R E S E A R C H
Open Access
The almost-periodic solutions of the
weakly coupled pendulum equations
Hepeng Li
1,2**Correspondence:
1School of Mathematical Sciences,
Fudan University, Shanghai, P.R. China
2Department of Mathematics,
Sichuan University of Arts and Science, Dazhou, P.R. China
Abstract
In this paper, it is proved that, for the networks of weakly coupled pendulum equations
d2x n dt2 +
λ
2
nsinxn=
Wn(xn–1,xn,xn–1), n∈Z,
there are many (positive Lebesgue measure) normally hyperbolic invariant tori which are infinite dimensional in both tangent and normal directions.
Keywords: Coupled pendulum equations; Normally hyperbolic invariant tori; Almost-periodic solutions; KAM theory
1 Introduction and main result
In the last several decades, models of (infinitely) many coupled oscillators have found di-verse 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 vari-ety of biological processes [13–15], etc. The research of these systems has brought out-standing 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 de-rive [35], shunted model of electrical rotator [36], and many other applications. Among those interesting models are the networks of weakly coupled pendulum equations
d2x
n
dt2 +λ 2
nsinxn=Wn(xn–1,xn,xn+1), n∈Z, (1.1)
whereλn,n∈Zare constants and Wnare 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
H=H(u,v) =1 2
n∈Z λn
u2n+v2n+P(u,v), (1.2)
wherePis of short range.
(i) Introducing the action-angle variables onun,vnfor∀n∈Z, then the Hamiltonian
system (1.2) takes the form
H=H(I,θ) =
n∈Z
ωnIn+P(I,θ), (1.3)
where tangent frequenciesωn=λn,n∈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 onun,vnforn∈ {1, 2, . . . ,m}and by abuse of
notation to rearrange the subscript ofun,vn,λn, we get
0→0; –n→–n;m+n→n, forn≥1.
Then Hamiltonian (1.2) is of the form
H=H(I,θ) =
m
n=1
ωnIn+
1 2
n∈Z λn
u2n+v2n+P(I,θ,u,v). (1.4)
Here tangent frequenciesωarem-dimensional and normal frequencies are infinite di-mensional. Kuksin [45], Pöschel [46, 47], and Wayne [48] (in alphabetic order) concluded that, under the multiplicity ofλnequals to 1 for∀n∈Zand some non-resonant
condi-tions (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 ofλnare the
(iii) For the case that both tangent frequencies ωand 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 ex-istence of almost-periodic solutions for the networks of weakly coupled pendulum equa-tions (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 x2j+1,j∈Zand the transformation (x2j–π,x2˙ j) = (1/
2λ2j(uj–vj),
λ2j/2(uj+vj)),j∈Z,
we transform the Hamiltonian of Eq. (1.1) into the form
H=H(I,θ,u,v) =
j∈Z
ωjIj+
j∈Z
λ2jujvj+P(I,θ,u,v), (1.5)
where P is of short range. Obviously, both tangent frequencies ω= (ωj)j∈Z and nor-mal frequencies= (λ2j)j∈Zare infinite dimensional belonging to case (iii). Meanwhile,
(uj,vj)j∈Z= (0, 0)j∈Zis a hyperbolic equilibrium point on the normal direction. From this,
we then mainly study the persistence of normally hyperbolic invariant tori which are in-finite dimensional in both tangent and normal directions for this Hamiltonian system in this issue.
Before starting our theorem, we define the norm
|x|∞:=sup
j∈Z|xj|
inCZand give the following assumptions: (A1)λn,n∈Zsatisfy
λ2j+1=λ> 0, λ2j≥ |j|–N, |λ2i–λ2j| ≥|i|–N–|j|–N, i,j∈Z\{0},N> 0.
(A2)Wnsatisfies
Wn=W
xn+1–xn+ (–1)nπ
e–34|n|1+α–Wxn–xn–1– (–1)nπe–34|n–1|1+α (1.6)
withα> 0 (arbitrarily small), andW=O(|x|3) is real analytic in the strip domain{x∈C:
|Imx|<δ0}for some constantδ0> 0. The expression 1
2y
2+λ2(1 –cosx) =hwithλ2
2 ≤h≤λ
2denotes a simple closed curve
which encloses (0, 0) in the (x,y)-plane. Letρ=ρ(h) be the area enclosed by(h), i.e.,
ρ(h) =
1
2y2+λ2(1–cosx)=h y dx.
