Lagrangian Stability of a Class of Second-Order Periodic Systems

Abstract

We study the following second-order periodic system: 𝑥^′′+𝑉^′(𝑥)+𝑝(𝑡)=0 where 𝑉(𝑥) has a singularity and 𝑝(𝑡)=𝑝(𝑡+1). Under some assumptions on the 𝑉(𝑥) and 𝑝(𝑡), by Moser's twist theorem we obtain the existence of quasiperiodic solutions and boundedness of all the solutions.

1. Introduction and Main Result

In the early 1960s, Littlewood [1] asked whether or not the solutions of the Duffing-type equations 𝑥^′′+𝑔(𝑥)=𝑒(𝑡),where𝑒(𝑡+1)=𝑒(𝑡) (1.1) are bounded for all time, that is, whether there are resonances that might cause the amplitude of the oscillations to increase without bound. Littlewood suggested studying the following two cases: (i)superlinear case: 𝑔(𝑥)/𝑥→+∞ as 𝑥→±∞,(ii)sublinear case: sign(𝑥)⋅𝑥→+∞ and 𝑔(𝑥)/𝑥→0 as 𝑥→±∞.

The first positive result of boundedness of solutions in the superlinear case (i) was due to Morris [2]. By means of KAM theorem, Morris proved that every solution of the second-order system (1.1) is bounded if 𝑔(𝑥)=2𝑥³ and 𝑒(𝑡) is piecewise continuous and periodic. This result relies on the fact that the nonlinearity 2𝑥³ can guarantee the twist condition of KAM theorem. Later, several authors (see [3–5]) improved the Morris’s result and obtained similar results for a large class of superlinear function 𝑔(𝑥).

In 1999, the first result in the sublinear case was proved by Küpper and You [6] in the study of 𝑥^′′+|𝑥|^𝛼−1𝑥=𝑒(𝑡), (1.2) where 0<𝛼<1 and 𝑒(𝑡)∈𝐶^∞(𝑇). The authors transform (1.2) into a perturbation of an integrable Hamiltonian system and then prove that the Poincaré map of the transformed system is close to a so-called twist map. So, the Moser’s twist theorem guarantees the boundedness of all solutions of (1.2). The general sublinear case was considered by Liu [7] under certain reasonable conditions.

The Littlewood problem for singular potentials is known to be challenging, and there are only very a few results. Recently, Capietto et al. [8] studied 𝑥^′′+𝑉′(𝑥)=𝑝(𝑡), (1.3) with 𝑝(𝑡) is a 𝜋-periodic function and 𝑉=(1/2)𝑥²₊+(1/(1−𝑥²₋)^𝜈)−1, where 𝑥₊=max{𝑥,0},𝑥₋=max{−𝑥,0} and 𝜈>2 is a positive integer. Under the Lazer-Leach assumption that 11+2^𝜋₀𝑝𝑡₀+𝜃sin𝜃𝑑𝜃>0,∀𝑡₀∈𝑅, (1.4) they prove the boundedness of solutions and the existence of quasiperiodic solution by Moser’ twist theorem. It is the first time that the equation of the boundedness of all solution is treated in case of a singular potential.

In this paper, We consider the following sublinearly growing potential: 𝑥^′′+𝑉^′(𝑥)=𝑝(𝑡), (1.5) where 𝑉(𝑥)=𝑥₊^𝛼+1+(1/1−𝑥²₋)−1,0<𝛼<1.

Our main result is the following theorem.

Theorem 1.1. If 𝑝(𝑡)∈𝐶⁶ is 1-periodic continuous, then all the solutions of (1.5) are bounded.

The idea for proving the boundedness of solutions of (1.5) is as follows. By means of transformation theory, (1.5) is, outside of a large disc 𝒟={(𝑥,𝑥̇)∈ℝ²∶𝑥²+𝑥̇²≤𝑟²} in the (𝑥,𝑥̇)-plane, transformed into a perturbation of an integrable Hamiltonian system. Then, Poincaré map of the transformed system is close to a so-called twist map in ℝ⧵𝒟. The Moser’s twist theorem [9] guarantees the existence of arbitrarily large invariant curves diffeomorphic to circles and surrounding the origin in the (𝑥,𝑥̇)-plane. Every such curves is the base of a time-periodic and flow-invariant cylinder in the extended phase space (𝑥,𝑥̇,𝑡)∈ℝ²×ℝ, which confines the solutions in the interior and which leads to a bound of these solutions.

The paper is organized as follows. In Sections 2.1 and 2.2, we give action-angle variables and some estimates which is useful for our proof. In Section 2.3, we will give an asymptotic expression of the Poincaré map and prove the main result by Moser’s twist theorem [9].

2. Proof of Theorem

2.1. Action-Angle Variables and Some Estimates

Without loss of generality and for brevity of arguments, we assume that the average value of 𝑝(𝑡) vanishes; that is, ∫¹₀𝑝(𝑡)𝑑𝑡=0. Hence the function 𝑃(𝑡)=∫^𝑠₀𝑝(𝑠)𝑑𝑠 is also 1-periodic in 𝑡 and is in 𝐶⁶.

System (1.5) is equivalent to the planar Hamiltonian system 𝑥^′=𝑦+𝑃(𝑡),𝑦^′=−𝑉^′(𝑥), (2.1) where Hamiltonian is ℋ(𝑥,𝑦,𝑡)=(1/2)𝑦²+𝑉(𝑥)+𝑦𝑃(𝑡).

In order to introduce action and angle variables, we first consider the auxiliary autonomous system 𝑥^′=𝑦,𝑦^′=−𝑉^′(𝑥), (2.2) which is integrable with the Hamiltonian 𝐻₀=1𝑥,𝑦2𝑦²+𝑉(𝑥). (2.3) The closed curves 𝐻₀(𝑥,𝑦)=ℎ>0 are just the integral curves of (2.2). Denote by 𝑇₀(ℎ) the time period of the integral curve Γ_ℎ: 𝐻₀(𝑥,𝑦)=ℎ and by 𝐼₀ the area enclosed by the closed curve Γ_ℎ. Let 𝛼_ℎ=ℎℎ+1,𝛽_ℎ=ℎ^1/(1+𝛼). (2.4) Then, 𝑉(−𝛼_ℎ)=𝑉(𝛽_ℎ)=ℎ.

It is easy to see that 𝐼₀(ℎ)=2^𝛽ℎ_−𝛼ℎ2((𝑠𝑇ℎ−𝑉))𝑑𝑠,∀ℎ>0,₀(ℎ)=𝐼^′₀(ℎ)=2^𝛽ℎ_−𝛼ℎ12(ℎ−𝑉(𝑠))𝑑𝑠,∀ℎ>0. (2.5) Denote 𝐼₊(ℎ)=2^𝛽ℎ₀2(ℎ−𝑉(𝑠))𝑑𝑠,𝐼₋(ℎ)=2^𝛼ℎ₀2((𝑇ℎ−𝑉−𝑠))𝑑𝑠,₊(ℎ)=2^𝛽ℎ₀12(ℎ−𝑉(𝑠))𝑑𝑠,𝑇₋(ℎ)=2^𝛼ℎ₀12(ℎ−𝑉(−𝑠))𝑑𝑠. (2.6) Then, 𝐼₀(ℎ)=𝐼₋(ℎ)+𝐼₊(ℎ),𝑇₀(ℎ)=𝑇₋(ℎ)+𝑇₊(ℎ). (2.7)

The following estimates on the functions 𝐼₊, 𝐼₋, and 𝐼₀ and 𝑇₊, 𝑇₋, and 𝑇₀ are crucial for this paper. We first estimate 𝐼₊ and 𝑇₊. Since 𝐼₊ is the area enclosed by the closed curve Γ_ℎ and 𝑦-axis when 𝑥≥0, we can easily prove that 𝐼₊(ℎ)=2^𝛽ℎ₀2ℎ−𝑥^𝛼+1𝑑𝑥=2^{ℎ1/(1+𝛼)}₀2ℎ−𝑥^𝛼+1𝑑𝑥. (2.8)

Let 𝑥=𝑡ℎ^1/(1+𝛼), then we get 𝐼₊(ℎ)=2¹₀2ℎ1−𝑡^1+𝛼ℎ^1/(1+𝛼)𝑑𝑡=2√2ℎ^{(1/2)+1/(1+𝛼)1}₀√1−𝑡^1+𝛼𝑑𝑡. (2.9) Since 𝑇₊(ℎ)=𝐼^′₊(ℎ), we have 𝑇₊(ℎ)=2√212+1ℎ𝛼+1^{−(1/2)+1/(1+𝛼)1}₀√1−𝑡^1+𝛼𝑑𝑡. (2.10)

We now give the estimates on the function 𝐼₋ and 𝑇₋.

