AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation35406310.1155/2011/354063354063Research ArticleOn the Reducibility for a Class of Quasi-Periodic Hamiltonian Systems with Small Perturbation ParameterLiJia1XuJunxiang1RogovchenkoYuri V.1Department of MathematicsSoutheast UniversityNanjing 210096Chinaseu.ac.bd20112407201120110112201018042011250520112011Copyright © 2011 Jia Li and Junxiang Xu.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system x˙=JxH, where H(x,t,ε)=(1/2)β(x12+x22)+F(x,t,ε) with β0,xF(0,t,ε)=O(ε) and xxF(0,t,ε)=O(ε) as ε0. 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.

1. Introduction

We first give some definitions and notations for our problem. A function f(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={θCl/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))1i,jn is called analytic quasi-periodic on Dρ.

If f(t) is analytic quasi-periodic on Dρ, we can write it as Fourier series: f(t)=kZlfkeik,ωt. Define a norm of f by fρ=kZl|fk|e|k|ρ. It follows that |fk|fρe-|k|ρ. If the matrix function Q(t) is analytic quasi-periodic on Dρ, we define the norm of Q by Qρ=n×max1i,jnqijρ. It is easy to verify Q1Q2ρQ1ρQ2ρ. The average of Q(t) is denoted by [Q]=([qij])1i,jn, where [qij]=limT12T-TTqij(t)dt. For the existence of the above limit, see .

Denote D(r,ρ,ε0)={(x,θ,ε)Cn×(Cl2πZl)×C|x|r,  θDρ,  |ε|ε0}, where x=(x1,x2,,xn) and |x|=|x1|+|x2|++|xn|.

Let f(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 f(x,t,ε)=m=0kZlfmk(x)εmeik,ωt. Define a norm by fD(r,ρ,ε0)=m=0kZl|fmk|rε0meρ|k|, where |fmk|r=sup|x|r|fmk(x)|. Note that f1f2D(r,ρ,ε0)f1D(r,ρ,ε0)f2D(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 if A(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 , 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 , Jorba and Simó considered the reducibility of the following linear system: ẋ=(A+εQ(t))x,xRn, 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: |k,ω-1+λi-λj|α|k|τ, for all kZl{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: ddε(λi0(ε)-λj0(ε))|ε=00,ij, then authors proved that for sufficiently small ε0>0, there exists a nonempty Cantor subset E(0,ε0), such that for εE, 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 , to study one-dimensional linear Schrödinger equation d2qdt2+Q(ωt)q=Eq, Eliasson considered the following equivalent two-dimensional quasi-periodic Hamiltonian system: ṗ=(E-Q(ωt))q,q̇=p, where Q is an analytic quasi-periodic function and E is an energy parameter. The result in  implies that for almost every sufficiently large E, the quasi-periodic system (1.11) is reducible. Later, in  the author considered the almost reducibility of linear quasi-periodic systems.  Recently, the similar problem was considered by Her and You . 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 ΛR. In , Her and You considered one-parameter families of quasi-periodic linear equations ẋ=(A(λ)+g(ω1t,,ωlt,λ))x, where ACω(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 A𝒜, the system (1.12) is reducible for almost all λΛ.

In 1996, Jorba and Simó extended the conclusion of the linear system to the nonlinear case. In , Jorba and Simó considered the quasi-periodic system ẋ=(A+εQ(t))x+εg(t)+h(x,t),xRn, 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 εE.

In , 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: ẋ=Ax+f(x,t,ε),xR2, where A has a pair of pure imaginary eigenvalues ±-1ω0 with ω00 satisfying the nonresonance conditions |k,ω|α|k|τ,|k,ω-2ω0|α|k|τ for all kZl{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 ε0. They proved that either of the following two results holds:

for ε(0,ε0), the system (1.14) is reducible to ẏ=By+O(y) as y0;

there exists a nonempty Cantor subset E(0,ε0), such that for εE the system (1.14) is reducible to ẏ=By+O(y2) as y0.

Note that the result (1) happens when the eigenvalue of the perturbed matrix of A 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 , in this paper we consider the Hamiltonian system and we have a better result.

