\times\mathcal{O}_{m}}\leq C(m)\epsilon_{m}\). At the same time,
Thus
By the definition of \(\rho^{m}\), when \(\vert j \vert \leq L_{m}\), we have
Hence, if \(\vert k \vert \leq L_{m}\), we have
For the case \(L_{m}< \vert k \vert \leq L_{m+1}\), by Assumption \((l.2)\), we know that \(P^{m}\) makes no contribution to \(R^{m}_{k00}(\theta,\xi)\). Thus we see that in this case the factor \(\epsilon_{m}\) in (3.9) can be replaced by \(e^{-\frac{3}{2}( \vert k \vert -1)^{1+\alpha}}\) and by the definition of \(\rho^{m}\):
Also, by Assumption \((l.2)\), if \(\vert k \vert >L_{m+1}, R^{m}_{k00}(\theta,\xi )=0\). From (3.10) and (3.11), for β small enough, we have \(\vert R^{m}_{j00}(\theta,\xi) \vert ^{\mathcal {D}_{m}\times\mathcal{O}_{m}}\leq(\epsilon_{m})^{2/9}, \vert j \vert \leq L_{m+1}\). Thus
The case of \(R^{m}_{0jj}(0,\xi)\) can be proved in a similar way.
By the Cauchy estimate, we have
If \(\vert i \vert > L_{m+1} \text{ or } \vert j \vert > L_{m+1}\), \(\partial_{\xi_{i}} R^{m}_{j00}(0,\xi)=0\) is obvious.
For (b). Let \((I,\theta,u,v)\in\mathcal{D}_{m+1}, P_{m}=\epsilon _{m}\mathcal{P}_{m}\), and \(\upsilon=(\epsilon_{m})^{1/12}\). Then \(((\frac {z}{\upsilon})^{2}I,\theta,(\frac{z}{\upsilon})u,(\frac{z}{\upsilon })v)\) \(\in\mathcal{D}_{m}\) for \(z\in\mathbb{C}, \vert z \vert \leq1\). Let us consider the function \(z\mapsto\mathcal{P}_{m}((\frac{z}{\upsilon })^{2}I,\theta,(\frac{z}{\upsilon})u,(\frac{z}{\upsilon})v)\) and its Taylor series at zero:
From \(\vert P_{m} \vert ^{\mathcal{D}_{m}\times\mathcal{O}_{m}}\leq C(m)\epsilon_{m}\), we have \(\vert \mathcal{P}_{m} \vert ^{\mathcal{D}_{m}\times\mathcal{O}_{m}}\leq C(m)\). Thus \(\vert h_{k} \vert ^{\mathcal{D}_{m}\times\mathcal{O}_{m}}\leq C(m)\) for all k. Since \(P'_{3m}=\epsilon_{m}(h_{3}\upsilon^{3}+h_{4}\upsilon ^{4}+\cdots)\), then
For (c). From Assumptions \((l.1)\), \((l.2)\), the proof of (c) is obvious. □
Step 2. Truncation. Let
Then, by Lemma 3.3, the frequencies satisfy assumptions \((l.3)\) and \((l.4)\) with \(l = m + 1\). Write
here \(I=I(m), u=u(m), v=v(m)\). In addition, \(R^{m}_{0j00}=R^{m}_{j00}(0,\xi ), R^{m}_{00jj}=R^{m}_{0jj}(0,\xi)\) is obvious. Then we can write \(\mathcal{H}_{m}\) as
and the functions \(P_{2m}\) and \(P_{3m}\) are real analytic and depend only on \((I_{j},\theta_{j},u_{j},v_{j},\xi_{j})\) with \(\vert j \vert \leq L_{m+1}\).
Claim
Proof
From (3.15) we first consider
With Lemma 3.3(b), we obtain that
Thus the proof of the claim is complicated. □
Step 3. Derivation of the homological equations.
Proof
We look for a near-to-the-identity transformation \(\mathcal {C}_{m+1}\) so that (3.7) holds; such transformation will be determined by a generating function of the form
and assume that S is \(O(\epsilon_{m})\).