Lemma 2.1. One has ℎ^𝑛𝑑^𝑛𝑇₋(ℎ)𝑑ℎ^𝑛≤𝐶ℎ^−1/2,ℎ^𝑛𝑑^𝑛𝐼₋(ℎ)𝑑ℎ^𝑛≤𝐶ℎ^1/2, (2.11) where 𝑛=0,1,…,6, ℎ→+∞. Note that here and below, one always uses 𝐶, 𝐶₀, or 𝐶^′₀ to indicate some constants.

Proof. Now, we estimate the first inequality. We choose 𝑉(𝑠)/ℎ=𝜂 as the new variable of integration, then we have 𝑇₋(ℎ)=⁰_−𝛼ℎ12(ℎ−𝑉(𝑠))𝑑𝑠=¹₀√ℎ𝑉^′𝑠1𝜂,ℎ21−𝜂𝑑𝜂. (2.12) Since 𝑉(𝑠)=(1/(1−𝑠²))−1 and 𝑉(𝑠)/ℎ=𝜂, we have 𝑠=𝜂ℎ/(1+𝜂ℎ). By direct computation, we have 𝑉^′(𝑠)=2𝑠1−𝑠²²=2𝜂ℎ1+𝜂ℎ²1+𝜂ℎ, (2.13) then we get 𝑇₋^(𝑛)(ℎ)=(−3/2)!(−(3/2)−𝑛)!¹₀𝜂^𝑛2𝜂1−𝜂1+𝜂ℎ^(3/2)+𝑛𝑑𝜂,𝑛=0,1,…,6. (2.14) When 0≤𝜂≤ℎ⁻¹ and ℎ sufficient large, there exits 𝐶₀ such that 1−𝜂>𝐶₀, so we have ^ℎ−1₀𝜂^𝑛2𝜂1−𝜂1+𝜂ℎ^(3/2)+𝑛𝑑𝜂≤𝐶^ℎ−1₀𝜂^𝑛≤𝐶2𝜂1−𝜂𝑑𝜂𝐶₀^ℎ−1₀𝜂^𝑛−(1/2)𝑑𝜂≤𝐶ℎ^{−(1/2)−𝑛}. (2.15) Since ℎ^−2/3≤𝜂≤1, we have ℎ^1/3<1+ℎ^1/3≤1+𝜂ℎ≤1+ℎ, (2.16) then ¹_ℎ−2/3𝜂^𝑛2𝜂1−𝜂1+𝜂ℎ^(3/2)+𝑛𝑑𝜂≤𝐶¹_ℎ−2/3𝜂^𝑛ℎ^𝑛2𝜂1−𝜂ℎ^𝑛1+𝜂ℎ^𝑛1+𝜂ℎ^3/2𝑑𝜂≤𝐶¹_ℎ−2/312𝜂1−𝜂ℎ^𝑛1+𝜂ℎ^3/2𝑑𝜂≤𝐶¹_ℎ−2/312𝜂1−𝜂ℎ^𝑛ℎ^1/2𝑑𝜂≤𝐶ℎ^{−(1/2)−𝑛1}₀12𝜂1−𝜂𝑑𝜂≤𝐶ℎ^{−(1/2)−𝑛}. (2.17) Observing that there is 𝐶₀>0 such that 1−𝜂≥𝐶₀ when ℎ⁻¹≤𝜂≤ℎ^−2/3 and ℎ→+∞, we have ^ℎ−2/3_ℎ−1𝜂^𝑛2𝜂1−𝜂1+𝜂ℎ^(3/2)+𝑛𝑑𝜂≤𝐶₁ℎ^{−(3/2)−𝑛ℎ−2/3}_ℎ−112𝜂1−𝜂𝜂^3/2≤𝐶𝑑𝜂₁𝐶₀ℎ^{−(3/2)−𝑛ℎ−2/3}_ℎ−11𝜂²𝐶𝑑𝜂=¹𝐶₀ℎ^{−(3/2)−𝑛}1𝜂∣^ℎ−2/3_ℎ−1=𝐶₁𝐶₀ℎ^{−(3/2)−𝑛}ℎ−ℎ^2/3≤𝐶ℎ^{−(1/2)−𝑛}. (2.18) By (2.15)–(2.18), we have 𝑇₋^(𝑛)(ℎ)≤𝐶ℎ^{−(1/2)−𝑛}, 𝑛=0,1,…,6.
The proof of the second inequality is similar to the first one, so we only give the brief proof.
We choose 𝑉(𝑠)/ℎ=𝜂 as the new variable of integration, so we have 𝜕𝑠=𝜂𝜕ℎ𝑉^′,𝑠=𝜂ℎ,𝑉1+𝜂ℎ_′(𝑠)=2𝑠1−𝑠²²=2𝜂ℎ1+𝜂ℎ².1+𝜂ℎ (2.19) By direct computation, we have 𝐼₋(ℎ)=2⁰_−𝛼ℎ2(ℎ−𝑉(𝑠))𝑑𝑠=ℎ¹₀21−𝜂√𝜂1+𝜂ℎ^3/2𝑑𝜂. (2.20) By (2.20), we can easily get 𝐼₋^(𝑛)(ℎ)=𝐼^(𝑛)₋₁(ℎ)+𝐼^(𝑛)₋₂(ℎ)()!=𝑛−3/2(−(3/2)−𝑛+1)!¹₀21−𝜂√𝜂𝜂^𝑛−11+𝜂ℎ^{(3/2)+𝑛−1}+()!𝑑𝜂−3/2(−())!ℎ3/2−𝑛¹₀21−𝜂√𝜂𝜂^𝑛1+𝜂ℎ^(3/2)+𝑛𝑑𝜂, (2.21) where 𝑛=0,1,…,6.
By the similar way in estimating 𝑇₋^(𝑛)(ℎ), we get 𝐼^(𝑛)₋₁(ℎ)≤𝐶ℎ^(1/2)−𝑛,𝐼^(𝑛)₋₂(ℎ)≤𝐶ℎ^(1/2)−𝑛, (2.22) which means that 𝐼₋^(𝑛)(ℎ)≤𝐶ℎ^(1/2)−𝑛,𝑛=0,1,…,6. (2.23)
Thus, we complete the proof of Lemma 2.1.

Remark 2.2. It follows from (2.10) and Lemma 2.1 that lim_ℎ→+∞𝑇₋(ℎ)=0,lim_ℎ→+∞𝑇₊(ℎ)=+∞. (2.24) Thus, the time period 𝑇₀(ℎ) is dominated by 𝑇₊(ℎ) when ℎ is sufficiently large. By 𝑇₀(ℎ)=𝐼^′₀(ℎ), we know 𝐼₀(ℎ) is dominated by 𝐼₊(ℎ) when ℎ is sufficiently large.

Remark 2.3. It also follows from the definition of 𝐼₀(ℎ), 𝐼₋(ℎ), 𝐼₊(ℎ) and Remark 2.2 that ℎ^𝑛𝑑^𝑛𝐼₀(ℎ)𝑑ℎ^𝑛≤𝐶₀𝐼₀(ℎ),for𝑛≥1. (2.25) In particular, 𝑐ℎ^{(1/2)+1/(1+𝛼)}≤𝐼₀(ℎ)≤𝐶ℎ^{(1/2)+1/(1+𝛼)}. (2.26)

Remark 2.4. Note that ℎ=ℎ₀(𝐼₀) is the inverse function of 𝐼₀. By Remark 2.3, we have 𝐼^𝑛𝑑^𝑛ℎ(𝐼)𝑑𝐼^𝑛≤𝐶₀ℎ(𝐼),for𝑛≥1. (2.27)

We now carry out the standard reduction to the action-angle variables. For this purpose, we define the generating function 𝑆(𝑥,𝐼)=∫_Γ2(ℎ−𝑉(𝑠))𝑑𝑠, where Γ is the part of the the closed curve Γ_ℎ connecting the point on the 𝑦-axis and point (𝑥,𝑦).