2. Main ResultsTheorem 2.1.

Consider the following real two-dimensional Hamiltonian system ẋ=JxH,xR2, where H(x,t,ε)=(1/2)β(x12+x22)+F(x,t,ε) with β0, 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 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 ε0. Moreover, assume that β and ω satisfy |k,ω|α0|k|τ,|k,ω-2β|α0|k|τ for all kZl{0}, where α0>0 is a small constant and τ>l-1.

Then there exist a sufficiently small ε*(0,ε0] and a nonempty Cantor subset E*(0,ε*), such that for εE*, 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=JyH*, where H*(y,t,ε)=1/2β*(ε)(y12+y22)+F*(y,t,ε), where F*(y,t,ε)=O(y3) as y0. Moreover, meas((0,ε*)E*)=o(ε*) as ε*0. Furthermore, β*(ε)=β+O(ε) and ϕ*-Idρ/2+ψ*ρ/2=O(ε), where Id is the 2-order unit matrix.

3. The Lemmas

The proof of Theorem 2.1 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. Let R=(rij)1i,j2 be a Hamiltonian matrix. Then we have r11+r22=0. Define a matrix RA=(1/2)dJ with d=r12-r21. LetB=12(11-1--1). It is easy to verifyB-1RAB=12diag(-1d,--1d),B-1(R-RA)B=12(0σ--1κσ+-1κ0), 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: Ṗ=AP-PA+R(t), where A=β(ε)J with |β(ε)|>μ, μ>0 is a constant, and R(t)=(rij(t))1i,j2 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 εE(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 εE, the small divisors conditions (2.3) and the following small divisors conditions hold: |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 Pρ1cαsvRρ,εεPρ1cα2sv(Rρ+εεRρ), where v=τ+l, v=2τ+l and c>0 is a constant.

Remark 3.2.

The subset E of (0,ε*) is usually a Cantor set and so the derivative with respect to ε should be understood in the sense of Whitney .

Proof.

Let P¯=B-1PB, where B is defined by (3.1). Similarly, define A¯,R¯,R¯A. Then (3.3) becomes P¯̇=A¯P¯-P¯A¯+R¯(t), where 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 of P¯ and R¯ into (3.6). Then it follows that p¯ii0=0,p¯iik=r¯iik/(k,ω-1) for k0, and p¯ijk=r¯ijk-1(k,ω±2β)for  ij. Since R¯ is analytic on Dρ, we have |R¯k|R¯ρe-|k|ρ. So it follows P¯ρ-skZl|P¯k|e|k|  (ρ-s)cαsvRρ. Note that here and below we always use c to indicate constants, which are independent of KAM steps.

Since A 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 εP/ερ1. We only consider p¯12 and p¯21 since p¯11 and p¯22 are easy.

For ij we have 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 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 xρ1cαsvgρ,εxɛρ1cα2sv(gρ+εgερ), where s,ρ1,v, v are defined in Lemma 3.1.

Proof.

Similarly, let x¯=B-1x,A¯=B-1AB and g¯(t)=B-1g(t). Then (3.11) becomes 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 x¯ik=g¯ik-1(k,ω+(-1)iβ). Using 2k in place of k in (3.4), we have |k,ω-β(ε)|α2|k|τ. Thus, in the same way as the proof of Lemma 3.1, we can estimate xρ1 and ɛɛxρ1. We omit the details.

The following lemma is used in the estimate of Lebesgue measure for the parameter ɛ in the case of non-degeneracy.

Lemma 3.4.

Let ψ(ε)=σεN+εNf(ε), where N is a positive integer and f satisfies that f(ε)0 as ε0 and |f(ε)|c for ε(0,ε*). Let ϕ(ε)=k,ω-2β-ψ(ε). Let O={ε(0,ε*)|ϕ(ε)|α|k|τ,  k0}, where τ2τ+l, α(1/2)α0,σ0. Suppose that the small condition (2.4) holds. Then when ε* is sufficiently small, one has meas(0,ε*)Ocαα02ε*N+1, where c is a constant independent of α0,α,ε*

Proof.

Let 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 εNα0/(4σ|k|τ), by (2.4) we have |ϕ(ε)||k,ω-2β|-|ψ(ε)|α|k|τ.