Inserting \(I=I'+\frac{\partial S}{\partial\theta}, u=u'+\frac {\partial S}{\partial v}\) into \(\mathcal{H}_{m}\), one finds
where \(P_{m+1}=P^{m+1}+\epsilon\sum_{L_{m+1}\leq \vert j \vert < L_{m+2}}f_{j}(I'+\frac{\partial S}{\partial\theta},\theta,u'+\frac {\partial S}{\partial v},v)\) and
Clearly, we hope to find the transformation S satisfying
i.e., the homological equation. □
3.5 Solutions to the homological equations and investigation of \(\mathcal{C}_{m+1}\)
We can solve (3.19) by means of Fourier series, and we find
Thus
In general, the sum in (3.21) will diverge. To cure this problem, we first reduce the infinite sum to a finite one. By the definition of \(P_{2m}\), we can restrict the sum in (3.21) to vectors \(k\in\mathbb{X}^{m+1}\) with
With these restrictions, the sum in (3.21) contains only a finite number of terms, and a simple estimate shows this number is bounded by \((2M_{m+1})^{(2L_{m+1})}\).
To prevent that the sum in (3.21) fails to be well defined, we exclude
and set \(\Pi_{m+1}=\Pi_{m}\setminus R^{m}\).
Now we start to estimate \(\Pi_{m+1}\). By the definition of \(\mathbb{X}^{m+1}\),
therefore we just need to consider the finite dimensional situation. Let \(\xi(m)=(\xi)_{ \vert j \vert \leq L_{m+1}}, \omega^{m+1}(m)=\{(\omega _{j}^{m+1})(\xi(m))\}_{ \vert j \vert \leq L_{m+1}},\Pi_{m}(m)=\Pi_{m}\cap(\prod_{ \vert j \vert \leq L_{m+1}}\mathbb{R}_{+}),\mathcal{O}_{m}(m)\) be the complex \(w_{m}\)-neighborhood of \(\Pi_{m}(m)\). In view of estimates \((l.3)\) with \(l=m+1\) and assumption (B2), it is easy to see that, if \(0<\epsilon <\epsilon^{*}<\frac{\delta_{a}}{2}\),
Moreover, by the inverse function theorem, there exists the inverse \((\omega^{m+1}(m))^{-1}(\omega(m))\) for \(\omega(m)\in\omega ^{m+1}(m)(\mathcal{O}_{m}(m))=^{\text{def}}\Omega_{m}\), and
Obviously, the Kolmogorov measure
then
here \(K_{3}=\max_{\xi(m)\in\Pi_{m}(m)} \vert \frac{\partial\omega ^{m+1}(m)}{\partial\xi(m)} \vert \). Hence
Since there are at most \((2M_{m+1})^{2L_{m+1}}\) vectors in \(\mathbb {X}^{m+1}\), we find that \(\operatorname{Prob}( \Pi_{m}\backslash\Pi_{m+1})\) is bounded by \((\epsilon _{m})^{\kappa}\) for some \(0 < \kappa< \gamma\), and the bound on \(\operatorname{Prob}(\Pi_{m+1})\) follows.
We can bound the denominators in (3.21) only if \(\xi\in\Pi _{m+1}\). However, note that for any \(\xi'\in\mathcal{O}_{m+1}\), we can write
Since
thus, for ϵ small enough, one gets
Therefore
remains valid on the domain \(\mathcal{O}_{m+1}\).
For these preparations, we now can estimate the transformation \(S=S(I',\theta,u',v)\). For the estimates, we decompose \(P_{2m}=R^{0}+R^{1}+R^{2}\), where \(R^{j}\) comprises \(\vert q+\bar{q} \vert =j\); and furthermore,
where \(R^{ij}\) depend on \(\theta, \xi\), and \(R^{00}\) depends in addition on I. With a similar decomposition of S, it suffices to discuss each term individually. In the following we do this for \(\dot{S}=S^{10}\) and \(\ddot{S}=S^{11}\).
Consider the term \(\dot{S}=S^{10}\), and the corresponding coefficient of Ṡ is given by
By the small divisor assumptions, we have
and thus \(\vert \dot{S}_{k,j} \vert ^{\mathcal{O}_{m+1}}\leq2(\epsilon _{m})^{-\gamma} \vert \dot{R}_{k,j} \vert ^{\mathcal{O}_{m}}\). Hence
Therefore
Consider now the term \(\ddot{S}=S^{11}\), and the corresponding coefficient of S̈ is given by
Without loss of generality, let \(i>j\), and from the norm frequencies assumption, we get
Choose ϵ small enough such that
Thus
For \({\mathcal{O}_{m+1}}\subset\mathcal{O}_{m}\), the small divisor satisfies
Using this estimate, we see that
Hence
Therefore
The remaining terms of S can be estimated in the same line. Therefore we obtain
where we have written 2γ as γ by abuse of notation.
