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

Abstract

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system 𝑥̇=𝐽∇_𝑥𝐻, where 𝐻(𝑥,𝑡,𝜀)=(1/2)𝛽(𝑥²¹+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.

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,…,𝜔_𝑙) if 𝑓(𝑡)=𝐹(𝜔₁𝑡,𝜔₂𝑡,…,𝜔_𝑙𝑡) with 𝜃_𝑖=𝜔_𝑖𝑡, where 𝐹(𝜃₁,𝜃₂,…,𝜃_𝑙) is 2𝜋 periodic in all the arguments 𝜃_𝑗,𝑗=1,2,…,𝑙. If 𝐹(𝜃)(𝜃=(𝜃₁,𝜃₂,…,𝜃_𝑙)) is analytic on 𝐷_𝜌={𝜃∈𝐶^𝑙/2𝜋𝑍^𝑙∣|Im𝜃_𝑖|≤𝜌,𝑖=1,2,…,𝑙}, we call 𝑓(𝑡) analytic quasi-periodic on 𝐷_𝜌. If all 𝑞_𝑖𝑗(𝑡)(𝑖,𝑗=1,2…,𝑛) are analytic quasi-periodic on 𝐷_𝜌, then the matrix function 𝑄(𝑡)=(𝑞_𝑖𝑗(𝑡))_{1≤𝑖,𝑗≤𝑛} is called analytic quasi-periodic on 𝐷_𝜌.

If 𝑓(𝑡) is analytic quasi-periodic on 𝐷_𝜌, we can write it as Fourier series: 𝑓(𝑡)=_{𝑘∈𝑍𝑙}𝑓_𝑘𝑒^{i⟨𝑘,𝜔⟩𝑡}.(1.1) Define a norm of 𝑓 by ‖𝑓‖_𝜌=∑_{𝑘∈𝑍𝑙}|𝑓_𝑘|𝑒^|𝑘|𝜌. It follows that |𝑓_𝑘|≤‖𝑓‖_𝜌𝑒^{−|𝑘|𝜌}. If the matrix function 𝑄(𝑡) is analytic quasi-periodic on 𝐷_𝜌, we define the norm of 𝑄 by ‖𝑄‖_𝜌=𝑛×max_{1≤𝑖,𝑗≤𝑛}‖𝑞_𝑖𝑗‖_𝜌. It is easy to verify ‖𝑄₁𝑄₂‖_𝜌≤‖𝑄₁‖_𝜌‖𝑄₂‖_𝜌. The average of 𝑄(𝑡) is denoted by [𝑄]=([𝑞_𝑖𝑗])_{1≤𝑖,𝑗≤𝑛}, where 𝑞_𝑖𝑗=lim_𝑇→∞12𝑇^𝑇−𝑇𝑞_𝑖𝑗(𝑡)𝑑𝑡.(1.2) For the existence of the above limit, see [1].

Denote 𝐷𝑟,𝜌,𝜀₀=(𝑥,𝜃,𝜀)∈𝐶^𝑛×𝐶^𝑙2𝜋𝑍^𝑙×𝐶∣|𝑥|≤𝑟,𝜃∈𝐷^𝜌,|𝜀|≤𝜀₀,(1.3) where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) and |𝑥|=|𝑥₁|+|𝑥₂|+⋯+|𝑥_𝑛|.

Let 𝑓(𝑥,𝑡,𝜀) be analytic quasi-periodic of 𝑡 and analytic in 𝑥 and 𝜀 on 𝐷(𝑟,𝜌,𝜀₀). Then 𝑓(𝑥,𝑡,𝜀) can be expanded as 𝑓(𝑥,𝑡,𝜀)=^∞_𝑚₌₀_{𝑘∈𝑍𝑙}𝑓_𝑚𝑘(𝑥)𝜀^𝑚𝑒^{i⟨𝑘,𝜔⟩𝑡}.(1.4) Define a norm by ‖𝑓‖_{𝐷(𝑟,𝜌,𝜀0)}=^∞_𝑚₌₀_{𝑘∈𝑍𝑙}||𝑓_𝑚𝑘||_𝑟𝜀^𝑚0𝑒^𝜌|𝑘|,(1.5) where |𝑓_𝑚𝑘|_𝑟=sup_{|𝑥|≤𝑟}|𝑓_𝑚𝑘(𝑥)|. Note that ‖‖𝑓₁⋅𝑓₂‖‖_{𝐷(𝑟,𝜌,𝜀0)}≤‖‖𝑓₁‖‖_{𝐷(𝑟,𝜌,𝜀0)}⋅‖‖𝑓₂‖‖_{𝐷(𝑟,𝜌,𝜀0)}.(1.6)