We define the well-known map (𝜃,𝐼)→(𝑥,𝑦) by 𝑦=𝜕𝑆()𝜕𝑥𝑥,𝐼,𝜃=𝜕𝑆()𝜕𝐼𝑥,𝐼. (2.28) It is well-known that the map is symplectic, since 𝑑𝑥∧𝑑𝑦=𝑑𝑥∧𝑆_𝑥𝑥𝑑𝑥+𝑆_𝑥𝐼𝑑𝐼=𝑆_𝑥𝐼𝑆𝑑𝑥∧𝑑𝐼,𝑑𝜃∧𝑑𝐼=_𝐼𝑥𝑑𝑥+𝑆_𝐼𝐼𝑑𝐼∧𝑑𝐼=𝑆_𝐼𝑥𝑑∧𝑑𝐼. (2.29) From the above discussion, we can easily get 1𝜃=𝑇₀ℎ𝑥,𝑦^𝑥_−𝛼ℎ12ℎ(𝑠)1𝑥,𝑦−𝑉𝑑𝑠,if𝑦>0,1−𝑇₀ℎ𝑥,𝑦^𝑥_−𝛼ℎ12ℎ𝑥,𝑦−𝑉(𝑠)𝑑𝑠,if𝑦<0. (2.30)𝐼𝑥,𝑦=𝐼₀ℎ𝑥,𝑦=2^𝛽ℎ_−𝛼ℎ2ℎ𝑥,𝑦−𝑉(𝑠)𝑑𝑠. (2.31) In the new variables (𝜃,𝐼), the system (2.1) becomes 𝜃^′=𝜕𝐻𝜕𝐼,𝐼^′=−𝜕𝐻𝜕𝜃, (2.32) where 𝐻(𝜃,𝐼,𝑡)=ℎ₀(𝐼)+𝐻₁(𝐼,𝜃,𝑡), (2.33) where ℎ₀(𝐼) is the inverse function of 𝐼₀(𝐻) and 𝐻₁(𝐼,𝜃,𝑡)=𝑦(𝐼,𝜃)𝑃(𝑡).

In order to estimate 𝐻₁(𝐼,𝜃,𝑡), we need the estimate on the functions 𝑥(𝐼,𝜃) and 𝑦(𝐼,𝜃). For this purpose, we first give some definitions which are very similar to those in [4].

Define a function 𝐿 in terms of ℎ and 𝑉 by 𝐿()ℎ𝑥,𝐼=−^𝐼𝐼ℎ_𝐼+ℎ_𝐼ℎ𝑊^′−12, (2.34) and a linear differential operator acting on functions of 𝑥,𝐼, according to ℒ𝑓=ℎ_𝐼ℎ𝑓𝑉𝑉_𝑥𝑥−12𝑓+𝑓_𝐼, (2.35) where 𝑓(𝑥,𝐼) is a smooth function, and we denote ℒ^𝑛=ℒ∘⋯∘ℒ_𝑛.

The following equality (its proof can be found in [4]) is crucial for the proof of the following lemmas: 𝑑𝑑𝐼^𝑥_−𝛼ℎ𝑓()1𝑠,𝐼ℎ−𝑉(𝑠)𝑑𝑠=^𝑥_−𝛼ℎℒ𝑓1(𝑠)()ℎ−𝑉𝑑𝑠+𝑓𝑥,𝐼^𝑥_−𝛼ℎ𝐿()1𝑠,𝐼ℎ−𝑉(𝑠)𝑑𝑠. (2.36)

Before giving the estimates on 𝑥(𝐼,𝜃) and 𝑦(𝐼,𝜃), we now prove some lemmas which will be used frequently in the following proof.

Lemma 2.5. Suppose that there is a constant 𝐶₀ such that |𝑔(𝑥,𝐼)|≤𝐶₀𝐼^−𝑘, then one can find a constant 𝐶^′₀ and 𝐶^′₁ such that, for −𝛼_ℎ≤𝑥≤0, 𝑉^′(𝑥)^𝑥_−𝛼ℎ𝑔(𝑠,𝐼)ℎ−𝑉(𝑠)𝑑𝑠≤𝐶^′₀𝐼^−𝑘ℎ−𝑉(𝑥), (2.37)ℎ−𝑉(𝑥)^𝑥_−𝛼ℎ𝑔(𝑠,𝐼)ℎ−𝑉(𝑥)≤𝐶^′₁𝐼^−𝑘𝛼_ℎ+𝑥. (2.38)

Proof. We now prove (2.37). Let 𝐺(𝑥,𝐼)=𝐼^−𝑘ℎ−𝑉(𝑥)𝑉^′(𝑥),𝐹(𝑥,𝐼)=^𝑥_−𝛼ℎ𝑔()1𝑠,𝐼ℎ−𝑉(𝑠)𝑑𝑠, (2.39) then 𝐹−𝛼_ℎ,𝐼=lim_{𝑥→−𝛼ℎ}𝐺(𝑥,𝐼)=0. (2.40) By direct calculation, 𝜕𝐹(𝑥,𝐼)=𝑔()1𝜕𝑥𝑥,𝐼ℎ−𝑉(𝑥)≤𝐶₀𝐼^−𝑘1(𝑥),()ℎ−𝑉𝜕𝐺𝑥,𝐼𝜕𝑥=𝐼^−𝑘𝑉^′(𝑥)²+(ℎ−𝑉(𝑥))𝑉^′′(𝑥)𝑉^′(𝑥)²1(𝑥).ℎ−𝑉 (2.41) Since 𝑉^′′(𝑥)=1−𝑥²²+4𝑥²1−𝑥²1−𝑥²⁴>0, (2.42) we have 𝑉^′(𝑥)²+(ℎ−𝑉(𝑥))𝑉^′′(𝑥)𝑉^′(𝑥)²>0. (2.43) By |𝑔(𝑥,𝐼)|≤𝐶₀𝐼^−𝑘, there is 𝐶^′₀=2𝐶₀ such that −𝐶^′₀𝐼^−𝑘𝑉^′(𝑥)²+(ℎ−𝑉(𝑥))𝑉^′′(𝑥)𝑉^′(𝑥)²1((𝑥≤()ℎ−𝑉))𝜕𝐹𝑥,𝐼/𝜕𝑥2ℎ−𝑉(𝑥)≤𝐶^′₀𝐼^−𝑘𝑉^′(𝑥)²+(ℎ−𝑉(𝑥))𝑉^′′(𝑥)𝑉^′(𝑥)²1(𝑥),ℎ−𝑉 (2.44) that is −𝐶^′₀𝜕𝐺(𝑥,𝐼)≤()𝜕𝑥𝜕𝐹𝑥,𝐼𝜕𝑥≤𝐶^′₀𝜕𝐺(𝑥,𝐼)𝜕𝑥, (2.45) which means that −𝐶^′₀𝐺(𝑥,𝐼)≤𝐹(𝑥,𝐼)≤𝐶^′₀𝐺(𝑥,𝐼). (2.46) That is, 𝑉^′(𝑥)^𝑥_−𝛼ℎ𝑔()1𝑠,𝐼ℎ−𝑉(𝑥)≤𝐶𝐼^−𝑘ℎ−𝑉(𝑥). (2.47) Thus, we complete the proof (2.37).
Now, we prove (2.38). Let 𝐺(𝑥,𝐼)=𝐼^−𝑘𝛼_ℎ+𝑥(𝑥)()=ℎ−𝑉,𝐹𝑥,𝐼^𝑥_−𝛼ℎ𝑔()1𝑠,𝐼ℎ−𝑉(𝑠)𝑑𝑠. (2.48) Then, we have 𝐹−𝛼_ℎ,𝐼=lim_{𝑥→−𝛼ℎ}𝐺(𝑥,𝐼)=0. (2.49) By direct computation, we have 𝜕𝐹(𝑥,𝐼)=𝑔()1𝜕𝑥𝑥,𝐼(ℎ−𝑉(𝑥))≤𝐶₀𝐼^−𝑘1(𝑥),()ℎ−𝑉𝜕𝐺𝑥,𝐼𝜕𝑥=𝐼^−𝑘𝑉1+^′(𝑥)𝛼_ℎ+𝑥2((𝑥1ℎ−𝑉))(𝑥).ℎ−𝑉 (2.50) Since 𝑉^′′(𝑥)>0 and ℎ=𝑉(−𝛼_ℎ+𝑥), it follows that for 𝑉^′(𝑥)𝛼_ℎ2((𝑥≤1ℎ−𝑉))2, (2.51) so for 𝐶^′₁>2𝐶₀+1, we have −𝐶^′₁𝜕𝐺(𝑥,𝐼)≤()𝜕𝑥𝜕𝐹𝑥,𝐼𝜕𝑥≤𝐶^′₁𝜕𝐺(𝑥,𝐼)𝜕𝑥, (2.52) which means that −𝐶^′₁𝐺(𝑥,𝐼)≤𝐹(𝑥,𝐼)≤𝐶^′₁𝐺(𝑥,𝐼). (2.53) By the definition of 𝐺(𝑥,𝐼), we have ℎ−𝑉(𝑥)𝐹(𝑥,𝐼)≤𝐶^′₁𝐼^−𝑘𝛼_ℎ+𝑥. (2.54) Thus, we complete the proof of (2.38) and Lemma 2.5.

By Lemma 2.5, we have the following Lemma which is important to our estimation.