Thus, we only consider the case that ε*NεN(α0/(4σ|k|τ)). We have |k|(α0/(4σε*N))1/τ=K. Since |ϕ(ε)|=|ψ(ε)|σ2εN-1α08|k|τε*, we have meas(Ok)((2α)/|k|τ)×((8|k|τε*)/α0)=(16αε*)/(|k|τ-τα0). So meas((0,ɛ*)0)|k|Kmeas(Ok)16αα0ε*|k|K1|k|τ-τcαα0ε*Kl-τ+τcαα02ε*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 system ẋ=JxH, where H(x,t,ε)=12β(x12+x22)+F(x,t,ε)with  β0. 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)=JxF(x,t,ε). Assume that f(0,t,ε)=O(ε2m0) and xf(0,t,ε)=O(εm0) as ε0, where m0 is a positive integer. Let Q(t,ε)=xf(0,t,ε)=km0Qk(t)εk. Suppose there exists m0k2m0-1 such that [Qk]A0 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 εE*, there exists a quasi-periodic symplectic transformation x=ϕ*(t)y+ψ*(t) with the frequencies ω, which changes the Hamiltonian system to ẏ=JyH*, where H*(y,t,ε)=12β*(ε)(y12+y22)+F*(y,t,ε), where F*(y,t,ε)=O(y3) as y0. Moreover, meas((0,ε*)E*)=O(ε*m0+1) as ε*0. Furthermore, β*(ε)=β+O(εm0) and ϕ*-Idρ/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]A0 for some m0k2m0-1 is a non-degeneracy condition.

Consider the following Hamiltonian system 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 fDαrε̃ and εεfDαrε̃. Let Q(t,ε)=xf(0,t,ε),g(t,ε)=f(0,t,ε) and 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 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 gραrε̃,Qραε̃,hD3αrε̃. Moreover, we have εεgραrε̃,εεQραε̃,εεhD3αrε̃.

Now we want to construct the symplectic change of variables x=Ty=eP(t)y to (3.26), where P is a Hamiltonian matrix to be defined later. Then we have 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 ẏ=(A++R-Ṗ+A+P-PA+)y+Qy+e-Pg(t,ε)+e-Ph(ePy,t,ε), where 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 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 εE+ the small divisors conditions hold: |k,ω-2β+(ε)|α+|k|τ,kZl{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: Pρ-scQρα+svcε̃sv,εPερ-scα+2sv(Qρ+εQερ)cε̃α+sv, where v=τ+l,  v=2τ+l and c>0 is a constant. Then the system (3.30) becomes ẏ=A+y+f(y,t,ε), where f=Qy+e-Pg(t,ε)+e-Ph(ePy,t,ε).

By Lemma 3.3, 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 x̲ρ-2scgρ-sα+svcrε̃sv,εx̲ερ-2scα+2sv(gρ-s+εgερ-s)crε̃α+sv.

Under the symplectic change of variables y=T′′x+=x̲+x+, the Hamiltonian system (3.35) is changed to ẋ+=A+x++f+(x+,t,ε), where A+=β+J and f+=QT′′+e-PhTT′′.

Let the symplectic transformation T=TT′′. 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 Pρ-2s1/2, then ϕ-Iρ-2scɛ̃sv,εεϕρ-2scε̃α+sv,ψρ-2scrɛ̃sv,εεψρ-2scrε̃α+sv. Under the symplectic change of variables x=Tx+, the Hamiltonian system (3.24) becomes (3.37).

Below we give the estimates for A+ and f+. Obviously, it follows that A+(ε)-A=[Q]A=β̂(ε)J and |β+(ε)-β(ε)|=|β̂(ε)|cαε̃,|ε(β+(ε)-β(ε))|=|εβ̂(ε)|cαε̃. By (3.38) we have 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 η1/8. If cε̃/α+sv+vη, it follows that x̲ρ-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))|4ηrr. By all the estimates (3.27), (3.28), (3.34), and (3.36), and using usual technique of KAM estimate, we have f+D+cε̃2s2vηr+cαrε̃η2(cε̃s2v+cαη)r+ε̃,εεf+D+cε̃2α+sv+vηr+cαrε̃η2(cε̃αsv+v+cαη)r+ε̃. Let α+=α/2 and η=cε̃/(α2sv+v). Then we have f+D+cα+r+ηε̃=α+r+ε̃+,  ε̃+=cηε̃. Similarly, we have εεf+D+α+r+ε̃+.