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 [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: 𝑥̇=(𝐴+𝜀𝑄(𝑡))𝑥,𝑥∈𝑅^𝑛,(1.7) where 𝐴 is an 𝑛×𝑛 constant matrix with 𝑛 different eigenvalues 𝜆₁,𝜆₂,…,𝜆_𝑛 and 𝑄(𝑡) is analytic quasi-periodic with respect to 𝑡 with frequencies 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑙). Here 𝜀 is a small perturbation parameter. Suppose that the following nonresonance conditions hold: |||⟨𝑘,𝜔⟩√−1+𝜆_𝑖−𝜆_𝑗|||≥𝛼||𝑘||^𝜏,(1.8) for all 𝑘∈𝑍^𝑙⧵{0}, where 𝛼>0 is a small constant and 𝜏>𝑙−1. Assume that 𝜆^0𝑗(𝜀)(𝑗=1,2,…,𝑛) are eigenvalues of 𝐴+𝜀[𝑄]. If the following non-degeneracy conditions hold: 𝑑𝑑𝜀𝜆^0𝑖(𝜀)−𝜆^0𝑗(𝜀)|||_𝜀=0≠0,∀𝑖≠𝑗,(1.9) 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 𝑑²𝑞𝑑𝑡²+𝑄(𝜔𝑡)𝑞=𝐸𝑞,(1.10) Eliasson considered the following equivalent two-dimensional quasi-periodic Hamiltonian system: 𝑝̇=(𝐸−𝑄(𝜔𝑡))𝑞,𝑞̇=𝑝,(1.11) 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 𝐶^𝜔(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 𝑥̇=𝐴(𝜆)+𝑔𝜔₁𝑡,…,𝜔_𝑙𝑡,𝜆𝑥,(1.12) where 𝐴∈𝐶^𝜔(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 𝐶^𝜔(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 𝑥̇=(𝐴+𝜀𝑄(𝑡))𝑥+𝜀𝑔(𝑡)+ℎ(𝑥,𝑡),𝑥∈𝑅^𝑛,(1.13) where 𝐴 has 𝑛 different nonzero eigenvalues 𝜆_𝑖. 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: 𝑥̇=𝐴𝑥+𝑓(𝑥,𝑡,𝜀),𝑥∈𝑅²,(1.14) where 𝐴 has a pair of pure imaginary eigenvalues ±√−1𝜔₀ with 𝜔₀≠0 satisfying the nonresonance conditions ||⟨𝑘,𝜔⟩||≥𝛼||𝑘||^𝜏,||⟨𝑘,𝜔⟩−2𝜔₀||≥𝛼||𝑘||^𝜏(1.15) for all 𝑘∈𝑍^𝑙⧵{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 subset 𝐸⊂(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 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 [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 𝑥̇=𝐽∇_𝑥𝐻,𝑥∈𝑅²,(2.1) where 𝐻(𝑥,𝑡,𝜀)=(1/2)𝛽(𝑥²¹+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,…,𝜔_𝑙) and real analytic with respect to x and 𝜀 on 𝐷(𝑟,𝜌,𝜀₀), and 𝐽=−0110.(2.2) 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 ||⟨𝑘,𝜔⟩||≥𝛼⁰||⟨𝑘,𝜔⟩−2𝛽||||≥𝑘||^𝜏𝛼,₀(2.3)||𝑘||^𝜏(2.4) for all 𝑘∈𝑍^𝑙⧵{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 𝜀∈𝐸_∗, there exists an analytic quasi-periodic symplectic transformation 𝑥=𝜙_∗(𝑡)𝑦+𝜓_∗(𝑡) on 𝐷_𝜌/2 with the frequencies 𝜔, which changes (2.1) into the Hamiltonian system 𝑦̇=𝐽∇_𝑦𝐻_∗, where 𝐻_∗(𝑦,𝑡,𝜀)=1/2𝛽_∗(𝜀)(𝑦²¹+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 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=(𝑟_𝑖𝑗)_{1≤𝑖,𝑗≤2} be a Hamiltonian matrix. Then we have 𝑟₁₁+𝑟₂₂=0. Define a matrix 𝑅_𝐴=(1/2)𝑑𝐽 with 𝑑=𝑟₁₂−𝑟₂₁. Let𝐵=1√2√1−1−√1−1.(3.1) It is easy to verify𝐵⁻¹𝑅_𝐴𝐵=12diag√𝐵−1𝑑,−√−1𝑑₋₁,𝑅−𝑅_𝐴𝐵=12⎛⎜⎜⎝0𝜎^−√−1𝜅^𝜎^+√−1𝜅^0⎞⎟⎟⎠,(3.2) 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),(3.3) where 𝐴=𝛽(𝜀)𝐽 with |𝛽(𝜀)|>𝜇, 𝜇>0 is a constant, and 𝑅(𝑡)=(𝑟_𝑖𝑗(𝑡))_{1≤𝑖,𝑗≤2} is a real analytic quasi-periodic Hamiltonian matrix on 𝐷_𝜌 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: ||⟨𝑘,𝜔⟩−2𝛽(𝜀)||≥𝛼||𝑘||^𝜏′,(3.4) 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 ‖𝑃‖_𝜌1≤𝑐𝛼𝑠^𝑣‖𝑅‖_𝜌,‖‖𝜀𝜕_𝜀𝑃‖‖_𝜌1≤𝑐𝛼²𝑠^𝑣′‖𝑅‖_𝜌+‖‖𝜀𝜕_𝜀𝑅‖‖_𝜌,(3.5) 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 [9].