Then we can see thatρ(h) > 0,ρ(h)= 0 for anyh∈[λ2
2,λ 2].
Equation (1.1) can be regarded as a perturbation of the following system:
d2x 2j+1 dt2 +λ
2sinx2
j+1= 0, j∈Z, (1.7a)
d2x2
j
dt2 –λ 2
For anyη= (hj)j∈Z,hj∈[λ
2
2,λ2], then, by the fact that 1
2y2+λ2(1 –cosx) =hj,j∈Zis a first integral of (1.7a),j∈Z(hj) is an invariant torus with the frequenciesω(η) = (H0(ρ(hj)))j∈Z
for (1.7a), where H0 is the inverse of ρ=ρ(h). Observe that (π, 0) is an equilibrium of (1.7b). Thus,
T(η) =
j∈Z
(hj)× j∈Z
(π, 0)
is an invariant torus with the frequenciesω(η) for (1.7a)–(1.7b). Therefore, any solution of (1.7a)–(1.7b) starting fromT(η) is a trivial breather for (1.7a)–(1.7b). Our goal is to show that the torus T(η) remains under the small perturbation. Here is our main re-sult which expresses that there does persist a large Cantor sub-family of rotationalZ-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
= [λ2
2,λ2]Z,there is a positive constant∗sufficiently small such that,when0 <<∗, there are a setS⊂withProb(S)arbitrarily close to one(depending on),a family of Z-tori
T[S] =
η∈S
T(η)⊂
η∈
T(η)
overS,and an analytic embedding
:T[S]→RZ×TZ×RZ×RZ,
which is a higher order perturbation of the inclusion map0:η∈T(η)→RZ×TZ×
RZ×RZrestricted toT[S],such that the restrictionto eachT(η)in the family is an embedding of a rotationalZ-torus for(1.1).Moreover,any solution of(1.1)starting from
(T(η))is an almost-periodic breather of frequenciesω∗with|ω∗–ω|∞=O(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.
Letx˙n=yn. Then (1.1) is a Hamiltonian system with its Hamiltonian
H=
j∈Z
1 2y
2
2j+1+λ2(1 –cosx2j+1)
+
j∈Z
1 2y
2
2j– (1 +cosx2j)λ22j
+
j∈Z
W(x2j+2–x2j+1–π)e– 3
2|j+12|1+α+W(x
2j+1–x2j+π)e–
=
We now carry out the standard reduction to action-angle variables. To construct the map (x,y)→(θ,ρ), whereρandθare action and angle variables, respectively, we letH0(ρ) be the value of the function 1
2y
This implies thatis symplectic. Thus, HamiltonianHis transformed into
By assumption (A2), there exists the inverseH–1
0 ofH0. Let [μ,ν] =H0–1([
λ2
2,λ2]). For any
ξ= (ξj)j∈Z∈[μ,ν]Z, letρ=I+ξ, whereI= (Ij)j∈Z. ExpandH0(ξj+Ij) inξjby Taylor’s
for-mula:
H0(ξj+Ij) =H0(ξj) +H0(ξj)Ij+O
|Ij|2
, j∈Z.
Letω= (H0(ξj))j∈Z,= [μ,ν]Z, and from transformation (2.4), we can denote
W(x2j+2–x2j+1–π)e– 3
2|j+12|1+α+W(x2j+1–x2j+π)e–32|j|1+α=fj(Ij,θj,uj,uj+1,vj,vj+1,ξj).
Thenξ∈and (2.5) can be written as
H=
j∈Z
ωjIj+
j∈Z
λ2jujvj+
j∈Z
O|uj–vj|4
+
j∈Z O|Ij|2
+
j∈Z
fj(Ij,θj,uj,uj+1,vj,vj+1,ξj), (2.6)
where the constantj∈ZH0(ξj) is omitted since it does not affect the dynamics.
