# On the Reducibility for a Class of Quasi-Periodic Hamiltonian Systems with Small Perturbation Parameter

Authors: Jia Li; Junxiang Xu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian systemx˙=J∇xH, where H(x,t,ε)=(1/2)β(x12+x22)+F(x,t,ε) with β≠0,∂xF(0,t,ε)=O(ε) and ∂xxF(0,t,ε)=O(ε) as ε→0. Without any nondegeneracy condition with respect to ε, we prove that for most of the sufficiently small ε, by a quasi-periodic symplectic transformation, it can be reduced to a quasi-periodic Hamiltonian system with an equilibrium.


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## Body

## 1. Introduction

We first give some definitions and notations for our problem. A functionf(t) is called a quasi-periodic function with frequencies ω=(ω1,ω2,…,ωl) if f(t)=F(ω1t,ω2t,…,ωlt) with θi=ωit, where F(θ1,θ2,…,θl) is 2π periodic in all the arguments θj,j=1,2,…,l. If F(θ)(θ=(θ1,θ2,…,θl)) is analytic on Dρ={θ∈Cl/2πZl∣|Imθi|≤ρ,i=1,2,…,l}, we call f(t) analytic quasi-periodic on Dρ. If all qij(t)(i,j=1,2…,n) are analytic quasi-periodic on Dρ, then the matrix function Q(t)=(qij(t))1≤i,j≤n is called analytic quasi-periodic on Dρ.Iff(t) is analytic quasi-periodic on Dρ, we can write it as Fourier series: (1.1)f(t)=∑k∈Zlfkei〈k,ω〉t.
Define a norm of f by ∥f∥ρ=∑k∈Zl|fk|e|k|ρ. It follows that |fk|≤∥f∥ρe-|k|ρ. If the matrix function Q(t) is analytic quasi-periodic on Dρ, we define the norm of Q by ∥Q∥ρ=n×max1≤i,j≤n∥qij∥ρ. It is easy to verify ∥Q1Q2∥ρ≤∥Q1∥ρ∥Q2∥ρ. The average of Q(t) is denoted by [Q]=([qij])1≤i,j≤n, where (1.2)[qij]=lim⁡T→∞⁡12T∫-TTqij(t)dt.
For the existence of the above limit, see [1].Denote(1.3)D(r,ρ,ε0)={(x,θ,ε)∈Cn×(Cl2πZl)×C∣|x|≤r,θ∈Dρ,|ε|≤ε0},
where x=(x1,x2,…,xn) and |x|=|x1|+|x2|+⋯+|xn|.Letf(x,t,ε) be analytic quasi-periodic of t and analytic in x and ɛ on D(r,ρ,ε0). Then f(x,t,ε) can be expanded as (1.4)f(x,t,ε)=∑m=0∞∑k∈Zlfmk(x)εmei〈k,ω〉t.
Define a norm by (1.5)‖f‖D(r,ρ,ε0)=∑m=0∞∑k∈Zl|fmk|rε0meρ|k|,
where |fmk|r=sup|x|≤r|fmk(x)|. Note that (1.6)‖f1⋅f2‖D(r,ρ,ε0)≤‖f1‖D(r,ρ,ε0)⋅‖f2‖D(r,ρ,ε0).Problems
The reducibility on the linear differential system has been studied for a long time. The well-known Floquet theorem tells us that ifA(t) is a T-periodic matrix, then the linear system ẋ=A(t)x is always reducible to the constant coefficient one by a T-periodic change of variables. However, this cannot be generalized to the quasi-periodic system. In [2], Johnson and Sell considered the quasi-periodic system ẋ=A(t)x, where A(t) is a quasi-periodic matrix. Under some “full spectrum” conditions, they proved that ẋ=A(t)x is reducible. That is, there exists a quasi-periodic nonsingular transformation x=ϕ(t)y, where ϕ(t) and ϕ(t)-1 are quasi-periodic and bounded, such that ẋ=A(t)x is transformed to ẏ=By, where B is a constant matrix.
In [3], Jorba and Simó considered the reducibility of the following linear system:
(1.7)ẋ=(A+εQ(t))x,x∈Rn,
where A is an n×n constant matrix with n different eigenvalues λ1,λ2,…,λn and Q(t) is analytic quasi-periodic with respect to t with frequencies ω=(ω1,ω2,…,ωl). Here ε is a small perturbation parameter. Suppose that the following nonresonance conditions hold:
(1.8)|〈k,ω〉-1+λi-λj|≥α|k|τ,
for all k∈Zl∖{0}, where α>0 is a small constant and τ>l-1. Assume that λj0(ε)(j=1,2,…,n) are eigenvalues of A+ε[Q]. If the following non-degeneracy conditions hold:
(1.9)ddε(λi0(ε)-λj0(ε))|ε=0≠0,∀i≠j,
then authors proved that for sufficiently small ε0>0, there exists a nonempty Cantor subset E⊂(0,ε0), such that for ε∈E, the system (1.7) is reducible. Moreover, meas((0,ε0)∖E)=o(ε0).