Proof. Let 𝑃=𝐵⁻¹𝑃𝐵, where 𝐵 is defined by (3.1). Similarly, define 𝐴,𝑅,𝑅_𝐴. Then (3.3) becomes ̇‌𝑃=𝐴𝑃−𝑃𝐴+𝑅(𝑡),(3.6) where 𝐴=diag√−1𝛽,−√−1𝛽.(3.7) 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 𝑟₁₁=−𝑟₂₂ with [𝑟_𝑖𝑖]=0.
Insert the Fourier series of 𝑃 and 𝑅 into (3.6). Then it follows that 𝑝^0𝑖𝑖=0,𝑝^𝑘𝑖𝑖=𝑟^𝑘𝑖𝑖/(⟨𝑘,𝜔⟩√−1) for 𝑘≠0, and 𝑝^𝑘𝑖𝑗=𝑟^𝑘𝑖𝑗√−1(⟨𝑘,𝜔⟩±2𝛽)for𝑖≠𝑗.(3.8) Since 𝑅 is analytic on 𝐷_𝜌, we have |𝑅_𝑘|≤‖𝑅‖_𝜌𝑒^{−|𝑘|𝜌}. So it follows ‖‖𝑃‖‖_𝜌−𝑠≤_{𝑘∈𝑍𝑙}||𝑃_𝑘||𝑒^{|𝑘|(𝜌−𝑠)}≤𝑐𝛼𝑠^𝑣‖𝑅‖_𝜌.(3.9) 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 𝑝₁₁=−𝑝₂₂ 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 𝑝₁₂ and 𝑝₂₁ since 𝑝₁₁ and 𝑝₂₂ are easy.
For 𝑖≠𝑗 we have 𝑑𝑝^𝑘𝑖𝑗(𝜀)𝑑𝜀=±2𝛽^(𝜀)𝑟^𝑘𝑖𝑗−(⟨𝑘,𝜔⟩±2𝛽)𝑟^{𝑘𝑖𝑗'}(𝜀)−√−1(⟨𝑘,𝜔⟩±2𝛽)².(3.10)
Then, in the same way as above we obtain the estimate for ‖𝜀(𝜕𝑃/𝜕𝜀)‖_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),(3.11) where 𝐴 is the same as in Lemma 3.1, and 𝑔 is real analytic quasi-periodic in 𝑡 on 𝐷_𝜌 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 ‖𝑥‖_𝜌1≤𝑐𝛼𝑠^𝑣‖𝑔‖_𝜌,‖‖‖𝜀𝜕𝑥𝜕𝜀‖‖‖_𝜌1≤𝑐𝛼²𝑠^𝑣′‖𝑔‖^𝜌+‖‖‖𝜀𝜕𝑔𝜕𝜀‖‖‖_𝜌,(3.12) 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 𝑥̇‌=𝐴𝑥+𝑔(𝑡),(3.13) where 𝐴=diag(√−1𝛽,−√−1𝛽). Expanding 𝑥=(𝑥₁,𝑥₂) and 𝑔=(𝑔₁,𝑔₂) into Fourier series and using (3.13), we have 𝑥^𝑘𝑖=𝑔^𝑘𝑖√−1⟨𝑘,𝜔⟩+(−1)^𝑖𝛽.(3.14) Using 2𝑘 in place of 𝑘 in (3.4), we have ||⟨𝑘,𝜔⟩−𝛽(𝜀)||≥𝛼2||𝑘||^𝜏′.(3.15) Thus, in the same way as the proof of Lemma 3.1, we can estimate ‖𝑥‖_𝜌1 and ‖𝜀𝜕_𝜀𝑥‖_𝜌1. We omit the details.

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

Lemma 3.4. Let 𝜓(𝜀)=𝜎𝜀^𝑁+𝜀^𝑁𝑓(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 𝑂=𝜀∈0,𝜀_∗∣||𝜙(𝜀)||≥𝛼||𝑘||^𝜏′,∀𝑘≠0,(3.16) 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 meas0,𝜀_∗⧵𝑂≤𝑐𝛼𝛼²⁰𝜀^∗𝑁+1,(3.17) where 𝑐 is a constant independent of 𝛼₀,𝛼,𝜀_∗

