About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 354063, 17 pages
http://dx.doi.org/10.1155/2011/354063
Research Article

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

Department of Mathematics, Southeast University, Nanjing 210096, China

Received 1 December 2010; Revised 18 April 2011; Accepted 25 May 2011

Academic Editor: Yuri V.Β Rogovchenko

Copyright Β© 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.

Abstract

We consider the following real two-dimensional nonlinear analytic quasi-periodic Hamiltonian system Μ‡ π‘₯ = 𝐽 βˆ‡ π‘₯ 𝐻 , where 𝐻 ( π‘₯ , 𝑑 , πœ€ ) = ( 1 / 2 ) 𝛽 ( π‘₯ 2 1 + π‘₯ 2 2 ) + 𝐹 ( π‘₯ , 𝑑 , πœ€ ) with 𝛽 β‰  0 , πœ• π‘₯ 𝐹 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ ) and πœ• π‘₯ π‘₯ 𝐹 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ ) as πœ€ β†’ 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 ( 𝑑 ) is called a quasi-periodic function with frequencies πœ” = ( πœ” 1 , πœ” 2 , … , πœ” 𝑙 ) if 𝑓 ( 𝑑 ) = 𝐹 ( πœ” 1 𝑑 , πœ” 2 𝑑 , … , πœ” 𝑙 𝑑 ) with πœƒ 𝑖 = πœ” 𝑖 𝑑 , where 𝐹 ( πœƒ 1 , πœƒ 2 , … , πœƒ 𝑙 ) is 2 πœ‹ periodic in all the arguments πœƒ 𝑗 , 𝑗 = 1 , 2 , … , 𝑙 . If 𝐹 ( πœƒ ) ( πœƒ = ( πœƒ 1 , πœƒ 2 , … , πœƒ 𝑙 ) ) is analytic on 𝐷 𝜌 = { πœƒ ∈ 𝐢 𝑙 / 2 πœ‹ 𝑍 𝑙 ∣ | I m πœƒ 𝑖 | ≀ 𝜌 , 𝑖 = 1 , 2 , … , 𝑙 } , we call 𝑓 ( 𝑑 ) analytic quasi-periodic on 𝐷 𝜌 . If all π‘ž 𝑖 𝑗 ( 𝑑 ) ( 𝑖 , 𝑗 = 1 , 2 … , 𝑛 ) are analytic quasi-periodic on 𝐷 𝜌 , then the matrix function 𝑄 ( 𝑑 ) = ( π‘ž 𝑖 𝑗 ( 𝑑 ) ) 1 ≀ 𝑖 , 𝑗 ≀ 𝑛 is called analytic quasi-periodic on 𝐷 𝜌 .

If 𝑓 ( 𝑑 ) is analytic quasi-periodic on 𝐷 𝜌 , we can write it as Fourier series:  𝑓 ( 𝑑 ) = π‘˜ ∈ 𝑍 𝑙 𝑓 π‘˜ 𝑒 i ⟨ π‘˜ , πœ” ⟩ 𝑑 . ( 1 . 1 ) Define a norm of 𝑓 by β€– 𝑓 β€– 𝜌 = βˆ‘ π‘˜ ∈ 𝑍 𝑙 | 𝑓 π‘˜ | 𝑒 | π‘˜ | 𝜌 . It follows that | 𝑓 π‘˜ | ≀ β€– 𝑓 β€– 𝜌 𝑒 βˆ’ | π‘˜ | 𝜌 . If the matrix function 𝑄 ( 𝑑 ) is analytic quasi-periodic on 𝐷 𝜌 , we define the norm of 𝑄 by β€– 𝑄 β€– 𝜌 = 𝑛 Γ— m a x 1 ≀ 𝑖 , 𝑗 ≀ 𝑛 β€– π‘ž 𝑖 𝑗 β€– 𝜌 . It is easy to verify β€– 𝑄 1 𝑄 2 β€– 𝜌 ≀ β€– 𝑄 1 β€– 𝜌 β€– 𝑄 2 β€– 𝜌 . The average of 𝑄 ( 𝑑 ) is denoted by [ 𝑄 ] = ( [ π‘ž 𝑖 𝑗 ] ) 1 ≀ 𝑖 , 𝑗 ≀ 𝑛 , where ξ€Ί π‘ž 𝑖 𝑗 ξ€» = l i m 𝑇 β†’ ∞ 1 ξ€œ 2 𝑇 𝑇 βˆ’ 𝑇 π‘ž 𝑖 𝑗 ( 𝑑 ) 𝑑 𝑑 . ( 1 . 2 ) For the existence of the above limit, see [1].