Some related problems were considered by Eliasson in [4, 5]. In the paper [4], to study one-dimensional linear Schrödinger equation
(1.10)d2qdt2+Q(ωt)q=Eq,
Eliasson considered the following equivalent two-dimensional quasi-periodic Hamiltonian system:
(1.11)ṗ=(E-Q(ωt))q,q̇=p,
where Q is an analytic quasi-periodic function and E is an energy parameter. The result in [4] implies that for almost every sufficiently large E, the quasi-periodic system (1.11) is reducible. Later, in [5] the author considered the almost reducibility of linear quasi-periodic systems.  Recently, the similar problem was considered by Her and You [6]. Let Cω(Λ,gl(m,C)) be the set of m×m matrices A(λ) depending analytically on a parameter λ in a closed interval Λ⊂R. In [6], Her and You considered one-parameter families of quasi-periodic linear equations
(1.12)ẋ=(A(λ)+g(ω1t,…,ωlt,λ))x,
where A∈Cω(Λ,gl(m,C)), and g is analytic and sufficiently small. They proved that under some nonresonance conditions and some non-degeneracy conditions, there exists an open and dense set 𝒜 in Cω(Λ,gl(m,C)), such that for each A∈𝒜, the system (1.12) is reducible for almost all λ∈Λ.
In 1996, Jorba and Simó extended the conclusion of the linear system to the nonlinear case. In [7], Jorba and Simó considered the quasi-periodic system
(1.13)ẋ=(A+εQ(t))x+εg(t)+h(x,t),x∈Rn,
where A has n different nonzero eigenvalues λi. They proved that under some nonresonance conditions and some non-degeneracy conditions, there exists a nonempty Cantor subset E⊂(0,ε0), such that the system (1.13) is reducible for ε∈E.
In [8], the authors found that the non-degeneracy condition is not necessary for the two-dimensional quasi-periodic system. They considered the two-dimensional nonlinear quasi-periodic system:
(1.14)ẋ=Ax+f(x,t,ε),x∈R2,
where A has a pair of pure imaginary eigenvalues ±-1ω0 with ω0≠0 satisfying the nonresonance conditions
(1.15)|〈k,ω〉|≥α|k|τ,|〈k,ω〉-2ω0|≥α|k|τ
for all k∈Zl∖{0}, where α>0 is a small constant and τ>l-1. Assume that f(0,t,ε)=O(ε) and ∂xf(0,t,ε)=O(ε) as ε→0. They proved that either of the following two results holds:(1)
for∀ε∈(0,ε0), the system (1.14) is reducible to ẏ=By+O(y) as y→0;(2)
there exists a nonempty Cantor subsetE⊂(0,ε0), such that for ε∈E the system (1.14) is reducible to ẏ=By+O(y2) as y→0.
Note that the result (1) happens when the eigenvalue of the perturbed matrix ofA in KAM steps has nonzero real part. But the authors were interested in the equilibrium of the transformed system and obtained a small quasi-periodic solution for the original system.
Motivated by [8], in this paper we consider the Hamiltonian system and we have a better result.


## 2. Main Results

Theorem 2.1.
Consider the following real two-dimensional Hamiltonian system(2.1)ẋ=J∇xH,x∈R2,
where H(x,t,ε)=(1/2)β(x12+x22)+F(x,t,ε) with β≠0, F(x,t,ε) is analytic quasi-periodic with respect to t with frequencies ω=(ω1,ω2,…,ωl) and real analytic with respect to x and ε on D(r,ρ,ε0), and
(2.2)J=(01-10).
Here ε∈(0,ε0) is a small parameter. Suppose that ∂xF(0,t,ε)=O(ε) and ∂xxF(0,t,ε)=O(ε) as ε→0. Moreover, assume that β and ω satisfy
(2.3)|〈k,ω〉|≥α0|k|τ,(2.4)|〈k,ω〉-2β|≥α0|k|τ
for all k∈Zl∖{0}, where α0>0 is a small constant and τ>l-1.