Proof. Let 𝑂_𝑘=𝜀∈0,𝜀_∗∣||𝜙(𝜀)||<𝛼||𝑘||^𝜏′.(3.18) By assumption, if 𝜀_∗ is sufficient small, we have that |𝜓(𝜀)|≤2𝜎𝜀^𝑁 and |𝜓^(𝜀)|≥(𝜎/2)𝜀^𝑁−1 for 𝜀∈(0,𝜀_∗). If 𝜀^𝑁≤𝛼₀/(4𝜎|𝑘|^𝜏), by (2.4) we have ||𝜙(𝜀)||≥||⟨𝑘,𝜔⟩−2𝛽||−||𝜓(𝜀)||≥𝛼||𝑘||^𝜏′.(3.19)
Thus, we only consider the case that 𝜀^𝑁∗≥𝜀^𝑁≥(𝛼₀/(4𝜎|𝑘|^𝜏)). We have |𝑘|≥(𝛼₀/(4𝜎𝜀^𝑁∗))^1/𝜏=𝐾. Since ||𝜙^(𝜀)||=||𝜓^(𝜀)||≥𝜎2𝜀^𝑁−1≥𝛼⁰8||𝑘||^𝜏𝜀_∗,(3.20) we have meas(𝑂_𝑘)≤((2𝛼)/|𝑘|^𝜏′)×((8|𝑘|^𝜏𝜀_∗)/𝛼₀)=(16𝛼𝜀_∗)/(|𝑘|^{𝜏′−𝜏}𝛼₀). So meas0,𝜀_∗⧵0≤_{||𝑘||≥𝐾}meas𝑂_𝑘≤16𝛼𝛼₀𝜀_∗||𝑘_||≥𝐾1||𝑘||^{𝜏′−𝜏}≤𝑐𝛼𝛼₀𝜀_∗𝐾^{𝑙−𝜏′+𝜏}≤𝑐𝛼𝛼²⁰𝜀^∗𝑁+1,(3.21) 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=𝐽∇_𝑥𝐻, where 𝐻(𝑥,𝑡,𝜀)=12𝛽𝑥²¹+𝑥²²+𝐹(𝑥,𝑡,𝜀)with𝛽≠0.(3.22) 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 𝜀 on 𝐷(𝑟,𝜌,𝜀₀). Let 𝑓(𝑥,𝑡,𝜀)=𝐽∇_𝑥𝐹(𝑥,𝑡,𝜀). Assume that 𝑓(0,𝑡,𝜀)=𝑂(𝜀^2𝑚0) and 𝜕_𝑥𝑓(0,𝑡,𝜀)=𝑂(𝜀^𝑚0) as 𝜀→0, where 𝑚₀ is a positive integer. Let 𝑄(𝑡,𝜀)=𝜕_𝑥𝑓(0,𝑡,𝜀)=∑_𝑘≥𝑚0𝑄_𝑘(𝑡)𝜀^𝑘. Suppose there exists 𝑚₀≤𝑘≤2𝑚₀−1 such that [𝑄_𝑘]_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 𝐻_∗(𝑦,𝑡,𝜀)=12𝛽^∗(𝜀)𝑦²¹+𝑦²²+𝐹_∗(𝑦,𝑡,𝜀),(3.23) 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(𝜀^𝑚0) and ‖𝜙_∗−𝐼𝑑‖_𝜌/2+‖𝜓_∗‖_𝜌/2=𝑂(𝜀^𝑚0).