Now we need to introduce the domain of the definition for HamiltonianH. Set
D=(I,θ,u,v,ξ)∈CZ×CZ×CZ×CZ×CZ:|Ij|<ρj0,|Imθj|<δ0,
|uj|<0j,|vj|<j0, andξj–ξj<wfor someξ∈,j∈Z
, (2.7)
hereρj0=14μe–|j|1+α,0j =14
ρj0.fj(I,θ,u,v),j∈Zare real analytic on the domainDand
satisfy
sup D
fj(I,θ,u,v,ξ)≤Kexp
–3 2
|j|– 11+α
, j∈Z, (2.8)
for someK> 0. Letting
P(I,θ,u,v,ξ) =
j∈Z
fj(Ij,θj,uj,uj+1,vj,vj+1,ξj),
Q(I,u,v) =
j∈Z
O|uj–vj|4
+
j∈Z
O|Ij|2
.
Then Hamiltonian (2.6) is of the form
H=H(I,θ,u,v,ξ) =
j∈Z
ωjIj+
j∈Z
λ2jujvj+Q(I,u,v) +P(I,θ,u,v,ξ), (2.9)
where the HamiltonianHsatisfies the following conditions: (B1)His real analytic inD.
(B2) (Non-degenerate) There are constantsδb>δa> 0 such that on some complex
neigh-borhood of
δa≤∂ωj ∂ξj
(B3) The inequality
|P| ≤K1
holds on the domainD, whereK1is a positive constant.
3 KAM theorem and its iterative lemma 3.1 Statement of KAM theorem
LetTˆZ=CZ/(2πZ)Z. Define the phase space
P=CZ× ˆTZ×CZ×CZ(I,θ,u,v).
We now consider a small perturbation
H=H0+P(I,θ,u,v,ξ), ξ∈, (3.1)
of an infinite dimensional Hamiltonian in the parameter-dependent normal form
H0=
j∈Z ωjIj+
j∈Z
λ2jujvj, ξ∈, (3.2)
on the phase spaceP with the symplectic structure
j∈Z
dθj∧dIj+
j∈Z
duj∧dvj. (3.3)
The Hamiltonian equations of motion ofH0are as follows:
˙
θ=ω, I˙= 0, u˙=v, v˙= –u,
here=diag(λ2j)j∈Z. Hence, for eachξ ∈, there is an infinite dimensional invariant torus:T0Z=TZ× {0} × {0} × {0}forH0.
Our aim in this issue is to prove the persistence of the torusT0Zunder the small pertur-bationPfor “most”ξ∈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 ∗ such that, if 0 < <∗, then there are a set ∞ ⊂ with Prob(∞) arbitrarily close to one (depending on ), an analytic torus embeddingC∞ : TZ×
∞ →P, and a mapω∞ :∞ →RZ such that, for each ξ ∈∞,the map C∞ restricted to TZ× {ξ}is an analytic embedding of rotational torus with frequencies ω∞
satisfying|ω∞–ω|∞<1/6for the Hamiltonian H defined by(2.9).
3.2 Iterative constants and iterative domains
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 proce-dure run, we need the following iterative constants and iterative domains.
1.m=(
5
4)m,mbounds the size of the interaction aftermiterations.
2.δm+1=δm–bm=δm–δ0/[64(m+ 1)2],δmmeasures the size of the analyticity domain
in the angular variables aftermiterations, andbm is the amount by which the domain
shrinks in the (m+ 1)th step.
3.wm= (m)2γ,wmmeasures the size of the analyticity domain in the frequency space. γ is a small positive constant.
4.Lm={2(1 +β)|lnm|/3}1/1+α;Lmdetermines the size of the region we must consider
at themth iterative step. Hereβis a small positive constant,αis the constant in (1.6). 5.Mm+1= 3|lnm|/(2bm),Mm determines the number of Fourier coefficients we must
consider at themth step of the iteration,bmis defined in (2).
6.
ρjm+1= 2–3ρjm, if|j|>Lm+1,
= 2–3ρLm
m+1, if|j| ≤Lm+1,
ρmmeasures the size of the analyticity domain for the action variables. 7.
jm+1= 2–3jm, if|j|>Lm+1,
= 2–3Lm
m+1, if|j| ≤Lm+1,
mmeasures the size of the analyticity domain for variablesu,v. 8.
ˆ
ωmj = 0, if|j|>Lm,
=
m–1
n=n(j)
(n)2/9, if|j| ≤Lm
[Here,n(j) is defined byLn(j)<|j| ≤Ln(j)+1]. 9.
ηmij =min
m–1
n=n(i) (n)1/6,
m–1
n=n(j) (n)1/6
if|i| ≤Lm, and|j| ≤Lm,
= 0, otherwise.