Then there exist a sufficiently smallε*∈(0,ε0] and a nonempty Cantor subset E*⊂(0,ε*), such that for ε∈E*, there exists an analytic quasi-periodic symplectic transformation x=ϕ*(t)y+ψ*(t) on Dρ/2 with the frequencies ω, which changes (2.1) into the Hamiltonian system ẏ=J∇yH*, where H*(y,t,ε)=1/2β*(ε)(y12+y22)+F*(y,t,ε), where F*(y,t,ε)=O(y3) as y→0. Moreover, meas((0,ε*)∖E*)=o(ε*) as ε*→0. Furthermore, β*(ε)=β+O(ε) and ∥ϕ*-Id∥ρ/2+∥ψ*∥ρ/2=O(ε), where Id is the 2-order unit matrix.


## 3. The Lemmas

The proof of Theorem2.1 is based on KAM-iteration. The idea is the same as [7, 8]. When the non-degeneracy conditions do not happen, the small parameter ε is not involved in the nonresonance conditions. So without deleting any parameter, the KAM step will be valid. Once the non-degeneracy conditions occur at some step, they will be kept for ever and we can apply the results with the non-degeneracy conditions. Thus, after infinite KAM steps, the transformed system is convergent to a desired form.We first give some lemmas. LetR=(rij)1≤i,j≤2 be a Hamiltonian matrix. Then we have r11+r22=0. Define a matrix RA=(1/2)dJ with d=r12-r21. Let(3.1)B=12(11-1--1).
It is easy to verify(3.2)B-1RAB=12diag⁡(-1d,--1d),B-1(R-RA)B=12(0σ′--1κ′σ′+-1κ′0),
where σ′=2r11 and κ′=r21+r12.In the same way as in [7, 8], in KAM steps we need to solve linear homological equations. For this purpose we need the following lemma.Lemma 3.1.
Consider the following equation of the matrix:(3.3)Ṗ=AP-PA+R(t),
where A=β(ε)J with |β(ε)|>μ, μ>0 is a constant, and R(t)=(rij(t))1≤i,j≤2 is a real analytic quasi-periodic Hamiltonian matrix on Dρ with frequencies ω. Suppose β(ε) and R are smooth with respect to ε and |εβ′(ε)|≤c0 for ε∈E⊂(0,ε*), where c0 is a constant. Note that here and below the dependence of ε is usually implied and one does not write it explicitly for simplicity. Assume [R]A=0, where [R] is the average of R. Suppose that for ε∈E, the small divisors conditions (2.3) and the following small divisors conditions hold:
(3.4)|〈k,ω〉-2β(ε)|≥α|k|τ′,
where τ′>2τ+l. Let 0<s<ρ and ρ1=ρ-s. Then there exists a unique real analytic quasi-periodic Hamiltonian matrix P(t) with frequencies ω, which solves the homological linear equation (3.3) and satisfies
(3.5)∥P∥ρ1≤cαsv∥R∥ρ,‖ε∂εP‖ρ1≤cα2sv′(‖R‖ρ+‖ε∂εR‖ρ),
where v=τ′+l, v′=2τ′+l and c>0 is a constant.Remark 3.2.
The subsetE of (0,ε*) is usually a Cantor set and so the derivative with respect to ε should be understood in the sense of Whitney [9].Proof.
LetP¯=B-1PB, where B is defined by (3.1). Similarly, define A¯,R¯,R¯A. Then (3.3) becomes
(3.6)P¯̇=A¯P¯-P¯A¯+R¯(t),
where
(3.7)A¯=diag⁡(-1β,--1β).
Moreover, R¯A and R¯-R¯A have the same forms as (3.2) and (3.2), respectively
Noting that[R]A=0, we have [R¯]A=0. Write P¯=(p¯ij)i,j and R¯=(r¯ij)i,j. Obviously, we have r¯11=-r¯22 with [r¯ii]=0.
Insert the Fourier series ofP¯ and R¯ into (3.6). Then it follows that p¯ii0=0,p¯iik=r¯iik/(〈k,ω〉-1) for k≠0, and
(3.8)p¯ijk=r¯ijk-1(〈k,ω〉±2β)fori≠j.
Since R¯ is analytic on Dρ, we have |R¯k|≤∥R¯∥ρe-|k|ρ. So it follows
(3.9)‖P¯‖ρ-s≤∑k∈Zl|P¯k|e|k|(ρ-s)≤cαsv‖R‖ρ.
Note that here and below we always use c to indicate constants, which are independent of KAM steps.