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 [𝑄_𝑘]_𝐴≠0 for some 𝑚₀≤𝑘≤2𝑚₀−1 is a non-degeneracy condition.
Consider the following Hamiltonian system 𝑥̇=𝐴𝑥+𝑓(𝑥,𝑡,𝜀),(3.24) 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 ℎ(𝑥,𝑡,𝜀)=𝑓(𝑥,𝑡,𝜀)−𝑔(𝑡,𝜀)−𝑄(𝑡,𝜀)𝑥.(3.25) 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),(3.26) where 𝐴₊=𝐴+[𝑄]_𝐴=𝛽₊(𝜀)𝐽 and 𝑅=𝑄−[𝑄]_𝐴. By assumption we have ‖𝑔‖_𝜌≤𝛼𝑟𝜀̃,‖𝑄‖_𝜌≤𝛼𝜀̃,‖ℎ‖_𝐷≤3𝛼𝑟𝜀̃.(3.27) Moreover, we have ‖‖𝜀𝜕_𝜀𝑔‖‖_𝜌≤𝛼𝑟𝜀̃,‖‖𝜀𝜕_𝜀𝑄‖‖_𝜌≤𝛼𝜀̃,‖‖𝜀𝜕_𝜀ℎ‖‖_𝐷≤3𝛼𝑟𝜀̃.(3.28)
Now we want to construct the symplectic change of variables 𝑥=𝑇^𝑦=𝑒^𝑃(𝑡)𝑦 to (3.26), where 𝑃 is a Hamiltonian matrix to be defined later. Then we have 𝑦̇=𝑒^−𝑃𝐴₊+𝑅−𝑃̇𝑒^𝑃+𝑒^−𝑃𝑃̇𝑒^𝑃−𝑑+𝑒_−𝑃𝑑𝑡𝑒^𝑃(𝑡)𝑦𝑔(𝑡,𝜀)+𝑒^−𝑃ℎ𝑒^𝑃𝑦,𝑡,𝜀.(3.29) Let 𝑊=𝑒^𝑃−𝐼−𝑃 and 𝑊=𝑒^−𝑃−𝐼−𝑃. Then the system (3.29) becomes 𝑦̇=𝐴₊+𝑅−𝑃̇+𝐴₊𝑃−𝑃𝐴₊𝑦+𝑄^𝑦+𝑒^−𝑃𝑔(𝑡,𝜀)+𝑒^−𝑃ℎ𝑒^𝑃𝑦,𝑡,𝜀,(3.30) where 𝑄′=−−𝑃𝑃𝑅𝐴₊−𝑃̇+𝑅−𝑃̇𝑃−𝑃𝐴₊+𝑅−𝑃̇𝑃+𝑊𝐴+𝑅−𝑃̇𝑊+𝐴₊+𝑅−₊𝑃̇𝑊+𝑅−𝑃̇𝑒^𝑃+𝑒^−𝑃𝑃̇𝑒^𝑃−𝑑𝑑𝑡𝑒^𝑃.(3.31)
We would like to have 𝑃̇−𝐴₊𝑃+𝑃𝐴₊=𝑅,(3.32) 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: ||⟨𝑘,𝜔⟩−2𝛽₊(𝜀)||≥𝛼⁺||𝑘||^𝜏′,∀𝑘∈𝑍^𝑙⧵{0},(3.33) 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: ‖𝑃‖_𝜌−𝑠≤𝑐‖𝑄‖^𝜌𝛼₊𝑠^𝑣≤𝑐𝜀̃𝑠^𝑣‖‖‖𝜀𝜕𝑃,𝜕𝜀‖‖‖_𝜌−𝑠≤𝑐𝛼²⁺𝑠^𝑣′‖𝑄‖^𝜌+‖‖‖𝜀𝜕𝑄𝜕𝜀‖‖‖_𝜌≤𝛼₊𝑐𝜀𝑠̃^𝑣′,(3.34) 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),(3.35) where 𝑓^=𝑄^𝑦+𝑒^−𝑃𝑔(𝑡,𝜀)+𝑒^−𝑃ℎ(𝑒^𝑃𝑦,𝑡,𝜀).
By Lemma 3.3, let us denote by 𝑥 the solution of 𝑥̇=𝐴₊𝑥+𝑔′(𝑡,𝜀) on 𝐷_𝜌−2𝑠, where 𝑔′=𝑒^−𝑃𝑔(𝑡,𝜀). Then, by Lemma 3.3 we have ‖‖𝑥‖‖_𝜌−2𝑠≤𝑐‖𝑔‖^𝜌−𝑠𝛼₊𝑠^𝑣≤𝑐𝑟𝜀̃𝑠^𝑣‖‖‖𝜀𝜕𝑥,𝜕𝜀‖‖‖_𝜌−2𝑠≤𝑐𝛼²⁺𝑠^𝑣′‖𝑔‖^𝜌−𝑠+‖‖‖𝜀𝜕𝑔𝜕𝜀‖‖‖_𝜌−𝑠≤𝛼𝑐₊𝑟𝑠𝜀̃^𝑣′.(3.36)