Lemma 2.6. One can find a constant 𝐶 such that, for −𝛼_ℎ≤𝑥≤0, 𝜕^𝑘_𝐼^𝑥_−𝛼ℎ𝐿(𝑥,𝐼)(𝑠)≤𝐶ℎ−𝑉𝑑𝑠𝑉^′(𝑥)𝐼^−(𝑘+1)ℎ−𝑉(𝑥), (2.55)𝜕^𝑘_𝐼^𝑥_−𝛼ℎ𝐿(𝑥,𝐼)(𝑠)≤𝐶ℎ−𝑉𝑑𝑠(𝑥)𝐼ℎ−𝑉^−(𝑘+1)𝛼_ℎ+𝑥, (2.56) where 0≤𝑘≤6.

Proof. When 𝑘=1, we have 𝜕_𝐼^𝑥_−𝛼ℎ𝐿1(𝑠)=ℎ−𝑉𝑑𝑠^𝑥_−𝛼ℎℒ(𝐿)1ℎ−𝑉(𝑠)𝑑𝑠+𝐿^𝑥_−𝛼ℎ𝐿1ℎ−𝑉(𝑠)𝑑𝑠. (2.57) By the definitions of 𝐿 and ℒ, we have |𝐿|≤𝐶𝐼⁻¹ and |ℒ(𝐿)|≤𝐶𝐼⁻². By (2.37), we obtain ^𝑥_−𝛼ℎ𝐿1(𝑠)𝐶ℎ−𝑉𝑑𝑠≤𝑉^′𝐼⁻²√ℎ−𝑉. (2.58) Suppose that 𝑘=𝑙, we have 𝜕^𝑙_𝐼^𝑥_−𝛼ℎ𝐿1(𝑠)≤𝐶ℎ−𝑉𝑑𝑠𝑉^′𝐼^−(𝑙+1)√ℎ−𝑉. (2.59) We now proven that when 𝑘=𝑙+1, 𝜕_𝐼^𝑙+1𝑥_−𝛼ℎ𝐿1(𝑠)≤𝐶ℎ−𝑉𝑑𝑠𝑉^′𝐼^−(𝑙+2)√ℎ−𝑉. (2.60) By direct computation, we have 𝜕_𝐼^𝑙+1𝑥_−𝛼ℎ𝐿(𝑠)=𝜕ℎ−𝑉𝑑𝑠^𝑛_𝐼ℒ^𝑚𝑓𝜕_𝐼^{𝑙−𝑚−𝑛𝑥}_−𝛼ℎ𝐿(𝑠),ℎ−𝑉𝑑𝑠^𝑥_−𝛼ℎℒ^𝑙+1(𝐿)(𝑠),ℎ−𝑉𝑑𝑠 (2.61) where {𝑓₁,𝑓₂,…,𝑓_𝑛} denotes linear combination of functions 𝑓₁,…,𝑓_𝑛 with integer coefficients and 0≤𝑚+𝑛≤𝑙.
Since |ℒ(𝐿)|≤𝐶𝐼⁻², we have ℒ^𝑙+1(𝐿)≤𝐶𝐼^−(𝑙+2). By (2.37), we obtain ^𝑥_−𝛼ℎℒ^𝑙+1(𝐿)(𝑠)≤𝐶ℎ−𝑉𝑑𝑠𝑉^′𝐼^−(𝑙+2)√ℎ−𝑉. (2.62) By direct computation, we get 𝜕^𝑛ℒ^𝑚(𝐿)≤𝐶𝐼^{−(𝑚+𝑛+1)}. (2.63) By assumption (2.59), we have 𝜕_𝐼^{𝑙−𝑚−𝑛𝑥}_−𝛼ℎ𝐿(𝑠)≤𝐶ℎ−𝑉𝑑𝑠𝑉^′𝐼^{−(𝑙−𝑚−𝑛+1)}√ℎ−𝑉. (2.64) So, we have 𝜕^𝑛_𝐼ℒ^𝑚𝑓𝜕_𝐼^{𝑙−𝑚−𝑛𝑥}_−𝛼ℎ𝐿(𝑠)≤𝐶ℎ−𝑉𝑑𝑠𝑉^′𝐼^−(𝑙+2)√ℎ−𝑉. (2.65) By (2.62) and (2.65), we have 𝜕_𝐼^𝑙+1𝑥_−𝛼ℎ𝐿(𝑠)≤𝐶ℎ−𝑉𝑑𝑠𝑉^′𝐼^−(𝑙+2)√ℎ−𝑉. (2.66) Thus, we have proved (2.55).
The inequality (2.56) can be proved by (2.38), and the process of proof is similar to that of (2.55), so we omit it.
Thus, we have proved Lemma 2.6.

Now, we give the estimates of 𝑥(𝐼,𝜃) and 𝑦(𝐼,𝜃).

Lemma 2.7. For 𝐼 sufficient large and −𝛼_ℎ≤𝑥<0, the following estimates hold: 𝐼^𝑛𝜕^𝑛𝑥(𝐼,𝜃)𝜕𝐼^𝑛≤𝐶|𝑥(𝐼,𝜃)+1|,𝐼^𝑛𝜕^𝑛𝑦(𝐼,𝜃)𝜕𝐼^𝑛≤𝐶𝑦(𝐼,𝜃),for0≤𝑛≤6. (2.67)