SinceA and R(t) are real matrices, it is easy to obtain that P(t) is also a real matrix. Obviously, it follows that p¯11=-p¯22 and the trace of the matrix P¯ is zero. So is the trace of P. Thus, P is a Hamiltonian matrix.
Now we estimate∥ε∂P/∂ε∥ρ1. We only consider p¯12 and p¯21 since p¯11 and p¯22 are easy.
Fori≠j we have
(3.10)dp¯ijk(ε)dε=±2β′(ε)r¯ijk-(〈k,ω〉±2β)r¯ijk(ε)'--1(〈k,ω〉±2β)2.
Then, in the same way as above we obtain the estimate for∥ε(∂P/∂ɛ)∥ρ1.The following lemma will be used for the zero order term in KAM steps.Lemma 3.3.
Consider the equation(3.11)ẋ=Ax+g(t),
where A is the same as in Lemma 3.1, and g is real analytic quasi-periodic in t on Dρ with frequencies ω and smooth with respect to ε. Suppose that the small divisors conditions (3.4) hold. Then there exists a unique real analytic quasi-periodic solution x(t) with frequencies ω, which satisfies
(3.12)∥x∥ρ1≤cαsv∥g∥ρ,‖ε∂x∂ɛ‖ρ1≤cα2sv′(‖g‖ρ+‖ε∂g∂ε‖ρ),
where s,ρ1,v, v′ are defined in Lemma 3.1.Proof.
Similarly, letx¯=B-1x,A¯=B-1AB and g¯(t)=B-1g(t). Then (3.11) becomes
(3.13)x¯̇=A¯x¯+g¯(t),
where A¯=diag(-1β,--1β). Expanding x¯=(x¯1,x¯2) and g¯=(g¯1,g¯2) into Fourier series and using (3.13), we have
(3.14)x¯ik=g¯ik-1(〈k,ω〉+(-1)iβ).
Using 2k in place of k in (3.4), we have
(3.15)|〈k,ω〉-β(ε)|≥α2|k|τ′.
Thus, in the same way as the proof of Lemma 3.1, we can estimate ∥x∥ρ1 and ∥ɛ∂ɛx∥ρ1. We omit the details.The following lemma is used in the estimate of Lebesgue measure for the parameterɛ in the case of non-degeneracy.Lemma 3.4.
Letψ(ε)=σεN+εNf(ε), where N is a positive integer and f satisfies that f(ε)→0 as ε→0 and |f′(ε)|≤c for ε∈(0,ε*). Let ϕ(ε)=〈k,ω〉-2β-ψ(ε). Let
(3.16)O={ε∈(0,ε*)∣|ϕ(ε)|≥α|k|τ′,∀k≠0},
where τ′≥2τ+l, α≤(1/2)α0,σ≠0. Suppose that the small condition (2.4) holds. Then when ε* is sufficiently small, one has
(3.17)meas⁡(0,ε*)∖O≤cαα02ε*N+1,
where c is a constant independent of α0,α,ε*Proof.
Let(3.18)Ok={ε∈(0,ε*)∣|ϕ(ε)|<α|k|τ′}.
By assumption, if ε* is sufficient small, we have that |ψ(ε)|≤2σεN and |ψ′(ε)|≥(σ/2)εN-1 for ε∈(0,ε*). If εN≤α0/(4σ|k|τ), by (2.4) we have
(3.19)|ϕ(ε)|≥|〈k,ω〉-2β|-|ψ(ε)|≥α|k|τ′.
Thus, we only consider the case thatε*N≥εN≥(α0/(4σ|k|τ)). We have |k|≥(α0/(4σε*N))1/τ=K. Since
(3.20)|ϕ′(ε)|=|ψ′(ε)|≥σ2εN-1≥α08|k|τε*,
we have meas(Ok)≤((2α)/|k|τ′)×((8|k|τε*)/α0)=(16αε*)/(|k|τ′-τα0). So
(3.21)meas⁡((0,ɛ*)∖0)≤∑|k|≥Kmeas⁡(Ok)≤16αα0ε*∑|k|≥K1|k|τ′-τ≤cαα0ε*Kl-τ′+τ≤cαα02ε*N+1,
where c is a constant independent of α0,α, and ε*.Below we give a lemma with the non-degeneracy conditions.Lemma 3.5.
Consider the real nonlinear Hamiltonian systemẋ=J∇xH, where
(3.22)H(x,t,ε)=12β(x12+x22)+F(x,t,ε)withβ≠0.