Note that KAM steps only make sense for the small parameter ε 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  and prove its convergence.

At the initial step, let f0=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*]|f(0,t,ε)|cε2m0,|xf(0,t,ε)|cεm0,|εεf(0,t,ε)|cε2m0,|εεxf(0,t,ε)|cεm0. Moreover, |h(x,t,ε)|c|x|2,|εεh(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 |f0|D0α0r0ε̃0,|εεf0|D0α0r0ε̃0. For n1, let 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 ϕn-Iρn+1cε̃nsnv,ψnρn+1crnε̃nsnv. Let Tn=T0T1Tn-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 |βn+1(ε)-βn(ε)|cαnε̃n,|ε(βn+1(ε)-βn(ε))|cαnε̃n,fnDnαnrnε̃n.

Convergence

By the above definitions we have ηn/ηn-1=cε̃n/ε̃n-1=cηn-1. Thus, we have ηncηn-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 cη01/2.  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 TnT* 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 for fn and Cauchy’s estimate, we have |fn(0,t,ε)|αnrnε̃n0 and |xfn(0,t,ε)|αnε̃n0 as n. Let fnf*. Then it follows that f*(x,t,ε)=O(x2).

By the estimates (3.50) for βn we have βnβ*. 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 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 m0N2m0-1 such that [Q]A=δεN+O(εN+1) with δ0. 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*=n1En. By αn=α0/2n, it follows that meas((0,ε*)-E*)cε*N+1/α0. Thus Lemma 3.5 is proved.

4. Proof of Theorem <xref ref-type="statement" rid="thm2.1">2.1</xref>

As we pointed previously, once the non-degeneracy conditions are satisfied in some KAM step, the proof is complete by Lemma 3.5. 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 ẋ=Ax+f(x,t,ε), where A=βJ and f=JxF. 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 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 Lemma 3.5 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 n1 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 f1D1α0r1ε̃1.

At nth step, we consider the Hamiltonian system 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, fnDnα0rnε̃n. Suppose Qn(t,ε)=xfn(0,t,ε)=O(ε2n-1),fn(0,t,ε)=O(ε2n).

Since Qn is analytic with respect to ε, it follows that Qn=k=2n-1Qnkεk. Truncating the above power series of ε, we let 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=T1T2Tn-1.

By the transformation x=Tny, the system (4.3) is changed to ẏ=Ay+fn+1(y,t,ε), where fn+1=Q̃n·Tn+Qn·Tn+e-Pn·hnTn=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 Q̃n=Qn-Rn=k2nQnkεk 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.

Let r1=ε,ρ1=3ρ/4,ε1=ε0 and s1=ρ/16.

Define sn+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).

If cε̃n/sn2vηn<(1/8), it follows that fn+1Dn+1(α0ε̃ne-(4/3)n+(cε̃nsnv)2)ηnrn+cα0rnε̃nηn2α0rn+1ε̃n+1, where ε̃n+1=cηnε̃n. Moreover, it is easy to see xfn+1(0,t,ε)=O(ε2n),fn+1(0,t,ε)=O(ε2n+1). Now we verify cε̃n/sn2vηn<1/8. Let Gn=cε̃n/sn2v. By Gn=ce-(4/3)n-116vGn-1, it follows that 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=Gnηn.

Note that (crnε̃n/snv)0 and (cε̃n/(ηnsnv))0 as n, and ε̃ncsn2vGn. Let ε*=n1(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 function H(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 , σ(ε) is the modified term of the normal frequency. Then by some technique as in , we can also prove Theorem 2.1.

Acknowledgments

The authors would like to thank the reviewers’s suggestions about this revised version. This work was supported by the National Natural Science Foundation of China (11071038) and the Natural Science Foundation of Jiangsu Province (BK2010420).

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