Using the similar discussion in the proof of Lemma (a) and the Cauchy estimate, on the domain \(\mathcal{D}^{2}_{m}\times{\mathcal {O}_{m+1}}\), one has
Choose \(\beta,\gamma\) small enough, thus
By the analytic inverse function theorems, the equations
can be inverted to yield an analytic and invertible canonical transformation on the domain \(\mathcal{D}_{m+1}\times\mathcal{O}_{m+1}\). More precisely, we have
maps \(\mathcal{D}_{m+1}\times\mathcal{O}_{m+1}\) into \(\mathcal {D}_{m}\times\mathcal{O}_{m}\). Furthermore, on the domain \(\mathcal {D}_{m+1}\times\mathcal{O}_{m+1}\), we get
Since \(\mathcal{C}^{m+1}=\mathcal{C}^{m}\circ\mathcal{C}_{m+1}\),
and (3.38) imply the bounds on \(\mathcal{C}^{m+1}\) stated in (3.6) with \(l=m+1\). In addition, on the domain \(\mathcal{D}_{m+1}\times\mathcal {O}_{m+1}\), by Cauchy estimate and (3.38), we obtain
Making use of the same way to the other three terms, on the domain \(\mathcal{D}_{m+1}\times\mathcal{O}_{m+1}\), one gets
By virtue of the fact that S does not depend on \((I_{j},\theta _{j},u_{j},v_{j})\) with \(\vert j \vert >L_{m+1}\), we see that \(\mathcal{C}_{m+1}\), and hence \(\mathcal{C}^{m+1}\) will reduce to the identity at these sites. Then with these bounds we conclude that
It remains to verify \(l.2\) with \(l=m+1\). Write \(\mathcal {H}_{m+1}=\mathcal{H}_{m}\circ\mathcal{C}_{m+1}(I',\theta ',u',v')=\mathcal{H}_{m}(I'+\Xi,\theta'+\Theta,u'+\Delta ,v'+\Upsilon)\), which is in turn equal to
with \(P^{m+1}\) having the form
Now we estimate \(P^{m+1}\). From (3.38), the first term in (3.42) is bounded on the domain \(\mathcal{D}_{m+1}\times \mathcal{O}_{m+1}\) by
The second term in (3.42) can be similarly estimated on \(\mathcal{D}_{m+1}\times\mathcal{O}_{m+1}\) by \((\epsilon _{m})^{\frac{3}{2}}\). Finally, the fourth term in (3.42) is bounded by using similar discussion as that in the proof of Lemma 3.3(a) on the domain \(\mathcal{D}_{m+1}\times\mathcal{O}_{m+1}\), and we find it is less than
Thus, with (3.17), we obtain that
completing the verification of \(l.2\) with \(l=m+1\) and the proof of Lemma 3.2.
4 Proof of the theorem
Proof of Theorem 3.1
The proof is finished by running Lemma 3.2. Obviously, the Hamiltonian H defined by (2.9) satisfies conditions \((l.1)\)–\((l.6)\), with \(l = 0\). Thus, the iterative lemma (Lemma 3.2) works. Inductively, we get the following sequences:
Let
By (3.41), \(l.2\), and \(l.3\), we conclude that \(\mathcal{H}_{m},\mathcal{C}^{m}\) converges uniformly on the domain \(\mathcal{D}_{\infty}\times\Pi _{\infty}\), and
Thus, \(\mathbb{T}^{\mathbb{Z}}\times\{0\}\times\{0\}\times\{0\}\) is an embedding torus with rotational frequencies \(\omega^{\infty}\in \Pi_{\infty}\) of the Hamiltonian \(\mathcal{H}_{\infty}\). Returning to the original Hamiltonian H, it has an embedding torus \(\mathcal{C}^{\infty}(\mathbb{T}^{\mathbb{Z}}\times \{0\}\times\{0\}\times\{0\})\) with frequencies \(\omega^{\infty}\). This proves the theorem. □
Proof of Theorem 1.1
By Sect. 2, Theorem 1.1 is just a corollary of Theorem 3.1. □
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The author thanks Professor Xiaoping Yuan, his advisor, for his constant encouragement and guidance.
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Li, H. The almost-periodic solutions of the weakly coupled pendulum equations. Adv Differ Equ 2018, 157 (2018). https://doi.org/10.1186/s13662-018-1604-0
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DOI: https://doi.org/10.1186/s13662-018-1604-0