Denote 𝐷 ξ€· π‘Ÿ , 𝜌 , πœ€ 0 ξ€Έ = ξ‚» ( π‘₯ , πœƒ , πœ€ ) ∈ 𝐢 𝑛 Γ— ξ‚΅ 𝐢 𝑙 2 πœ‹ 𝑍 𝑙 ξ‚Ά Γ— 𝐢 ∣ | π‘₯ | ≀ π‘Ÿ , πœƒ ∈ 𝐷 𝜌 , | πœ€ | ≀ πœ€ 0 ξ‚Ό , ( 1 . 3 ) where π‘₯ = ( π‘₯ 1 , π‘₯ 2 , … , π‘₯ 𝑛 ) and | π‘₯ | = | π‘₯ 1 | + | π‘₯ 2 | + β‹― + | π‘₯ 𝑛 | .

Let 𝑓 ( π‘₯ , 𝑑 , πœ€ ) be analytic quasi-periodic of 𝑑 and analytic in π‘₯ and πœ€ on 𝐷 ( π‘Ÿ , 𝜌 , πœ€ 0 ) . Then 𝑓 ( π‘₯ , 𝑑 , πœ€ ) can be expanded as 𝑓 ( π‘₯ , 𝑑 , πœ€ ) = ∞  π‘š = 0  π‘˜ ∈ 𝑍 𝑙 𝑓 π‘š π‘˜ ( π‘₯ ) πœ€ π‘š 𝑒 i ⟨ π‘˜ , πœ” ⟩ 𝑑 . ( 1 . 4 ) Define a norm by β€– 𝑓 β€– 𝐷 ( π‘Ÿ , 𝜌 , πœ€ 0 ) = ∞  π‘š = 0  π‘˜ ∈ 𝑍 𝑙 | | 𝑓 π‘š π‘˜ | | π‘Ÿ πœ€ π‘š 0 𝑒 𝜌 | π‘˜ | , ( 1 . 5 ) where | 𝑓 π‘š π‘˜ | π‘Ÿ = s u p | π‘₯ | ≀ π‘Ÿ | 𝑓 π‘š π‘˜ ( π‘₯ ) | . Note that β€– β€– 𝑓 1 β‹… 𝑓 2 β€– β€– 𝐷 ( π‘Ÿ , 𝜌 , πœ€ 0 ) ≀ β€– β€– 𝑓 1 β€– β€– 𝐷 ( π‘Ÿ , 𝜌 , πœ€ 0 ) β‹… β€– β€– 𝑓 2 β€– β€– 𝐷 ( π‘Ÿ , 𝜌 , πœ€ 0 ) . ( 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 ( 𝑑 ) is a 𝑇 -periodic matrix, then the linear system Μ‡ π‘₯ = 𝐴 ( 𝑑 ) π‘₯ 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 Μ‡ π‘₯ = 𝐴 ( 𝑑 ) π‘₯ , where 𝐴 ( 𝑑 ) is a quasi-periodic matrix. Under some “full spectrum” conditions, they proved that Μ‡ π‘₯ = 𝐴 ( 𝑑 ) π‘₯ is reducible. That is, there exists a quasi-periodic nonsingular transformation π‘₯ = πœ™ ( 𝑑 ) 𝑦 , where πœ™ ( 𝑑 ) and πœ™ ( 𝑑 ) βˆ’ 1 are quasi-periodic and bounded, such that Μ‡ π‘₯ = 𝐴 ( 𝑑 ) π‘₯ is transformed to Μ‡ 𝑦 = 𝐡 𝑦 , where 𝐡 is a constant matrix.
In [3], Jorba and Simó considered the reducibility of the following linear system: Μ‡ π‘₯ = ( 𝐴 + πœ€ 𝑄 ( 𝑑 ) ) π‘₯ , π‘₯ ∈ 𝑅 𝑛 , ( 1 . 7 ) where 𝐴 is an 𝑛 Γ— 𝑛 constant matrix with 𝑛 different eigenvalues πœ† 1 , πœ† 2 , … , πœ† 𝑛 and 𝑄 ( 𝑑 ) is analytic quasi-periodic with respect to 𝑑 with frequencies πœ” = ( πœ” 1 , πœ” 2 , … , πœ” 𝑙 ) . Here πœ€ is a small perturbation parameter. Suppose that the following nonresonance conditions hold: | | | √ ⟨ π‘˜ , πœ” ⟩ βˆ’ 1 + πœ† 𝑖 βˆ’ πœ† 𝑗 | | | β‰₯ 𝛼 | | π‘˜ | | 𝜏 , ( 1 . 