10.{m}∞m=0: be a sequence of compact subsets ofRZ+with
0⊃1⊃ · · · ⊃m⊃m+1⊃ · · ·,
11. Dl
m ={(I,θ,u,v)∈CZ×CZ×CZ×CZ:|Ij|< ρmj
2l ,|Imθj|<δm– (1 – 21l)bm,|uj|< mj
2l ,|vj|< mj
2l ,j∈Z},l= 0, 1, 2; and denoteDm=Dm0.
12.Om={ξ∈CZ:|ξj–ξj|<wm,j∈Z, for someξ∈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,Lm ∞, and at themth stage of our iterative
procedure, we consider only sitesj:|j| ≤Lm.
As a standard way of proving the theorem, we must give the iterative lemma.
Lemma 3.2 Consider a family of HamiltoniansHl(0≤l≤m):
Hl=ωl·I+lu·v+Q+Pl+
|j|≥Ll+1
fj(I,θ,u,v,ξ), (3.4)
where(I,θ,u,v,ξ)∈Dl,lu·v=
j∈Zljujvj.Write Pl:=P2l+P3l,where P3l=Pl–P2l
and
P2l= 2|p|+|q+¯q|≤2
Rlpq¯q(θ,ξ)Ip(l)uq(l)vq¯(l), (3.5)
in the usual multi-index notation,where
I(l) = (Ij)|j|≤Ll+1, u(l) = (uj)|j|≤Ll+1, v(l) = (vj)|j|≤Ll+1.
Assume that,for0≤l≤m,the following conditions hold true: (l.1)Hlis real analytic in the domainDl×Ol,H0=H;
(l.2)Pl=Pl+
Ll≤|j|<Ll+1fjand P
l depends only on(I
j,θj,uj,vj,ξj)with|j| ≤Ll,P0= |j|<L0fj(I,θ,u,v,ξ),and|Pl|Dl×Ol≤C(l)l;
(l.3)ωlj=ωjl–1+Rjl–100(0,ξ),l≥1,ω0=ω,and Rlj00–1(0,ξ) = 0with|j|>Ll,and|ωlj–ωj|Ol–1≤ ˆ
ωj,|∂ξi(ωjl–ωj)|Ol≤ηlij;
(l.4)lj=lj–1+Rl–1
0jj(0,ξ),l≥1,0=,and Rl0–1jj(0,ξ) = 0with|j|>Ll,|lj–j|Ol ≤ ˆ
ωj;
(l.5)Prob(l)≥1 –lj=0(j)κfor someκ> 0;
(l.6)WritingCl=C1◦ · · · ◦C
l= [I+l,θ+l,u+φl,v+ψl],we have
lj≤ l–1
n=n(j) (n)
8
9, l
j≤ l–1
n=n(j) (n)
2 9,
φlj≤ l–1
n=n(j) (n)
5
9, ψl
j≤ l–1
n=n(j) (n)
5 9,
(3.6)
andCl= identity at sites,j,with|j|>Ll.
Hm+1=Hm◦Cm+1=H0◦Cm+1is of the form
Hm+1=ωm+1·I+m+1u·v+Q+Pm+1+
|j|≥Lm+2
fj(I,θ,u,v,ξ) (3.7)
and satisfies all the above conditions(l.1)–(l.6)with l being replaced by m+ 1.
3.4 Derivation of homological equations
Step1. Splitting the perturbation. Let us consider the HamiltonianHm. Following Kuksin
[45] and Yuan [49], we split the perturbationPminto an “essential” partP2m(i.e.,l=min
(3.5)) which is linear inI, quadratic inu,v, and an unessential partP3m.
Lemma 3.3 If0 <<∗1,then the following estimates hold true:
(a)
Rmj00(0,ξ)Om≤(m)
2
9, Rm 0jj(0,ξ)
Om≤
(m)
2
9, |j| ≤Lm+1,
∂ξiR
m j00(0,ξ)
Om+1≤ (m)
1
6, |i|,|j| ≤Lm+1,
∂ξiR
m
j00(0,ξ) = 0, otherwise;
Rmj00(0,ξ) =Rm0jj(0,ξ) = 0, |j|>Lm+1.
(3.8)
(b) |P3m|Dm+1×Om≤C(m+ 1)
m+1;
(c) the functionsP2mandP3mare real analytic and depend only on(Ij,θj,uj,vj,ξj)with |j| ≤Lm+1.