Under the symplectic change of variables 𝑦=𝑇^𝑥₊=𝑥+𝑥₊, the Hamiltonian system (3.35) is changed to 𝑥̇₊=𝐴₊𝑥₊+𝑓₊𝑥₊,𝑡,𝜀,(3.37) where 𝐴₊=𝛽₊𝐽 and 𝑓₊=𝑄^⋅𝑇^+𝑒^−𝑃ℎ∘𝑇^∘𝑇^.(3.38)
Let the symplectic transformation 𝑇=𝑇^∘𝑇^. Then 𝑥=𝑇𝑥₊=𝜙(𝑡)𝑥₊+𝜓(𝑡), where 𝜙(𝑡)=𝑒^𝑃(𝑡) and 𝜓(𝑡)=𝑒^𝑃(𝑡)𝑥(𝑡). It is easy to obtain that if ‖𝑃‖_𝜌−2𝑠≤1/2, then ‖𝜙−𝐼‖_𝜌−2𝑠≤𝑐𝜀̃𝑠^𝑣,‖‖𝜀𝜕_𝜀𝜙‖‖_𝜌−2𝑠≤𝑐𝜀̃𝛼₊𝑠^𝑣′‖𝜓‖_𝜌−2𝑠,≤𝑐𝑟𝜀̃𝑠^𝑣,‖‖𝜀𝜕_𝜀𝜓‖‖_𝜌−2𝑠≤𝑐𝑟𝜀̃𝛼₊𝑠^𝑣′.(3.39) 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)||=|||𝜀𝛽^(𝜀)|||≤𝑐𝛼𝜀̃.(3.40) By (3.38) we have 𝑓₊𝑥₊,𝑡,𝜀=𝑄^(𝑡)𝑥₊+𝑥(𝑡)+𝑒^{−𝑃(𝑡)}ℎ𝑒^𝑃(𝑡)𝑥₊+𝑥(𝑡),𝑡,𝜀.(3.41) Let 𝜌₊=𝜌−2𝑠, and 𝑟₊=𝜂𝑟 with 𝜂≤1/8. If 𝑐𝜀̃/𝛼₊𝑠^𝑣+𝑣′≤𝜂, it follows that ‖𝑥‖_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 |𝑒^𝑃(𝑡)(𝑥₊+𝑥(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 ‖‖𝑓₊‖‖_𝐷+≤𝑐𝜀̃²𝑠^2𝑣𝜂𝑟+𝑐𝛼𝑟𝜀̃𝜂²≤𝑐𝜀̃𝑠^2𝑣‖‖𝜀𝜕_𝜀+𝑐𝛼𝜂𝑟₊𝜀̃,𝑓₊‖‖_𝐷+≤𝑐𝜀̃²𝛼₊𝑠^𝑣+𝑣′𝜂𝑟+𝑐𝛼𝑟𝜀̃𝜂²≤𝑐𝜀̃𝛼𝑠^𝑣+𝑣′+𝑐𝛼𝜂𝑟₊𝜀̃.(3.42) Let 𝛼₊=𝛼/2 and 𝜂=𝑐𝜀̃/(𝛼²𝑠^𝑣+𝑣′). Then we have ‖‖𝑓₊‖‖_𝐷+≤𝑐𝛼₊𝑟₊𝜂𝜀̃=𝛼₊𝑟₊𝜀̃₊,𝜀̃₊=𝑐𝜂𝜀̃.(3.43) Similarly, we have ‖‖𝜀𝜕_𝜀𝑓₊‖‖_𝐷+≤𝛼₊𝑟₊𝜀̃₊.(3.44)
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 [7] 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_∗]||𝜀𝜕_𝜀||𝑓(0,𝑡,𝜀)||≤𝑐𝜀^2𝑚0,||𝜕_𝑥𝑓(0,𝑡,𝜀)||≤𝑐𝜀^𝑚0,𝑓(0,𝑡,𝜀)||≤𝑐𝜀^2𝑚0,||𝜀𝜕_𝜀𝜕_𝑥𝑓(0,𝑡,𝜀)||≤𝑐𝜀^𝑚0.(3.45) Moreover, ||ℎ(𝑥,𝑡,𝜀)||≤𝑐|𝑥|²,||𝜀𝜕_𝜀ℎ(𝑥,𝑡,𝜀)||≤𝑐|𝑥|²,∀|𝑥|≤𝜀^𝑚0,∀𝜀∈0,𝜀_∗.(3.46) Let 𝑟₀=𝜀^𝑚0,𝜌₀=𝜌,𝑠₀=𝜌₀/8,𝐷₀=𝐷(𝑟₀,𝜌₀,𝜀_∗), and 𝜀̃₀=𝑐𝜀^𝑚0/𝛼₀. Then we have ||𝑓₀||_𝐷0≤𝛼₀𝑟₀𝜀̃₀,||𝜀𝜕_𝜀𝑓₀||_𝐷0≤𝛼₀𝑟₀𝜀̃₀.(3.47) For 𝑛≥1, let 𝛼_𝑛=𝛼^𝑛−12,𝑠^𝑛=𝑠^𝑛−12,𝜌^𝑛𝜂_𝑛−1=𝜌_𝑛−1−2𝑠_𝑛−1,=𝑐𝜀̃^𝑛−1𝛼^2𝑛−1𝑠^𝑣+𝑣′_𝑛−1,𝑟_𝑛=𝜂_𝑛−1𝑟_𝑛−1,𝜀̃_𝑛=𝑐𝜂_𝑛−1𝜀̃_𝑛−1.(3.48)
Then we have a sequence of quasi-periodic symplectic transformations {𝑇_𝑛} satisfying 𝑇_𝑛𝑥=𝜙_𝑛(𝑡)𝑥+𝜓_𝑛(𝑡) with ‖‖𝜙_𝑛−𝐼‖‖_𝜌𝑛+1≤𝑐𝜀̃^𝑛𝑠^𝑣𝑛,‖‖𝜓_𝑛‖‖_𝜌𝑛+1≤𝑐𝑟^𝑛𝜀̃_𝑛𝑠^𝑣𝑛.(3.49) Let 𝑇^𝑛=𝑇₀∘𝑇₁⋯∘𝑇_𝑛−1. Then under the transformation 𝑥=𝑇^𝑛𝑦 the Hamiltonian system 𝑥̇=𝐴₀𝑥+𝑓₀(𝑥,𝑡,𝜀) is changed to 𝑦̇=𝐴_𝑛𝑦+𝑓_𝑛(𝑦,𝑡,𝜀).