8 ) for all π‘˜ ∈ 𝑍 𝑙 ⧡ { 0 } , where 𝛼 > 0 is a small constant and 𝜏 > 𝑙 βˆ’ 1 . Assume that πœ† 0 𝑗 ( πœ€ ) ( 𝑗 = 1 , 2 , … , 𝑛 ) are eigenvalues of 𝐴 + πœ€ [ 𝑄 ] . If the following non-degeneracy conditions hold: 𝑑 ξ€· πœ† 𝑑 πœ€ 0 𝑖 ( πœ€ ) βˆ’ πœ† 0 𝑗 ξ€Έ | | | ( πœ€ ) πœ€ = 0 β‰  0 , βˆ€ 𝑖 β‰  𝑗 , ( 1 . 9 ) then authors proved that for sufficiently small πœ€ 0 > 0 , there exists a nonempty Cantor subset 𝐸 βŠ‚ ( 0 , πœ€ 0 ) , such that for πœ€ ∈ 𝐸 , the system (1.7) is reducible. Moreover, m e a s ( ( 0 , πœ€ 0 ) ⧡ 𝐸 ) = π‘œ ( πœ€ 0 ) .
Some related problems were considered by Eliasson in [4, 5]. In the paper [4], to study one-dimensional linear Schrödinger equation 𝑑 2 π‘ž 𝑑 𝑑 2 + 𝑄 ( πœ” 𝑑 ) π‘ž = 𝐸 π‘ž , ( 1 . 1 0 ) Eliasson considered the following equivalent two-dimensional quasi-periodic Hamiltonian system: Μ‡ 𝑝 = ( 𝐸 βˆ’ 𝑄 ( πœ” 𝑑 ) ) π‘ž , Μ‡ π‘ž = 𝑝 , ( 1 . 1 1 ) 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 𝐢 πœ” ( Ξ› , 𝑔 𝑙 ( π‘š , 𝐢 ) ) be the set of π‘š Γ— π‘š matrices 𝐴 ( πœ† ) depending analytically on a parameter πœ† in a closed interval Ξ› βŠ‚ 𝑅 . In [6], Her and You considered one-parameter families of quasi-periodic linear equations ξ€· 𝐴 ξ€· πœ” Μ‡ π‘₯ = ( πœ† ) + 𝑔 1 𝑑 , … , πœ” 𝑙 𝑑 , πœ† ξ€Έ ξ€Έ π‘₯ , ( 1 . 1 2 ) where 𝐴 ∈ 𝐢 πœ” ( Ξ› , 𝑔 𝑙 ( π‘š , 𝐢 ) ) , 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 π’œ in 𝐢 πœ” ( Ξ› , 𝑔 𝑙 ( π‘š , 𝐢 ) ) , such that for each 𝐴 ∈ π’œ , the system (1.12) is reducible for almost all πœ† ∈ Ξ› .
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 Μ‡ π‘₯ = ( 𝐴 + πœ€ 𝑄 ( 𝑑 ) ) π‘₯ + πœ€ 𝑔 ( 𝑑 ) + β„Ž ( π‘₯ , 𝑑 ) , π‘₯ ∈ 𝑅 𝑛 , ( 1 . 1 3 ) where 𝐴 has 𝑛 different nonzero eigenvalues πœ† 𝑖 . They proved that under some nonresonance conditions and some non-degeneracy conditions, there exists a nonempty Cantor subset 𝐸 βŠ‚ ( 0 , πœ€ 0 ) , such that the system (1.13) is reducible for πœ€ ∈ 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: Μ‡ π‘₯ = 𝐴 π‘₯ + 𝑓 ( π‘₯ , 𝑑 , πœ€ ) , π‘₯ ∈ 𝑅 2 , ( 1 . 