Proof For (a), we considerRm
j00(θ,ξ). Since|Pm|Dm×Om≤C(m)m, by Assumption (l.2), we
get that|P2m|Dm×Om≤C(m)
m. Therefore,
|j|≤Lm+1
Rmj00(θ,ξ)I(m)j
Dm×Om≤C(m)m.
For anykwith|k| ≤Lm+1, letI∗(m) satisfy
I∗(m)j= ⎧ ⎨ ⎩
1 2ρ
m k j=k,
0 otherwise.
Then||j|≤Lm+1Rm
j00(θ,ξ)I∗(m)j|Dm×Om≤C(m)m. At the same time,
|j|≤Lm+1
Rmj00(θ,ξ)I∗(m)j
Dm×Om=1 2ρ
m
kRmk00(θ,ξ) Dm×Om
.
Thus
By the definition ofρm, when|j| ≤L By the Cauchy estimate, we have
Step2. Truncation. Let
ωmj +1=ωmj +Rmj00(0,ξ), j∈Z, (3.13)
mj+1=mj +Rm0jj(0,ξ), j∈Z. (3.14)
Then, by Lemma 3.3, the frequencies satisfy assumptions (l.3) and (l.4) withl=m+ 1. Write
P2m=
|k|≤Mm+1
2|p|+|q+¯q|≤2
Rmkpqq¯(ξ)e √
–1k·θ
Ipuqvq¯–
j
Rm0j00Ij+Rm00jjujvj
ˆ
P2m=
|k|>Mm+1
2|p|+|q+¯q|≤2
Rmkpqq¯(ξ)e √
–1k·θIpuqvq¯, k∈Z2Lm+1,
P3m=P2ˆ m+P3m
(3.15)
hereI=I(m),u=u(m),v=v(m). In addition,Rm0j00=Rmj00(0,ξ),Rm00jj=Rm0jj(0,ξ) is obvious. Then we can writeHmas
Hm=ωm+1·I+m+1u·v+Q(I,u,v) +P2m+P3m
+
|j|≥Lm+1
fj(I,θ,u,v,ξ), (3.16)
and the functionsP2m andP3m are real analytic and depend only on (Ij,θj,uj,vj,ξj) with |j| ≤Lm+1.
Claim
|P3m|Dm+1×Om≤C(m+ 1)m+1.
Proof From (3.15) we first consider
|ˆP2m|Dm+1×Om≤
|k|>Mm+1
P2m(k)Dm×Ome|k|δm+1
≤
|k|>Mm+1
P 2m
Dm×Om
e–|k|bm
≤C(m+ 1)(m)2≤m+1.
With Lemma 3.3(b), we obtain that
|P3m|Dm+1×Om≤C(m+ 1)m+1. (3.17)
Thus the proof of the claim is complicated.
Proof We look for a near-to-the-identity transformationCm+1 so that (3.7) holds; such transformation will be determined by a generating function of the form
Iθ+uv+SI,θ,u,v,
Clearly, we hope to find the transformationSsatisfying
ωm+1·∂S
i.e., the homological equation.
3.5 Solutions to the homological equations and investigation ofCm+1 We can solve (3.19) by means of Fourier series, and we find
Skpqq¯=
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 ofP2m, we can restrict the sum in (3.21) to vectors
k∈Xm+1with
Xm+1≡k∈Z2Lm+1: 0 <|k|<M
m+1
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 (2Mm+1)(2Lm+1).
To prevent that the sum in (3.21) fails to be well defined, we exclude
Rm≡ξ∈m:∃k∈Xm+1s.t.k·ωm+1< (m)γ
, (3.23)
and setm+1=m\Rm.
Now we start to estimatem+1. By the definition ofXm+1,
Rm∩ |j|>Lm+1
R+
=∅,
therefore we just need to consider the finite dimensional situation. Letξ(m) = (ξ)|j|≤Lm+1,
ωm+1(m) ={(ωm+1
j )(ξ(m))}|j|≤Lm+1,m(m) =m ∩(
|j|≤Lm+1R+),Om(m) be the complex wm-neighborhood ofm(m). In view of estimates (l.3) withl=m+ 1 and assumption
(B2), it is easy to see that, if 0 <<∗<δa
2,
∂ωm+1(m)
∂ξ(m)
Om(m)≥δa
2.