Suppose that F(x,t,ε) is analytic quasi-periodic with respect to t with frequencies ω and real analytic with respect to x and ɛ on D(r,ρ,ε0). Let f(x,t,ε)=J∇xF(x,t,ε). Assume that f(0,t,ε)=O(ε2m0) and ∂xf(0,t,ε)=O(εm0) as ε→0, where m0 is a positive integer. Let Q(t,ε)=∂xf(0,t,ε)=∑k≥m0Qk(t)εk. Suppose there exists m0≤k≤2m0-1 such that [Qk]A≠0 and the nonresonance conditions (2.3) and (2.4) hold. Then, for sufficiently small ε*>0, there exists a nonempty Cantor subset E*⊂(0,ε*), such that for ε∈E*, there exists a quasi-periodic symplectic transformation x=ϕ*(t)y+ψ*(t) with the frequencies ω, which changes the Hamiltonian system to ẏ=J∇yH*, where
(3.23)H*(y,t,ε)=12β*(ε)(y12+y22)+F*(y,t,ε),
where F*(y,t,ε)=O(y3) as y→0. Moreover, meas((0,ε*)∖E*)=O(ε*m0+1) as ε*→0. Furthermore, β*(ε)=β+O(εm0) and ∥ϕ*-Id∥ρ/2+∥ψ*∥ρ/2=O(εm0).Proof
KAM Step
The proof is based on a modified KAM iteration. In spirit, it is very similar to [7, 8]. The important thing is to make symplectic transformations so that the Hamiltonian structure can be preserved. Note that [Qk]A≠0 for some m0≤k≤2m0-1 is a non-degeneracy condition.
Consider the following Hamiltonian system(3.24)ẋ=Ax+f(x,t,ε),
where A=β(ε)J and f is analytic quasi-periodic with respect to t with frequencies ω and real analytic with respect to x and ε on D=D(r,ρ,ε*).
Let∥f∥D≤αrε̃ and ∥ε∂εf∥D≤αrε̃. Let Q(t,ε)=∂xf(0,t,ε),g(t,ε)=f(0,t,ε) and
(3.25)h(x,t,ε)=f(x,t,ε)-g(t,ε)-Q(t,ε)x.
Then h is the higher-order term of f. Moreover, the matrix Q(t,ε) is Hamiltonian. Let [Q]A=β̂(ε)J.
The system (3.24) is written as
(3.26)ẋ=(A++R(t,ε))x+g(t,ε)+h(x,t,ε),
where A+=A+[Q]A=β+(ε)J and R=Q-[Q]A. By assumption we have
(3.27)‖g‖ρ≤αrε̃,‖Q‖ρ≤αε̃,‖h‖D≤3αrε̃.
Moreover, we have
(3.28)‖ε∂εg‖ρ≤αrε̃,‖ε∂εQ‖ρ≤αε̃,‖ε∂εh‖D≤3αrε̃.
Now we want to construct the symplectic change of variablesx=T′y=eP(t)y to (3.26), where P is a Hamiltonian matrix to be defined later. Then we have
(3.29)ẏ=(e-P(A++R-Ṗ)eP+e-P(ṖeP-ddteP(t)))y+e-Pg(t,ε)+e-Ph(ePy,t,ε).
Let W=eP-I-P and W̃=e-P-I-P. Then the system (3.29) becomes
(3.30)ẏ=(A++R-Ṗ+A+P-PA+)y+Q′y+e-Pg(t,ε)+e-Ph(ePy,t,ε),
where
(3.31)Q′=-P(R-Ṗ)+(R-Ṗ)P-P(A++R-Ṗ)P-P(A++R-Ṗ)W+(A++R-Ṗ)W+W̃(A++R-Ṗ)eP+e-P(ṖeP-ddteP).
We would like to have(3.32)Ṗ-A+P+PA+=R,
where R=Q-[Q]A. Suppose the small divisors conditions (2.3) hold. Let E+⊂(0,ε*) be a subset such that for ε∈E+ the small divisors conditions hold:
(3.33)|〈k,ω〉-2β+(ε)|≥α+|k|τ′,∀k∈Zl∖{0},
where τ′>2τ+l. By Lemma 3.1, we have a quasi-periodic Hamiltonian matrix P(t) with frequencies ω to solve the above equation with the following estimates:
(3.34)‖P‖ρ-s≤c‖Q‖ρα+sv≤cε̃sv,‖ε∂P∂ε‖ρ-s≤cα+2sv′(‖Q‖ρ+‖ε∂Q∂ε‖ρ)≤cε̃α+sv′,
where v=τ′+l,v′=2τ′+l and c>0 is a constant. Then the system (3.30) becomes
(3.35)ẏ=A+y+f′(y,t,ε),
where f′=Q′y+e-Pg(t,ε)+e-Ph(ePy,t,ε).