Moreover, 𝐴_𝑛(𝜀)=𝛽_𝑛(𝜀)𝐽 satisfies 𝐴_𝑛+1−𝐴_𝑛=[𝑄_𝑛]_𝐴 and ||𝛽_𝑛+1(𝜀)−𝛽_𝑛(𝜀)||≤𝑐𝛼_𝑛𝜀̃_𝑛,||𝜀𝛽^𝑛+1(𝜀)−𝛽^𝑛(𝜀)||≤𝑐𝛼_𝑛𝜀̃_𝑛,(3.50)‖‖𝑓_𝑛‖‖_𝐷𝑛≤𝛼_𝑛𝑟_𝑛𝜀̃_𝑛.(3.51)Convergence
By the above definitions we have 𝜂_𝑛/𝜂_𝑛−1=𝑐𝜀̃_𝑛/𝜀̃_𝑛−1=𝑐𝜂_𝑛−1. Thus, we have 𝜂_𝑛≤𝑐𝜂^2𝑛−1 and so 𝑐𝜂_𝑛≤(𝑐𝜂_𝑛−1)²≤(𝑐𝜂₀)^2𝑛. Note that 𝜂₀=𝑐𝜀̃₀/(𝛼²⁰𝑠^𝑣+𝑣′₀)≤𝑐𝜀^𝑚0/(𝛼²⁰𝜌^𝑣+𝑣′₀). Suppose that 𝜀_∗ is sufficiently small such that for 0<𝜀<𝜀_∗ we have 𝑐𝜂₀≤1/2.𝑇_𝑛 are affine, so are 𝑇^𝑛 with 𝑇^𝑛𝑥=𝜙^𝑛(𝑡)𝑥+𝜓^𝑛(𝑡). By the estimates (3.49) it is easy to prove that 𝜙^𝑛(𝑡) and 𝜓^𝑛(𝑡) are convergent and so 𝑇^𝑛 is actually convergent on the domain 𝐷(𝑟/2,𝜌/2). Let 𝑇^𝑛→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 for 𝑓_𝑛 and Cauchy’s estimate, we have |𝑓_𝑛(0,𝑡,𝜀)|≤𝛼_𝑛𝑟_𝑛𝜀̃_𝑛→0 and |𝜕_𝑥𝑓_𝑛(0,𝑡,𝜀)|≤𝛼_𝑛𝜀̃_𝑛→0 as 𝑛→∞. Let 𝑓_𝑛→𝑓_∗. Then it follows that 𝑓_∗(𝑥,𝑡,𝜀)=𝑂(𝑥²).
By the estimates (3.50) for 𝛽_𝑛 we have 𝛽_𝑛→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 𝐸_𝑛=𝜀∈0,𝜀_∗∣||⟨𝜔,𝑘⟩−2𝛽_𝑛(𝜀)||≥𝛼^𝑛||𝑘||^𝜏′.(3.52) Note that 𝛽_𝑛=𝛽₁+𝜓, where 𝜓=∑^{𝑛−1𝑗=1}𝛽_𝑗+1−𝛽_𝑗,𝛽₁=𝛽+𝛽̂, and 𝛽̂𝐽=[𝑄]_𝐴. Note that 𝜀̃₁=𝑐𝜀̃²⁰/(𝛼²⁰𝑠^𝑣+𝑣′₀) and 𝜀̃₀=𝑐𝜀^𝑚0/𝛼₀. By the estimates (3.50), we have 𝜓(𝜀)=𝑂(𝜀^2𝑚0) and 𝜀𝜓′(𝜀)=𝑂(𝜀^2𝑚0). By assumption, [𝑄]_𝐴 is analytic with respect to 𝜀 and there exists 𝑚₀≤𝑁≤2𝑚₀−1 such that [𝑄]_𝐴=𝛿𝜀^𝑁+𝑂(𝜀^𝑁+1) with 𝛿≠0. Thus, 𝛽₁(𝜀)=𝛽+𝛿𝜀^𝑁+𝑂(𝜀^𝑁+1). By Lemma 3.4, we have meas((0,𝜀_∗)−𝐸_𝑛)≤𝑐(𝛼_𝑛/𝛼²⁰)𝜀^∗𝑁+1. Let 𝐸_∗=⋂_𝑛≥1𝐸_𝑛. By 𝛼_𝑛=𝛼₀/2^𝑛, it follows that meas((0,𝜀_∗)−𝐸_∗)≤𝑐𝜀^∗𝑁+1/𝛼₀. Thus Lemma 3.5 is proved.