Moreover, by the inverse function theorem, there exists the inverse (ωm+1(m))–1(ω(m)) for
ω(m)∈ωm+1(m)(O
m(m)) =defm, and
∂(ωm+1(m))–1
∂ω(m)
m≤K2
δa
, hereK2=max
ξ∈
ω(ξ)∞.
Obviously, the Kolmogorov measure
Probω(m)∈ωm+1(m)–1m(m)
:k·ω(m)≤(m)γ
≤C(m)γ, k∈Xm+1,
then
Probξ(m)∈m(m) :k·ωm+1(m)≤(m)γ
≤CK3(m)γ, k∈Xm+1,
hereK3=maxξ(m)∈m(m)|∂ω
m+1(m)
∂ξ(m) |. Hence
Probξ∈m:k·ωm+1≤(m)γ
≤CK3(m)γ, k∈Xm+1.
Since there are at most (2Mm+1)2Lm+1 vectors inXm+1, we find thatProb(m\m+1) is bounded by (m)κfor some 0 <κ<γ, and the bound onProb(m+1) follows.
We can bound the denominators in (3.21) only ifξ∈m+1. However, note that for any
ξ∈Om+1, we can write
k·ωm+1ξ
Since
k·ωm+1(ξ) –k·ωm+1ξ≤
|j|≤Lm+1
|kj|
|i|≤Lm+1
∂ωmj +1
∂ξi
ξi–ξi
≤(m)2γ
|i|,|j|≤Lm+1
|kj|
∂(ωjm+1–ωj) ∂ξi
+∂ωj
∂ξi
≤(m)2γ
δb+ 4Lm+11/6 |j|≤Lm+1
|kj|
≤(m)2γδb+ 4Lm+11/6
Mm+1, (3.25)
thus, forsmall enough, one gets
k·ωm+1(ξ)–1k·ωm+1(ξ) –k·ωm+1ξ≤(m)γδb+ 4Lm+11/6
Mm+1≤ 1 2.
Therefore
k·ωm+1ξ≥1
2(m)
γ
(3.26)
remains valid on the domainOm+1.
For these preparations, we now can estimate the transformationS=S(I,θ,u,v). For the estimates, we decomposeP2m=R0+R1+R2, whereRjcomprises|q+q¯|=j; and
further-more,
R0=R00,
R1=R10,u(m) +R01,v(m),
R2=R20u(m),u(m) +R11u(m),v(m) +R02v(m),v(m),
(3.27)
whereRijdepend onθ,ξ, andR00depends in addition onI. With a similar decomposition ofS, it suffices to discuss each term individually. In the following we do this forS˙=S10and
¨
S=S11.
Consider the termS˙=S10, and the corresponding coefficient ofS˙is given by
˙
Sk,j=
˙
Rk,j √
–1k·ωm+1–m+1
j
, |j| ≤Lm+1,|k| ≤Mm+1. (3.28)
By the small divisor assumptions, we have
√
–1k·ω–mj+1Om+1≥min
(m)γ
2 , 1 2jN
≥(m)γ
and thus|˙Sk,j|Om+1≤2(m)–γ| ˙Rk,j|Om. Hence
|˙Sj|D
1
m×Om+1≤
|k|≤Mm+1
|˙Sk,j|Om+1e|k|(δm–
bm
2 )
≤2(m)–γ
|k|≤Mm+1
| ˙Rk,j|Ome|k|(δm–
bm
2 )
≤2(m)–γ
|k|≤Mm+1
| ˙Rj|Dm×Ome–|k|
bm
2
≤2
2 bm
2Lm+1
(m)–γ| ˙Rj|Dm×Om.
Therefore
S10,u(m+ 1) D1m×Om+1≤ |j|≤Lm+1
S˙juj
D1
m×Om+1
≤2
2 bm
2Lm+1 (m)–γ
|j|≤Lm+1
δjm| ˙Rj|Dm×Om
(by Cauchy estimate)≤C(m)Lm+1
2 bm
2Lm+1 (m)1–γ.
Consider now the termS¨=S11, and the corresponding coefficient ofS¨is given by
¨
Sk,ij=
¨
Rk,ij √
–1k·ωm+1+m+1
j –mi+1
, |i|,|j| ≤Lm+1,|k| ≤Mm+1. (3.29)
Without loss of generality, leti>j, and from the norm frequencies assumption, we get
mj+1–im+1Om≥j–N–1– 2(n(j))2/9≥Lm–N+1–1– 2(m)2/9.