Proof. We now prove the first inequality. It is sufficient to prove that 𝐼^𝑛𝜕^𝑛𝑥(𝐼,𝜃)𝜕𝐼^𝑛≤𝐶(1+𝑥). (2.68) Case 𝑘=1. Differentiating (2.30) by 𝐼 and noting 1/𝑇₀=ℎ_𝐼, we have 0=ℎ_𝐼𝐼^𝑥_−𝛼ℎ12ℎ𝑥,𝑦−𝑉(𝑠)𝑑𝑠+ℎ_𝐼𝜕_𝐼^𝑥_−𝛼ℎ12ℎ𝑥,𝑦−𝑉(𝑠)𝑑𝑠. (2.69) Now, we choose 𝑉(𝑠)/ℎ=𝜂 as the new variable of integration, so 𝜕_𝐼^𝑥_−𝛼ℎ12ℎ𝑥,𝑦−𝑉(𝑠)𝑑𝑠=𝜕_𝐼0^{𝑉(𝑥)/ℎ}√ℎ𝑉^′𝑠1𝜂,𝑡,ℎ2=𝑉1−𝜂𝑑𝜂_′𝑥_𝐼ℎ−𝑉ℎ_𝐼ℎ²ℎ^1/2𝑉^′12+1−𝜂^{0𝑉(𝑥)/ℎ}(1/2)ℎ^−1/2𝑉^′ℎ_𝐼−ℎ^1/2𝑉^′′(𝜕𝑠/𝜕𝐼)𝑉^′212=𝑥1−𝜂𝑑𝜂_𝐼2()−ℎℎ−𝑉ℎ2(^𝐼)𝑉ℎ−𝑉𝑉^′+ℎ_𝐼ℎ+₀^{𝑉(𝑥)/ℎ}(1/2)ℎ^−1/2𝑉^′ℎ_𝐼−ℎ^1/2𝑉^′′(𝜕𝑠/𝜕𝐼)𝑉^′2√ℎ2()𝑉ℎ−𝑉ℎ^′𝑑𝑠. (2.70) Observing that 𝜕𝑠=ℎ𝜕𝐼^𝐼𝜂𝑉^′,𝑊^′=1−𝑉𝑉^′′𝑉^′2, (2.71) and simplifying, we have ()=𝑥2.70^𝐼2()−ℎℎ−𝑉ℎ2(^𝐼)𝑉ℎ−𝑉𝑉^′+ℎ_𝐼ℎ^𝑥₀𝑊^′−12𝑑𝑠2(ℎ−𝑉). (2.72) By (2.69)–(2.72), we have 𝑥_𝐼(𝜃,𝐼)=2(ℎ−𝑉)^𝑥_−𝛼ℎ𝐿()1𝑠,𝐼2((𝑠ℎℎ−𝑉))𝑑𝑠+^𝐼ℎ𝑊(𝑥). (2.73) Since |𝐿|≤𝐶𝐼⁻¹, by Lemma 2.6, we have (2(ℎ−𝑉))^𝑥_−𝛼ℎ𝐿()1𝑠,𝐼2(ℎ−𝑉(𝑠))𝑑𝑠≤𝐶𝐼⁻¹𝛼_ℎ+𝑥. (2.74) We observe that 𝑊(𝑥)=12𝑥(1−𝑥)(1+𝑥), (2.75) where −1<𝑥<0, so |𝑊(𝑥)|≤𝐶(1+𝑥), which means that ℎ_𝐼ℎ𝑊(𝑥)≤𝐶𝐼⁻¹(1+𝑥). (2.76) By (2.74) and (2.76), we have |𝑥_𝐼|≤𝐼⁻¹(1+𝑥).
We suppose that 𝜕_𝐼^𝑘−1𝑥≤𝐼^{−(𝑘−1)}(1+𝑥), (2.77) where 1≤𝑘≤6. We will prove |𝜕^𝑘_𝐼𝑥|≤𝐼^−𝑘(1+𝑥).
For this purpose, we firstly estimate 2(ℎ−𝑉(𝑥)). We differentiate 2(ℎ−𝑉(𝑥)) in (2.73) and using (2.73), then we obtain 𝑑𝑑𝐼2((𝑥=ℎℎ−𝑉))^𝐼ℎ2((𝑥−𝑉ℎ−𝑉))^′(𝑥)2^𝑥_−𝛼ℎ𝐿()1𝑠,𝐼2(ℎ−𝑉(𝑠))𝑑𝑠. (2.78) Since |ℎ_𝐼/ℎ|≤𝐶𝐼⁻¹, we have ℎ_𝐼ℎ2(ℎ−𝑉(𝑥))≤𝐶𝐼⁻¹2(ℎ−𝑉(𝑥)). (2.79) By (2.37), we have 𝑉^′(𝑥)2^𝑥_−𝛼ℎ𝐿()1𝑠,𝐼2(ℎ−𝑉(𝑠))𝑑𝑠≤𝐶𝐼⁻¹2(ℎ−𝑉(𝑥)). (2.80) By (2.78)–(2.80), we have 𝜕_𝐼2(ℎ−𝑉(𝑥))≤𝐶𝐼⁻¹2(ℎ−𝑉(𝑥)). (2.81) We suppose that when𝑛≤𝑘−1, 𝜕^𝑛_𝐼2(ℎ−𝑉(𝑥))≤𝐶𝐼^−𝑛2(ℎ−𝑉(𝑥)). (2.82) We will prove that when𝑛=𝑘, 𝜕^𝑛_𝐼2(ℎ−𝑉(𝑥))≤𝐶𝐼^−𝑛2(ℎ−𝑉(𝑥)). (2.83) By direct computation, we have 𝜕^𝑘_𝐼2(ℎ−𝑉(𝑥))=𝑉^(𝑘1)𝜕^𝑙1_𝐼𝑥⋯𝜕^{𝑙𝑘1−1}_𝐼𝑥𝜕^𝑘3_𝐼^𝑥_−𝛼ℎ𝐿()1𝑠,𝐼2((𝑠,𝜕ℎ−𝑉))𝑑𝑠_𝑚𝐼ℎ_𝐼ℎ𝜕_𝐼^{𝑘−𝑚−1}2(ℎ−𝑉(𝑥)), (2.84) where 𝑙₁+⋯+𝑙_𝑘1−1=𝑘₂,𝑘₂+𝑘₃=𝑘−1 and 𝑘₂<𝑘₁<𝑘,𝑘₃<𝑘.
By the assume (2.77), we have 𝜕^𝑙1_𝐼𝑥⋯𝜕^{𝑙𝑘1−1}_𝐼𝑥≤(1+𝑥)^𝑘1−1𝐼^−𝑘2. (2.85) By (2.55), we get 𝜕^𝑘3_𝐼^𝑥_−𝛼ℎ𝐿()1𝑠,𝐼2((𝑠≤𝐶ℎ−𝑉))𝑑𝑠𝑉^′𝐼^−𝑘32(ℎ−𝑉(𝑥)) (2.86) By (2.85), (2.86) and noting the fact that (1+𝑥)^𝑘−1𝜕^𝑘_𝐼(𝑉)≤𝐶𝑉^′+(1+𝑥)^𝑘1+1,−1<𝑥<0, (2.87) we obtain 𝑉^(𝑘1)𝜕^𝑙1_𝐼𝑥⋯𝜕^{𝑙𝑘1−1}_𝐼𝑥𝜕^𝑘3_𝐼^𝑥_−𝛼ℎ𝐿()1𝑠,𝐼2((𝑠≤𝐶ℎ−𝑉))𝑑𝑠𝑉^′𝑉^(𝑘1)(1+𝑥)^𝑘1−1𝐼^{−(𝑘2+𝑘3+1)}2((𝑥𝑉ℎ−𝑉))≤^′+(1+𝑥)^𝑘1+1𝑉^′𝐼^−(𝑘)2(ℎ−𝑉(𝑥))≤𝐶𝐼^−(𝑘)2(ℎ−𝑉(𝑥)). (2.88) By assumption (2.82), we have 𝜕_𝐼^{𝑘−𝑚−1}2(ℎ−𝑉(𝑥))≤𝐶𝐼^{𝑘−𝑚−1}2(ℎ−𝑉(𝑥)). (2.89) By (2.89) and the fact that |𝜕^𝑚_𝐼(ℎ_𝐼/ℎ)|≤𝐶𝐼^−(𝑚+1), we have 𝜕^𝑚_𝐼ℎ_𝐼ℎ𝜕_𝐼^{𝑘−𝑚−1}2(ℎ−𝑉(𝑥))≤𝐶𝐼^−𝑘2(ℎ−𝑉(𝑥)). (2.90) So, by (2.88) and (2.90), we have proved (2.83).
By (2.56), we have 𝜕^𝑚_𝐼^𝑥_−𝛼ℎ𝐿(𝑥,𝐼)2((𝑠≤𝐶ℎ−𝑉))𝑑𝑠2((𝑥𝐼ℎ−𝑉))^−(𝑚+1)𝛼_ℎ+𝑥. (2.91) By (2.83) and (2.91), we have 𝜕^𝑛2(ℎ−𝑉(𝑥))𝜕^𝑚𝑥_−𝛼ℎ𝐿()1𝑠,𝐼2(ℎ−𝑉(𝑠))𝑑𝑠≤𝐶𝐼^{−(𝑚+𝑛+1)}𝛼_ℎ+𝑥≤𝐶𝐼^−(𝑘+1)𝛼_ℎ+𝑥. (2.92) By the assumption (2.77), the fact that 𝜕^𝑚3_𝐼(ℎ_𝐼/ℎ)≤𝐶𝐼^−(𝑚3+1), and noting that 𝜕_𝐼ℎ_𝐼ℎ𝑊=^𝑘𝑊^(𝑚1)𝜕^𝑙1_𝐼𝑥⋯𝜕^𝑙𝑚1_𝐼𝑥𝜕^𝑚3_𝐼ℎ_𝐼ℎ, (2.93) where 𝑚₂<𝑚₁,𝑙₁+⋯+𝑙_𝑚1=𝑚₂,𝑚₂+𝑚₃=𝑘, we have 𝜕_𝐼ℎ_𝐼ℎ𝑊≤^𝑘𝑊^(𝑚1)𝐼^{−(𝑙1+⋯+𝑙𝑚1)}(1+𝑥)^𝑚1𝐼^−(𝑚3+1)≤𝐶^𝑘𝑊^(𝑚1)(1+𝑥)^𝑚1𝐼^{−(𝑚2+𝑚3+1)}≤𝐶(1+𝑥)𝐼^−(𝑘+1). (2.94) By (2.92) and (2.94), we have 𝜕^𝑘_𝐼𝑥_𝐼=^𝑘𝜕^𝑛2(ℎ−𝑉(𝑥))𝜕^𝑚𝑥_−𝛼ℎ𝐿()1𝑠,𝐼2(ℎ−𝑉(𝑠))𝑑𝑠+𝜕^𝑘ℎ_𝐼ℎ𝑊(𝑥)≤𝐶𝐼^−(𝑘+1)(1+𝑥)≤𝐶𝐼^−(𝑘+1)(1+𝑥), (2.95) which means 𝜕^𝑘_𝐼𝑥≤𝐶𝐼^−𝑘(1+𝑥). (2.96)
We now prove 𝐼^𝑛𝜕^𝑛𝑦(𝐼,𝜃)𝜕𝐼^𝑛≤𝑐√𝐼. (2.97) Since ℎ(𝐼)=12𝑦²+𝑉(𝑥), (2.98) we have 𝑦=±2(ℎ(𝐼)−𝑉(𝑥)). (2.99) we have proved (2.83), so we have 𝐼^𝑛𝜕^𝑛𝑦(𝐼,𝜃)𝜕𝐼^𝑛≤𝐶⋅𝐼^−𝑛𝑦, (2.100) which means that 𝐼^𝑛𝜕^𝑛𝑦(𝐼,𝜃)𝜕𝐼^𝑛≤𝐶⋅ℎ₀(𝐼). (2.101) The proof of Lemma 2.7 is complete.

Remark 2.8. Lemma 2.7 also holds when 𝑥≥0. Since the idea and the process of the proof is more easily than that of Lemma 2.7, we omit the details.

Now, we give the estimate of 𝐻₁(𝐼,𝜃,𝑡).