By Lemma3.3, let us denote by x̲ the solution of ẋ=A+x+g′(t,ε) on Dρ-2s, where g′=e-Pg(t,ε). Then, by Lemma 3.3 we have
(3.36)‖x̲‖ρ-2s≤c‖g‖ρ-sα+sv≤crε̃sv,‖ε∂x̲∂ε‖ρ-2s≤cα+2sv′(‖g‖ρ-s+‖ε∂g∂ε‖ρ-s)≤crε̃α+sv′.
Under the symplectic change of variablesy=T′′x+=x̲+x+, the Hamiltonian system (3.35) is changed to
(3.37)ẋ+=A+x++f+(x+,t,ε),
where A+=β+J and
(3.38)f+=Q′⋅T′′+e-Ph∘T′∘T′′.
Let the symplectic transformationT=T′∘T′′. Then x=Tx+=ϕ(t)x++ψ(t), where ϕ(t)=eP(t) and ψ(t)=eP(t)x̲(t). It is easy to obtain that if ∥P∥ρ-2s≤1/2, then
(3.39)‖ϕ-I‖ρ-2s≤cɛ̃sv,‖ε∂εϕ‖ρ-2s≤cε̃α+sv′,‖ψ‖ρ-2s≤crɛ̃sv,‖ε∂εψ‖ρ-2s≤crε̃α+sv′.
Under the symplectic change of variables x=Tx+, the Hamiltonian system (3.24) becomes (3.37).
Below we give the estimates forA+ and f+. Obviously, it follows that A+(ε)-A=[Q]A=β̂(ε)J and
(3.40)|β+(ε)-β(ε)|=|β̂(ε)|≤cαε̃,|ε(β+′(ε)-β′(ε))|=|εβ′̂(ε)|≤cαε̃.
By (3.38) we have
(3.41)f+(x+,t,ε)=Q′(t)(x++x̲(t))+e-P(t)h(eP(t)(x++x̲(t)),t,ε).
Let ρ+=ρ-2s, and r+=ηr with η≤1/8. If cε̃/α+sv+v′≤η, it follows that ∥x̲∥ρ-2s≤(1/8)r. Let D+=D(r+,s+,ε*). Note that Q′ and h only consist of high-order terms of P and x, respectively. It is easy to see |eP(t)(x++x̲(t))|≤4ηr≤r. By all the estimates (3.27), (3.28), (3.34), and (3.36), and using usual technique of KAM estimate, we have
(3.42)‖f+‖D+≤cε̃2s2vηr+cαrε̃η2≤(cε̃s2v+cαη)r+ε̃,‖ε∂εf+‖D+≤cε̃2α+sv+v′ηr+cαrε̃η2≤(cε̃αsv+v′+cαη)r+ε̃.
Let α+=α/2 and η=cε̃/(α2sv+v′). Then we have
(3.43)‖f+‖D+≤cα+r+ηε̃=α+r+ε̃+,ε̃+=cηε̃.
Similarly, we have
(3.44)‖ε∂εf+‖D+≤α+r+ε̃+.
Note that KAM steps only make sense for the small parameterε satisfying small divisors conditions. However, by Whitney’s extension theorem, for convenience all the functions are supposed to be defined for ε on [0,ε*].KAM Iteration
Now we can give the iteration procedure in the same way as in [7] and prove its convergence.
At the initial step, letf0=f. Let f(x,t,ε)=f(0,t,ε)+∂xf(0,t,ε)x+h(x,t,ε). By assumption, if ε* is sufficiently small, we have that for all ε∈[0,ε*](3.45)|f(0,t,ε)|≤cε2m0,|∂xf(0,t,ε)|≤cεm0,|ε∂εf(0,t,ε)|≤cε2m0,|ε∂ε∂xf(0,t,ε)|≤cεm0.
Moreover,
(3.46)|h(x,t,ε)|≤c|x|2,|ε∂εh(x,t,ε)|≤c|x|2,∀|x|≤εm0,∀ε∈[0,ε*].
Let r0=ɛm0,ρ0=ρ,s0=ρ0/8,D0=D(r0,ρ0,ε*), and ε̃0=cεm0/α0. Then we have
(3.47)|f0|D0≤α0r0ε̃0,|ε∂εf0|D0≤α0r0ε̃0.