4. Proof of Theorem 2.1

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 𝜀 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),(4.1) where 𝐴=𝛽𝐽 and 𝑓=𝐽∇_𝑥𝐹. By Lemma 3.3, denote by 𝑥 the solution of 𝑥̇=𝐴𝑥+𝑓(0,𝑡,𝜀) on 𝐷_3𝜌/4. Under the change of variables 𝑥=𝑇₀𝑥₊=𝑥+𝑥₊, the Hamiltonian system (2.1) becomes 𝑥̇₊=𝐴𝑥₊+𝑓₁𝑥₊,𝑡,𝜀,(4.2) where 𝑓₁(𝑥₊,𝑡,𝜀)=𝑓(𝑥+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 𝐴_𝑛=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.
At 𝑛th step, we consider the Hamiltonian system 𝑥̇=𝐴𝑥+𝑓_𝑛(𝑥,𝑡,𝜀),(4.3) where 𝑓_𝑛 is analytic quasi-periodic with respect to 𝑡 with frequencies 𝜔 and real analytic with respect to 𝑥 and 𝜀 on 𝐷_𝑛=𝐷(𝑟_𝑛,𝜌_𝑛,𝜀_𝑛). Moreover, ‖𝑓_𝑛‖_𝐷𝑛≤𝛼₀𝑟_𝑛𝜀̃_𝑛. Suppose 𝑄_𝑛(𝑡,𝜀)=𝜕_𝑥𝑓_𝑛(0,𝑡,𝜀)=𝑂𝜀^2𝑛−1,𝑓^𝑛(0,𝑡,𝜀)=𝑂𝜀^2𝑛.(4.4)
Since 𝑄_𝑛 is analytic with respect to 𝜀, it follows that 𝑄_𝑛=^∞_𝑘=_2𝑛−1𝑄^𝑘𝑛𝜀^𝑘.(4.5) Truncating the above power series of 𝜀, we let 𝑅_𝑛(𝑡,𝜀)=^2𝑛−1_𝑘=_2𝑛−1𝑄^𝑘𝑛𝜀^𝑘,𝑄_𝑛=𝑄_𝑛−𝑅_𝑛.(4.6)
Because the non-degeneracy conditions do not happen in KAM steps, we must have [𝑅_𝑛]_𝐴=0. In the same way as the proof of Lemma 3.5, we have a quasi-periodic symplectic transformation 𝑇_𝑛 with 𝑇_𝑛𝑥=𝜙_𝑛(𝑡)𝑥+𝜓_𝑛(𝑡) satisfying (3.49). Let 𝑇^𝑛=𝑇₁∘𝑇₂⋯∘𝑇_𝑛−1.
By the transformation 𝑥=𝑇^𝑛𝑦, the system (4.3) is changed to 𝑦̇=𝐴𝑦+𝑓_𝑛+1(𝑦,𝑡,𝜀),(4.7) where 𝑓_𝑛+1=𝑄_𝑛⋅𝑇^𝑛+𝑄^𝑛⋅𝑇^𝑛+𝑒^−𝑃𝑛⋅ℎ_𝑛∘𝑇_𝑛=𝑄_𝑛(𝑥_𝑛+𝑦)+𝑄^𝑛(𝑥_𝑛+𝑦)+𝑒^−𝑃𝑛ℎ_𝑛(𝑒^𝑃𝑛(𝑥_𝑛+𝑦)).
The last two terms can be estimated similarly as those of (3.41). Note that 𝑄_𝑛=𝑄_𝑛−𝑅_𝑛=_𝑘≥2𝑛𝑄^𝑘𝑛𝜀^𝑘(4.8) 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 𝑠_𝑛+1=𝑠_𝑛/2,𝜌_𝑛+1=𝜌_𝑛−2𝑠_𝑛,𝜂_𝑛=(1/8)𝑒^−(4/3)𝑛,𝑟_𝑛+1=𝜂_𝑛𝑟_𝑛, 𝛿_𝑛=1−(2/3)^𝑛 and 𝜀_𝑛+1=𝛿_𝑛𝜀_𝑛. Let 𝐷_𝑛+1=𝐷(𝑟_𝑛+1,𝜌_𝑛+1,𝜀_𝑛+1).
If 𝑐𝜀̃_𝑛/𝑠^𝑛2𝑣≤𝜂_𝑛<(1/8), it follows that ‖‖𝑓_𝑛+1‖‖_𝐷𝑛+1≤𝛼₀𝜀̃_𝑛𝑒^−(4/3)𝑛+𝑐𝜀̃^𝑛𝑠^𝑣𝑛²𝜂^𝑛𝑟_𝑛+𝑐𝛼₀𝑟_𝑛𝜀̃_𝑛𝜂^2𝑛≤𝛼₀𝑟_𝑛+1𝜀̃_𝑛+1,(4.9) where 𝜀̃_𝑛+1=𝑐𝜂_𝑛𝜀̃_𝑛. Moreover, it is easy to see 𝜕_𝑥𝑓_𝑛+1(0,𝑡,𝜀)=𝑂𝜀^2𝑛,𝑓_𝑛+1(0,𝑡,𝜀)=𝑂𝜀^2𝑛+1.(4.10) Now we verify 𝑐𝜀̃_𝑛/𝑠^𝑛2𝑣≤𝜂_𝑛<1/8. Let 𝐺_𝑛=𝑐𝜀̃_𝑛/𝑠^𝑛2𝑣. By 𝐺_𝑛=𝑐𝑒^{−(4/3)𝑛−1}16^𝑣𝐺_𝑛−1, it follows that 𝐺_𝑛=𝑐16^𝑣^𝑛−1𝑒^{−[(4/3)𝑛−1+(4/3)𝑛−2+⋯+(4/3)1]}𝐺₁=𝑐16^𝑣^𝑛−1𝑒⁴𝑒^{−4(4/3)𝑛−1}𝐺₁.(4.11) Note that 𝐺₁=𝑐𝜀̃₁/𝑠^12𝑣. If 𝜀̃₁ is sufficiently small, we have 𝑐𝜀̃_𝑛/𝑠^𝑛2𝑣=𝐺_𝑛≤𝜂_𝑛.
Note that (𝑐𝑟_𝑛𝜀̃_𝑛/𝑠^𝑣𝑛)→0 and (𝑐𝜀̃_𝑛/(𝜂_𝑛𝑠^𝑣𝑛))→0 as 𝑛→∞, and 𝜀̃_𝑛≤𝑐𝑠^𝑛2𝑣𝐺_𝑛. 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 function 𝐻(𝑥,𝑡,𝜀)=⟨𝜔,𝐼⟩+(1/2)(𝛽_∗+𝜎(𝜀))(𝑥²¹+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.

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).