Choosesmall enough such that
2(m)2/9+ (m)γ
2
3(1 +β)|lnm+1|
N+1 1+α
≤1.
Thus
mj+1–mi +1Om≥(m)γ, i=j. (3.30)
ForOm+1⊂Om, the small divisor satisfies
√
–1k·ωm+1+mj +1–mi+1Om+1≥1
2(m)
γ
. (3.31)
Using this estimate, we see that
Hence
The remaining terms ofScan be estimated in the same line. Therefore we obtain
|S|D1m×Om+1≤C(m)L2
where we have written 2γ asγ by abuse of notation.
Using the similar discussion in the proof of Lemma (a) and the Cauchy estimate, on the domainD2
Chooseβ,γ small enough, thus
By the analytic inverse function theorems, the equations
Making use of the same way to the other three terms, on the domainDm+1×Om+1, one gets
jm+1–jm,φjm+1–φjm,ψjm+1–ψjm≤(m)16, |j| ≤Lm+1. (3.40)
By virtue of the fact that Sdoes not depend on (Ij,θj,uj,vj) with|j|>Lm+1, we see that Cm+1, and henceCm+1will reduce to the identity at these sites. Then with these bounds we conclude that
Cm+1–CmDm+1×Om+1
∞ ≤(m)
1
6. (3.41)
It remains to verify l.2 withl=m+ 1. WriteHm+1=Hm◦Cm+1(I,θ,u,v) =Hm(I+ ,θ+,u+,v+ϒ), which is in turn equal to
ωm+1·I+m+1u·v+QI,u,v+Pm+1+
|j|≥Lm+2 fj
I,θ,u,v
withPm+1having the form
|j|≤Lm+1
2Ij|j|+|j|2
+
|j|≤Lm+1
uj–vj–j+ϒj4–uj–vj
4 +P3m
+
|j|≤Lm+1
1
0
∂P2m
∂Ij
I+t,θ,u+,vj
+∂P2m
∂uj
I+,θ,u+t,vj
dt. (3.42)
Now we estimate Pm+1. From (3.38), the first term in (3.42) is bounded on the domain Dm+1×Om+1by
2Lm+1
2–3m–5τ(m+1) 2+2β
3 (m)1–γ+ (m)2–2γ≤(m)32. (3.43)
The second term in (3.42) can be similarly estimated onDm+1×Om+1by (m) 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 domainDm+1×Om+1, and we find it is less than
2Lm+1
C(m)(m)
1 3–β(
m)1–γ+C(m)(m)
2 3–β(
m)
2
3–β–γ≤(
m)
5 4 =
m+1. (3.44)
Thus, with (3.17), we obtain that
Pm+1Dm+1×Om+1≤
C(m+ 1)m+1
completing the verification ofl.2 withl=m+ 1 and the proof of Lemma 3.2.
4 Proof of the theorem
lemma (Lemma 3.2) works. Inductively, we get the following sequences:
Dm+1×m+1⊂Dm+1×m+1, Cm+1:D
m+1×m+1→D0,
Hm+1=Hm◦Cm+1=ωm+1·I+m+1u·v+Q+Pm+1+
|j|≥Lm+2 fj.
Let
∞= ∞
!
m=0
m,
D∞=
(I,θ,u,v)∈CZ×CZ×CZ×CZ:Ij=uj=vj= 0,|Imθj|< δ0
2,j∈Z
.
By (3.41),l.2, andl.3, we conclude thatHm,Cmconverges uniformly on the domainD∞× ∞, and
C∞= lim
m→∞C
m,
H∞=ω∞·I+∞u·v+Q.
Thus,TZ× {0} × {0} × {0}is an embedding torus with rotational frequenciesω∞∈∞
of the HamiltonianH∞. Returning to the original HamiltonianH, it has an embedding torusC∞(TZ× {0} × {0} × {0}) with frequenciesω∞. This proves the theorem.
Proof of Theorem1.1 By Sect. 2, Theorem 1.1 is just a corollary of Theorem 3.1.
Acknowledgements
The author thanks Professor Xiaoping Yuan, his advisor, for his constant encouragement and guidance.
Competing interests
The author declares that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 18 February 2018 Accepted: 17 April 2018
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