For n≥1, let
(3.48)αn=αn-12,sn=sn-12,ρn=ρn-1-2sn-1,ηn-1=cε̃n-1αn-12sn-1v+v′,rn=ηn-1rn-1,ε̃n=cηn-1ε̃n-1.
Then we have a sequence of quasi-periodic symplectic transformations{Tn} satisfying Tnx=ϕn(t)x+ψn(t) with
(3.49)‖ϕn-I‖ρn+1≤cε̃nsnv,‖ψn‖ρn+1≤crnε̃nsnv.
Let Tn=T0∘T1⋯∘Tn-1. Then under the transformation x=Tny the Hamiltonian system ẋ=A0x+f0(x,t,ε) is changed to ẏ=Any+fn(y,t,ε).
Moreover,An(ε)=βn(ε)J satisfies An+1-An=[Qn]A and
(3.50)|βn+1(ε)-βn(ε)|≤cαnε̃n,|ε(βn+1′(ε)-βn′(ε))|≤cαnε̃n,(3.51)‖fn‖Dn≤αnrnε̃n.Convergence
By the above definitions we haveηn/ηn-1=cε̃n/ε̃n-1=cηn-1. Thus, we have ηn≤cηn-12 and so cηn≤(cηn-1)2≤(cη0)2n. Note that η0=cε̃0/(α02s0v+v′)≤cεm0/(α02ρ0v+v′). Suppose that ε* is sufficiently small such that for 0<ε<ε* we have cη0≤1/2.Tn are affine, so are Tn with Tnx=ϕn(t)x+ψn(t). By the estimates (3.49) it is easy to prove that ϕn(t) and ψn(t) are convergent and so Tn is actually convergent on the domain D(r/2,ρ/2). Let Tn→T* and T*x=ϕ*(t)x+ψ*(t). It is easy to see that the estimates for ϕ* and ψ* in Theorem 2.1 hold.
Using the estimate forfn and Cauchy’s estimate, we have |fn(0,t,ε)|≤αnrnε̃n→0 and |∂xfn(0,t,ε)|≤αnε̃n→0 as n→∞. Let fn→f*. Then it follows that f*(x,t,ε)=O(x2).
By the estimates (3.50) for βn we have βn→β*. Thus, by the quasi-periodic symplectic transformation x=T*y, the original system is changed to ẏ=A*y+f*(y,t,ε) with A*=β*J.Estimate of Measure
Let(3.52)En={ε∈(0,ε*)∣|〈ω,k〉-2βn(ε)|≥αn|k|τ′}.
Note that βn=β1+ψ, where ψ=∑j=1n-1βj+1-βj,β1=β+β̂, and β̂J=[Q]A. Note that ε̃1=cε̃02/(α02s0v+v′) and ε̃0=cεm0/α0. By the estimates (3.50), we have ψ(ε)=O(ε2m0) and εψ′(ɛ)=O(ε2m0). By assumption, [Q]A is analytic with respect to ɛ and there exists m0≤N≤2m0-1 such that [Q]A=δεN+O(εN+1) with δ≠0. Thus, β1(ε)=β+δεN+O(εN+1). By Lemma 3.4, we have meas((0,ε*)-En)≤c(αn/α02)ε*N+1. Let E*=⋂n≥1En. By αn=α0/2n, it follows that meas((0,ε*)-E*)≤cε*N+1/α0. Thus Lemma 3.5 is proved.


## 4. Proof of Theorem2.1

As we pointed previously, once the non-degeneracy conditions are satisfied in some KAM step, the proof is complete by Lemma3.5. If the non-degeneracy conditions never happen, the small parameter ɛ does not involve into the small divisors and so the systems are analytic in ε. To prepare for KAM iteration, we need a preliminary step to change the original system to a suitable form.Preliminary Step
We first give the preliminary KAM step. Let(4.1)ẋ=Ax+f(x,t,ε),
where A=βJ and f=J∇xF. By Lemma 3.3, denote by x̲ the solution of ẋ=Ax+f(0,t,ε) on D3ρ/4. Under the change of variables x=T0x+=x̲+x+, the Hamiltonian system (2.1) becomes
(4.2)ẋ+=Ax++f1(x+,t,ε),
where f1(x+,t,ε)=f(x̲+x+,t,ε)-f(0,t,ε) satisfying f1(0,t,ε)=O(ε2) and ∂x+f1(0,t,ε)=O(ε).KAM Step
The next step is almost the same as the proof of Lemma3.5 and even more simple. In the KAM iteration, we only need consider the case that the non-degeneracy condition never happens. In this case, the normal frequency has no shift, which is equivalent to An=A for all n≥1 in the KAM steps in the above nondegenerate case. Moreover, the small divisors conditions are always the initial ones as (2.3) and (2.4) and are independent of the small parameter ε. Thus, we need not delete any parameter. Moreover, the analyticity in ε remains in the KAM steps, which makes the estimate easier.  At the first step, we consider ẋ=Ax+f1(x,t,ε). In the same way as the case of nondegenerate case, let r1=ε,ρ1=3ρ/4,ε1=ε0,D1=D(r1,ρ1,ε1), and ε̃1=cε/α0. Then we have ∥f1∥D1≤α0r1ε̃1.