Lemma 2.9. 𝐼^𝑘𝜕^𝑘_𝐼𝜕^𝑙_𝑡𝜕^𝑖_𝜃𝐻₁(𝐼,𝜃,𝑡)𝜕𝐼^𝑘𝜕𝑡^𝑙𝜕𝜃^𝑖≤𝐶⋅ℎ₀(𝐼),𝑘+𝑖≤7,𝑖=0,1. (2.102)

Proof. This lemma can be proved easily form the definition of 𝐻₁ and |𝑦(𝜃,𝐼)|≤2ℎ₀(𝐼).

2.2. New Action and Angle Variables

Now, we are concerned with the Hamiltonian system (2.32) with Hamiltonian function 𝐻(𝜃,𝐼,𝑡) given by (2.33). Note that 𝐼𝑑𝜃−𝐻𝑑𝑡=−(𝐻𝑑𝑡−𝐼𝑑𝜃). (2.103) This means that if one can solve 𝐼 form (2.32) as a function of 𝐻 (𝜃 and 𝑡 as parameters), then 𝑑𝐻𝑑𝜃=−𝜕𝐼(),𝜕𝑡𝑡,𝐻,𝜃𝑑𝑡𝑑𝜃=−𝜕𝐼()𝜕𝐻𝑡,𝐻,𝜃 (2.104) is also a Hamiltonian system with Hamiltonian function 𝐼, and now, the action, angle, and time variables are 𝐻, 𝑡, and 𝜃.

Form Remarks 2.3 and 2.4, we have 𝜕𝐻𝜕𝐼⟶1,as𝐼⟶+∞. (2.105) Hence, by the implicit function theorem, there is a function 𝐼(𝑡,𝐻,𝜃) such that 𝐻(𝜃,𝐼(𝑡,𝐻,𝜃),𝑡)=𝐻. (2.106) By Lemma 2.9, we have 𝐻₁(𝜃,𝐼,𝑡)𝐻₀(𝐼)⟶0,as𝐼→+∞. (2.107) So, there is a function 𝑅(𝑡,𝐻,𝜃) with |𝑅|≤(1/2)𝐻 such that 𝐼(𝑡,𝐻,𝜃)=𝐼₀(𝐻−𝑅(𝑡,𝐻,𝜃)),for𝐻⟶+∞. (2.108) Let 𝐼₁(𝑡,𝐻,𝜃)=𝐼₀(𝐻−𝑅(𝑡,𝐻,𝜃))−𝐼₀(𝐻)=^𝜋₀𝐼^′₀(𝐻−𝑠𝑅(𝑡,𝐻,𝜃))𝑅(𝑡,𝐻,𝜃)𝑑𝜃. (2.109) Then, 𝐼(𝑡,𝐻,𝜃)=𝐼₀(𝐻)+𝐼₁(𝑡,𝐻,𝜃). (2.110) From Remark 2.3, we have known the estimate of 𝐼₀(𝐻), so we need to give the estimate of 𝐼₁(𝑡,𝐻,𝜃). For this propose, we need firstly the following Lemma on the estimate of 𝑅(𝑡,𝐻,𝜃).

Lemma 2.10. The function 𝑅(𝑡,𝐻,𝜃) possesses the following estimates: 𝐻^𝑘𝜕^𝑘+𝑙𝑅(𝐻,𝑡,𝜃)𝜕^𝑘𝐻𝜕^𝑙𝑡≤𝐻^1/2, (2.111) for 𝑘+𝑙≤6.

Proof. From (2.108) and (2.110), it follows that 𝑅(𝑡,𝐻,𝜃)=𝐻₁𝜃,𝐼₀(𝐻−𝑅),𝑡. (2.112) When 𝑘+𝑙=0, by Lemma 2.9, we have |𝑅(𝑡,𝐻,𝜃)|=𝐻₁𝜃,𝐼₀(𝐻−𝑅),𝑡≤𝐶⋅ℎ₀𝐼₀()√𝐻−𝑅≤𝐶⋅√𝐻−𝑅≤𝐶⋅𝐻. (2.113) When 𝑘+𝑙=1, we first denote Δ=𝜕𝐻₁𝜕𝐼𝜃,𝐼₀(𝐻−𝑅),𝑡𝐼^′₀(𝐻−𝑅). (2.114) By Remark 2.3, we observe that 𝐼₀ is increasing and 𝐼₀12𝐻≤𝐼₀(𝐻)≤𝐼₀32𝐻≤𝐶⋅𝐼₀12𝐻. (2.115) By Lemma 2.9 and (2.115), we have |Δ|=𝜕𝐻₁𝜕𝐼𝜃,𝐼₀(𝐻−𝑅),𝑡⋅𝐼^′₀(1𝐻−𝑅)≤𝐶⋅𝐼₀()⋅𝐻−𝑅ℎ₀𝐼₀(𝐻−𝑅)⋅𝐼^′₀()1𝐻−𝑅≤𝐶⋅𝐼₀()⋅𝐻/2321𝐻⋅(3/2)𝐻⋅𝐼₀32𝐻≤𝐶⋅𝐻^−1/2≤12. (2.116) So, 11+Δ≥2. (2.117) By direct computation, we have 𝜕𝑅=𝜕𝐻𝜕𝐻₁/𝜕𝐼𝜃,𝐼₀(𝐻−𝑅),𝑡𝐼^′₀(𝐻−𝑅)1+Δ≤𝐶⋅𝜕𝐻₁𝜕𝐼𝜃,𝐼₀(𝐻−𝑅),𝑡𝐼^′₀(𝐻−𝑅)≤𝐶⋅𝐻^−1/2,𝜕𝑅=𝜕𝑡𝜕𝐻₁/𝜕𝑡𝐼₀(𝐻−𝑅)1+Δ≤𝐶⋅𝜕𝐻₁𝐼𝜕𝑡⁰(𝐻−𝑅)≤𝐶⋅𝐻^1/2. (2.118) By (2.118), we can easily get 𝐻^𝑘𝜕^𝑘+𝑙𝑅(𝐻,𝑡,𝜃)𝜕^𝑘𝐻𝜕^𝑙𝑡≤𝐻^1/2. (2.119) When 𝑘+𝑙≥2, one may get 𝜕^𝑘+𝑙𝑅(𝐻,𝑡,𝜃)𝜕^𝑘𝐻𝜕^𝑙𝑡=𝑐_{𝑛,𝑗1⋯𝑗𝑛}𝜕^𝑛𝐻₁𝜕^𝑛𝐼^𝑗1₀𝜕^𝑗1𝐼₀(𝐻−𝑅)𝜕𝐻^𝑗1⋯𝜕^𝑗𝑛𝐼₀(𝐻−𝑅)𝜕𝐻^𝑗𝑛, (2.120) where 1≤𝑛≤𝑘, 𝑗₁+⋯+𝑗_𝑛<𝑘. It is easy to verify 𝜕^𝑘+𝑙𝑅(𝐻,𝑡,𝜃)𝜕^𝑘𝐻𝜕^𝑙𝑡≤𝐶⋅𝐻^(1/2)−𝑘, (2.121) for 𝑘+𝑙≥2. This complete the proof.

Now, we give the estimates of 𝐼₁(𝑡,𝐻,𝜃).

Lemma 2.11. The function 𝐼₁(𝑡,𝐻,𝜃) possesses the following estimates: 𝐻^𝑘𝜕^𝑘+𝑙𝐼₁(𝐻,𝑡,𝜃)𝜕^𝑘𝐻𝜕^𝑙𝑡≤𝐻^1/(1+𝛼), (2.122) for 𝑘+𝑙≤6.

Proof. When 𝑘=𝑙=0. By Remark 2.3 and |𝑅|≤(1/2)𝐻, we have 𝐼^′₀((()𝐻−𝑠𝑅𝑡,𝐻,𝜃))≤𝐶⋅𝐻−𝑠𝑅^1/(1+𝛼)(𝐻−𝑠𝑅)^1/2)𝐻)≤𝐶⋅((3/2^1/(1+𝛼)((1/2)𝐻)^1/2≤𝐶⋅𝐻^{1/(1+𝛼)−(1/2)}. (2.123) By Lemma 2.10, we know |𝑅|≤𝐶⋅𝐻^1/2. (2.124) Since 𝐼₁(𝑡,𝐻,𝜃)=^𝜋₀𝐼^′₀(𝐻−𝑠𝑅(𝑡,𝐻,𝜃))𝑅(𝑡,𝐻,𝜃)𝑑𝜃, (2.125) it is easy to get that |𝐼₁|≤𝐶⋅𝐻^1/(1+𝛼).
When 𝑘+𝑙≥1. By direct computation, we have 𝜕^𝑘+𝑙𝐼₁(𝑡,𝐻,𝜃)𝜕𝐻^𝑘𝜕𝑡^𝑙=^𝜋₀𝜕^𝑘1+𝑙1𝐼^′₀(𝑡,𝐻,𝜃)𝜕𝐻^𝑘1𝜕𝑡^𝑙1𝜕^𝑘2+𝑙2𝑅(𝑡,𝐻,𝜃)𝜕𝐻^𝑘2𝜕𝑡^𝑙2𝑑𝑠, (2.126) where 𝑘₁+𝑘₂=𝑘, 𝑙₁+𝑙₂=𝑙. Now, we need to estimate the first term of the integrand. The following equality is important: 𝜕^𝑘1+𝑙1𝐼^′₀(𝑡,𝐻,𝜃)𝜕𝐻^𝑘1𝜕𝑡^𝑙1=𝐼₀^{(𝑝+𝑞+1)}()𝜕𝐻−𝑠𝑅^𝑚1𝑢𝜕𝐻^𝑚1⋯𝜕^𝑚𝑝𝑢𝜕𝐻^𝑚𝑝⋅𝜕^𝑗1+𝑛1𝑢𝜕𝐻^𝑗1𝜕𝑡^𝑛1⋯𝜕^{𝑗𝑞+𝑛𝑞}𝑢𝜕𝐻^𝑗𝑞𝜕𝑡^𝑛𝑞, (2.127) where 𝑢=𝐻−𝑠𝑅, 𝑝≤𝑚, 𝑞≤𝑛, 𝑛₁,…,𝑛_𝑞>0, 𝑚₁,…,𝑚_𝑝>0, 𝑛₁+⋯+𝑛_𝑞=𝑛, 𝑚₁+⋯+𝑚_𝑝+𝑗₁+⋯+𝑗_𝑞=𝑚. Assume that there are 𝛽(≤𝑝) members: 𝑚₁,…,𝑚_𝛽 in {𝑚₁,…,𝑚_𝑝} which equal to 1. Noting that 𝜕𝑢=𝜕𝐻1+𝑠𝜕𝑅𝜕𝜕𝐻≤𝐶,^𝑘𝑢𝜕𝐻^𝑘≤𝜕^𝑘𝑅𝜕𝐻^𝑘≤𝐶⋅𝐻^(1/2)−𝑘𝜕,𝑘>1,_𝑘+𝑙𝑢𝜕𝐻^𝑘𝜕𝑡^𝑙≤𝜕^𝑘+𝑙𝑅𝜕𝐻^𝑘𝜕𝑡^𝑙≤𝐶⋅𝐻^(1/2)−𝑘,𝑙>0. (2.128) By the above discussions, we have 𝜕^𝑘1+𝑙1𝐼^′₀(𝑡,𝐻,𝜃)𝜕𝐻^𝑘1𝜕𝑡^𝑙1()≤𝐶⋅𝐻−𝑠𝑅^{1/(1+𝛼)+(1/2)}(𝐻−𝑠𝑅)^𝑝+𝑞+1⋅𝐻^{(1/2)(𝑝−𝛽)−(𝑚𝛽+1+⋯+𝑚𝑝)}⋅𝐻^{(1/2)𝑞−(𝑗1+⋯+𝑗𝑞)}𝐻≤𝐶^{(1/2)+1/(1+𝛼)}𝐻^𝑝+𝑞+1⋅𝐻^{(1/2)(𝑝+𝑞−𝛽)}𝐻^{−(𝑚−𝛽)}≤𝐶𝐻^{−(1/2)+1/(1+𝛼)−𝑚}⋅𝐻^{(𝑝+𝑞−𝛽)((1/2)−1)}≤𝐶𝐻^{−(1/2)+1/(1+𝛼)−𝑚}. (2.129) By Lemma 2.10, we have known 𝐻^𝑘𝜕^𝑘+𝑙𝑅(𝐻,𝑡,𝜃)𝜕^𝑘𝐻𝜕^𝑙𝑡≤𝐻^1/2, (2.130) then we have 𝐻^𝑘𝜕^𝑘+𝑙𝐼₁(𝐻,𝑡,𝜃)𝜕^𝑘𝐻𝜕^𝑙𝑡≤𝐶𝐻^1/(1+𝛼). (2.131)

2.3. Proof of the Main Result

Up to now, we have given an equivalent form of (1.5), that is, the system (2.32), which is expressed in the action and angle variables (𝐻,𝑡). In this section, we first introduce some transformations such that in the transformed system, the perturbation terms of (2.32) depending on the new angle variable are very small if the new action variable is sufficiently large and then prove, by Moser’s twist theorem, the statement of Theorem 1.1.

Lemma 2.12. There is a canonical transformation Ψ∶(𝜆,𝜏)→(𝐻,𝑡) of the form Ψ∶𝐻=𝜆+𝑈(𝜏,𝜆,𝜃),𝑡=𝜏+𝑉(𝜏,𝜆,𝜃), (2.132) where the functions 𝑈 and 𝑉 are 1-periodic in 𝜃 and satisty 𝑈(𝜏,𝜆,𝜃)𝜆,𝑉(𝜏,𝜆,𝜃)⟶0as𝜆⟶∞, (2.133) uniformly for (𝜏,𝜃)∈𝑇² such that under this mapping, the system and the Hamiltonian function 𝐼 in (2.110) is changed into the form 𝑑𝜆𝑑𝜃=−𝜕𝒦(),𝜕𝜏𝜏,𝜆,𝜃𝑑𝜏=𝑑𝜃𝜕𝒦()𝜕𝐻𝜏,𝜆,𝜃, (2.134) where 𝒦(𝜏,𝜆,𝜃)=𝐼₀(𝜆)+𝐼₁(𝜆,𝜃)+ℳ(𝜏,𝜆,𝜃), (2.135) with 𝐼₁(𝜆,𝜃)=¹₀𝐼₁(𝑡,𝜆,𝜃)𝑑𝑡. (2.136) Moreover, the new perturbation ℳ possesses the estimate 𝜕^𝑘+𝑙𝜕𝜆^𝑘𝜕𝜏^𝑙ℳ(𝜏,𝜆,𝜃)≤𝐶⋅𝜆^{−𝑘+(1/2)}. (2.137)