Atnth step, we consider the Hamiltonian system
(4.3)ẋ=Ax+fn(x,t,ε),
where fn is analytic quasi-periodic with respect to t with frequencies ω and real analytic with respect to x and ɛ on Dn=D(rn,ρn,εn). Moreover, ∥fn∥Dn≤α0rnε̃n. Suppose
(4.4)Qn(t,ε)=∂xfn(0,t,ε)=O(ε2n-1),fn(0,t,ε)=O(ε2n).
SinceQn is analytic with respect to ε, it follows that
(4.5)Qn=∑k=2n-1∞Qnkεk.
Truncating the above power series of ε, we let
(4.6)Rn(t,ε)=∑k=2n-12n-1Qnkεk,Q̃n=Qn-Rn.
Because the non-degeneracy conditions do not happen in KAM steps, we must have[Rn]A=0. In the same way as the proof of Lemma 3.5, we have a quasi-periodic symplectic transformation Tn with Tnx=ϕn(t)x+ψn(t) satisfying (3.49). Let Tn=T1∘T2⋯∘Tn-1.
By the transformationx=Tny, the system (4.3) is changed to
(4.7)ẏ=Ay+fn+1(y,t,ε),
where fn+1=Q̃n·Tn′′+Qn′·Tn′′+e-Pn·hn∘Tn=Q̃n(x̲n+y)+Qn′(x̲n+y)+e-Pnhn(ePn(x̲n+y)).
The last two terms can be estimated similarly as those of (3.41). Note that
(4.8)Q̃n=Qn-Rn=∑k≥2nQnkεk
only consists of the higher order terms of ε. So, in the same way as [8, 10], we use the technique of shriek of the domain interval of ε to estimate the first term.
Letr1=ε,ρ1=3ρ/4,ε1=ε0  and s1=ρ/16.
Definesn+1=sn/2,ρn+1=ρn-2sn,ηn=(1/8)e-(4/3)n,rn+1=ηnrn, δn=1-(2/3)n and εn+1=δnεn. Let Dn+1=D(rn+1,ρn+1,εn+1).
Ifcε̃n/sn2v≤ηn<(1/8), it follows that
(4.9)‖fn+1‖Dn+1≤(α0ε̃ne-(4/3)n+(cε̃nsnv)2)ηnrn+cα0rnε̃nηn2≤α0rn+1ε̃n+1,
where ε̃n+1=cηnε̃n. Moreover, it is easy to see
(4.10)∂xfn+1(0,t,ε)=O(ε2n),fn+1(0,t,ε)=O(ε2n+1).
Now we verify cε̃n/sn2v≤ηn<1/8. Let Gn=cε̃n/sn2v. By Gn=ce-(4/3)n-116vGn-1, it follows that
(4.11)Gn=(c16v)n-1e-[(4/3)n-1+(4/3)n-2+⋯+(4/3)1]G1=(c16v)n-1e4e-4(4/3)n-1G1.
Note that G1=cε̃1/s12v. If ε̃1 is sufficiently small, we have cɛ̃n/sn2v=Gn≤ηn.
Note that(crnε̃n/snv)→0 and (cε̃n/(ηnsnv))→0 as n→∞, and ε̃n≤csn2vGn. Let ε*=∏n≥1(1-(2/3)n)ε0. Thus, in the same way as before we can prove the convergence of the KAM iteration for all ε∈(0,ε*) and obtain the result of Theorem 2.1. We omit the details.Remark 4.1.
As suggested by the referee, we can also introduce an outer parameter to consider the Hamiltonian functionH(x,t,ε)=〈ω,I〉+(1/2)(β*+σ(ε))(x12+x22)+F(x,t,ε), where (θ,I) are the angle variable and the action variable and x=(x1,x2) are a pair of normal variables. In the same way as in [11], σ(ε) is the modified term of the normal frequency. Then by some technique as in [11–13], we can also prove Theorem 2.1.



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*Source: 354063-2011-07-24.xml*