Proof. We will look for the required transformation Ψ by a generating function ℱ(𝑡,𝜆,𝜃) in the following way: 𝐻=𝜆+𝜕ℱ()𝜕𝑡𝑡,𝜆,𝜃,𝜏=𝑡+𝜕ℱ()𝜕𝜆𝑡,𝜆,𝜃, (2.138) where the function ℱ will be given later. Under Ψ, the transformed system of (2.104) is of the form 𝑑𝜆𝑑𝜃=−𝜕𝒦(),𝜕𝜏𝜏,𝜆,𝜃𝑑𝜏𝑑𝜃=−𝜕𝒦()𝜕𝜆𝜏,𝜆,𝜃, (2.139) where 𝒦(𝜏,𝜆,𝜃)=𝐼₀𝜆+𝜕ℱ𝜕𝑡+𝐼₁𝑡,𝜆+𝜕ℱ+𝜕𝑡,𝜃𝜕ℱ𝜕𝜃. (2.140) By Taylor’s formula, one can write 𝒦(𝜏,𝜆,𝜃)=𝐼₀(𝜆)+𝐼^′₀(𝜆)𝜕ℱ𝜕𝑡+𝐼₁(𝑡,𝜆,𝜃)+ℳ(𝜏,𝜆,𝜃), (2.141) where ℳ(𝜏,𝜆,𝜃)=𝜕ℱ+𝜕𝜃¹₀(1−𝑠)𝐼₀^′′𝜆+𝑠𝜕ℱ⋅𝜕𝑡𝜕ℱ𝜕𝜃²𝑑𝑠+¹₀𝜕𝐼₁𝜕𝐻𝑡,𝜆+𝑠𝜕ℱ⋅𝜕𝑡,𝜃𝜕ℱ𝜕𝑡𝑑𝑠. (2.142) We chooseℱℱ(𝑡,𝜆,𝜃)=−^𝑡₀1𝐼^′₀(𝜆)⋅𝐼₁(𝑡,𝜆,𝜃)−𝐼₁(𝜆,𝜃). (2.143) Then, 𝒦 is of the form (2.135).
We now show that ℳ satisfies (2.137). From Remark 2.3 and Lemma 2.11, it follows that 𝜕^{𝑘+𝑘+𝑖}𝜕𝜆^𝑘𝜕𝑡^𝑙𝜕𝜃^𝑖ℱ(𝑡,𝜆,𝜃)≤𝐶⋅𝜆^{−𝑘+(1/2)}, (2.144) for 𝑘+𝑖+𝑙≤6 and 𝑖=0,1. In particular 𝜕²ℱ(𝜕𝜆𝜕𝑡𝑡,𝜆,𝜃)≤𝐶𝜆^−1/2≤12, (2.145) if 𝜆≫1. So we can solve the second equation of (2.138) for 𝑡, 𝑡=𝜏+𝑉(𝜏,𝜆,𝜃), (2.146) where the function 𝑉 satisfies 𝑉(𝜏,𝜆,𝜃)=−𝜕ℱ()𝜕𝜆𝜏+𝑉,𝜆,𝜃. (2.147) Set 𝑈(𝜏,𝜆,𝜃)=𝜕ℱ()𝜕𝑡𝜏+𝑉,𝜆,𝜃. (2.148) Then, the canonical transformation Ψ is of the form (2.132). Moreover, similar to the proof of [5, Lemma  2], we can verify that 𝜕^𝑘+𝑙𝜕𝜆^𝑘𝜕𝜏^𝑙𝑈(𝜏,𝜆,𝜃)≤𝐶⋅𝜆^{−𝑘+(1/2)},𝜕^𝑘+𝑙𝜕𝜆^𝑘𝜕𝜏^𝑙𝑉(𝜏,𝜆,𝜃)≤𝐶⋅𝜆^{−𝑘−(1/2)}, (2.149) for 𝑘+𝑙≤5 and 𝑈/𝜆,𝑉→0 as 𝜆→+∞. Let 𝜙₁(𝜏,𝜆,𝜃)=𝜕ℱ(),𝜙𝜕𝜃𝜏+𝑉,𝜆,𝜃₂(𝜏,𝜆,𝜃)=¹₀(1−𝑠)𝐼₀^′′(𝜆+𝑠𝑈)⋅𝑈²𝜙𝑑𝑠,₃(𝜏,𝜆,𝜃)=¹₀𝜕𝐼₁()𝜕𝐻𝜏+𝑉,𝜆+𝑠𝑈,𝜃⋅𝑈𝑑𝑠. (2.150) It is not difficult to prove that 𝜕^𝑘+𝑙𝜕𝜆^𝑘𝜕𝜏^𝑙𝜙₁(𝜏,𝜆,𝜃)≤𝐶⋅𝜆^{−𝑘+(1/2)},𝜕^𝑘+𝑙𝜕𝜆^𝑘𝜕𝜏^𝑙𝜙₁(𝜏,𝜆,𝜃)≤𝐶⋅𝜆^{−𝑘+(1/(1+𝛼))−(1/2)},𝜕^𝑘+𝑙𝜕𝜆^𝑘𝜕𝜏^𝑙𝜙₁(𝜏,𝜆,𝜃)≤𝐶⋅𝜆^{−𝑘+(1/(1+𝛼))−(1/2)}, (2.151) for 𝑘+𝑙≤5. Note that 0<𝛼<1, we have 1/(1+𝛼)−1/2<1/2.
Hence, we have 𝜕^𝑘+𝑙𝜕𝜆^𝑘𝜕𝜏^𝑙ℳ(𝜏,𝜆,𝜃)≤𝜆^{−𝑘+(1/2)}. (2.152) The proof of Lemma 2.12 is complete.
For 𝜆₀>0, we denote by 𝐴_𝜆0 the domain 𝐴_𝜆0=(𝜆,𝜏,𝜃)∣𝜆≥𝜆₀,(𝜏,𝜃)∈𝑇². (2.153)

Lemma 2.13. The Poincaré mapping 𝑃 of (2.134) has the intersection property on 𝐴_𝐻0; that is, if Γ is an embedded circle in 𝐴_𝐻0 homotopic to a circle 𝜆= const. in 𝐴_𝐻0, then 𝑃(Γ)⋂Γ≠∅.

Proof. The proof can be found in [5].
Definite a diffeomorphism Ψ₁∶𝐴_𝐻0×𝕊¹→𝐴_𝐻0×𝕊¹𝜈=𝐼^′₀(𝐻),𝜏=𝜏,𝜃=𝜃. (2.154) Then, the system (2.134) under the transformation Ψ₁ becomes 𝑑𝜈𝑑𝜃=𝑓₁(𝜆,𝑡,𝜃),𝑑𝑡𝑑𝜃=𝜈+𝑓₂(𝜆,𝑡,𝜃), (2.155) where 𝑓₁(𝜏,𝜈,𝜃)=−𝐼₀^′′(𝜆)𝜕ℳ()𝜕𝜏𝜏,𝜆,𝜃,𝑓₂()=𝜕𝐼𝜏,𝜈,𝜃¹()+𝜕𝜆𝜆,𝜃𝜕ℳ()𝜕𝜆𝜏,𝜆,𝜃, (2.156) with 𝜆=𝜆(𝜈) defined through the transformation Ψ₁.
Now, we estimate 𝑓₁ and 𝑓₂. Since 𝑐⋅𝜆^{1/(1+𝛼)−(1/2)}≤𝐼^′₀(𝜆)≤𝐶⋅𝜆^{1/(1+𝛼)−(1/2)},𝜕𝐼₁𝜕𝜆≤𝐶⋅𝜆^{1/(1+𝛼)−1}, (2.157) we have 𝑐⋅𝜆^{2(1+𝛼)/(1−𝛼)}≤𝜆(𝜈)≤𝐶⋅𝜆^{2(1+𝛼)/(1−𝛼)}, (2.158) then 𝜆≫1⟺𝜈≫1. (2.159) Moreover, we have 𝜕^𝑘𝜕𝜈^𝑘𝜆(𝜈)≤𝐶⋅𝜈^−𝑘𝜆(𝜈). (2.160) When 0<𝛼<1, we have 𝜕^𝑘+𝑙𝜕𝜆^𝑘𝜕𝜏^𝑙ℳ(𝜏,𝜆,𝜃)≤𝐶⋅𝜆^{−𝑘+(1/2)},12<11+𝛼<1, (2.161) so 𝜕^𝑘+𝑙𝜕𝜈^𝑘𝜕𝜏^𝑙𝑓₁(𝜏,𝜈,𝜃)≤𝐶⋅𝜈^−𝑘⋅𝜆⁻²⋅𝜈^−𝑘𝐼₀(𝜆)⋅𝜆^1/2≤𝐶⋅𝜈^−𝑘𝜆^{−𝛼/(1+𝛼)}≤𝐶⋅𝜈^{−𝑘+(−4𝛼/(1−𝛼))}≤𝐶⋅𝜈^{−𝑘−𝜎},𝜕^𝑘+𝑙𝜕𝜈^𝑘𝜕𝜏^𝑙𝑓₂(𝜏,𝜈,𝜃)≤𝐶⋅𝜈^−𝑘⋅𝜆^−1/2+𝜆^{1/(1+𝛼)−1}≤𝐶⋅𝜈^−𝑘⋅𝜆^{1/(1+𝛼)−1}≤𝐶⋅𝜈^{−𝑘−(2𝛼/(1−𝛼))}≤𝐶⋅𝜈^{−𝑘−𝜎}, (2.162) for all 𝑘+𝑙≤4, where 𝜎=min{4𝛼/(1−𝛼),2𝛼/(1−𝛼)}=2𝛼/(1−𝛼)>0.
Since 𝑓₁ and 𝑓₂ are sufficiently small as 𝜈≫1, all solution of (2.155) exist for 0≤𝜃≤1 when the initial values 𝜈(0)=𝜆 are sufficiently large. Hence, the Poincaré map Φ associated to (2.155) is well defined on 𝐴_𝜆0 as 𝜆₀≫1. In fact, by integrating (2.155) from 𝜃=0 to 𝜃=𝜋, we see that Φ has the form Φ∶𝜏₁=𝜏₀+𝜇₀+Ξ₁𝑡₀,𝜆₀,𝜈₁=𝜈₀+Ξ₂𝑡₀,𝜆₀, (2.163) where Ξ₁ and Ξ₂ satisfy the same estimates as those of 𝑓₁ and 𝑓₂; that is, 𝜕^𝑘_𝜈𝜕^𝑙_𝜏Ξ_𝑖≤𝜈^−𝜎, (2.164) where 𝑖=1,2,0≤𝑘+𝑙≤4.
Since Φ satisfies all the assumptions of Moser’s twist theorem [9], from which we conclude that for any 𝜔≫1 satisfying 𝑝𝜔−𝑞≥𝑐₀𝑞^−5/2,𝑝𝑞∈ℚ. (2.165) There is an invariant curve Γ_𝜔 of Φ which is conjugated to pure rotation of the circle with rotation number 𝜔. Tracing back to the system (2.32), Γ_𝜔 gives rise to an invariant closed curve of the Poincaré map Φ of (2.32) with rotation number 1/𝜔 which surrounds and is arbitrarily far away form the origin. Hence, all solutions of (1.5) are bounded. This completes the proof of the Theorem.

Acknowledgments

The authors would like to thank the referees for many helpful suggsetions. The work was supported by the National Natural Science Foundation of China (11071038) and the Natural Science Foundation of Jiangsu Province (BK2010420).