1 4 ) where 𝐴 has a pair of pure imaginary eigenvalues Β± √ βˆ’ 1 πœ” 0 with πœ” 0 β‰  0 satisfying the nonresonance conditions | | | | β‰₯ 𝛼 ⟨ π‘˜ , πœ” ⟩ | | π‘˜ | | 𝜏 , | | ⟨ π‘˜ , πœ” ⟩ βˆ’ 2 πœ” 0 | | β‰₯ 𝛼 | | π‘˜ | | 𝜏 ( 1 . 1 5 ) for all π‘˜ ∈ 𝑍 𝑙 ⧡ { 0 } , where 𝛼 > 0 is a small constant and 𝜏 > 𝑙 βˆ’ 1 . Assume that 𝑓 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ ) and πœ• π‘₯ 𝑓 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ ) as πœ€ β†’ 0 . They proved that either of the following two results holds:(1)for βˆ€ πœ€ ∈ ( 0 , πœ€ 0 ) , the system (1.14) is reducible to Μ‡ 𝑦 = 𝐡 𝑦 + 𝑂 ( 𝑦 ) as 𝑦 β†’ 0 ;(2)there exists a nonempty Cantor subset 𝐸 βŠ‚ ( 0 , πœ€ 0 ) , such that for πœ€ ∈ 𝐸 the system (1.14) is reducible to Μ‡ 𝑦 = 𝐡 𝑦 + 𝑂 ( 𝑦 2 ) as 𝑦 β†’ 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 Μ‡ π‘₯ = 𝐽 βˆ‡ π‘₯ 𝐻 , π‘₯ ∈ 𝑅 2 , ( 2 . 1 ) where 𝐻 ( π‘₯ , 𝑑 , πœ€ ) = ( 1 / 2 ) 𝛽 ( π‘₯ 2 1 + π‘₯ 2 2 ) + 𝐹 ( π‘₯ , 𝑑 , πœ€ ) with 𝛽 β‰  0 , 𝐹 ( π‘₯ , 𝑑 , πœ€ ) is analytic quasi-periodic with respect to t with frequencies πœ” = ( πœ” 1 , πœ” 2 , … , πœ” 𝑙 ) and real analytic with respect to x and πœ€ on 𝐷 ( π‘Ÿ , 𝜌 , πœ€ 0 ) , and  ξƒͺ 𝐽 = 0 1 βˆ’ 1 0 . ( 2 . 2 ) Here πœ€ ∈ ( 0 , πœ€ 0 ) is a small parameter. Suppose that πœ• π‘₯ 𝐹 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ ) and πœ• π‘₯ π‘₯ 𝐹 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ ) as πœ€ β†’ 0 . Moreover, assume that 𝛽 and πœ” satisfy | | | | β‰₯ 𝛼 ⟨ π‘˜ , πœ” ⟩ 0 | | π‘˜ | | 𝜏 | | | | β‰₯ 𝛼 , ( 2 . 3 ) ⟨ π‘˜ , πœ” ⟩ βˆ’ 2 𝛽 0 | | π‘˜ | | 𝜏 ( 2 . 4 ) for all π‘˜ ∈ 𝑍 𝑙 ⧡ { 0 } , where 𝛼 0 > 0 is a small constant and 𝜏 > 𝑙 βˆ’ 1 .
Then there exist a sufficiently small πœ€ βˆ— ∈ ( 0 , πœ€ 0 ] and a nonempty Cantor subset 𝐸 βˆ— βŠ‚ ( 0 , πœ€ βˆ— ) , such that for πœ€ ∈ 𝐸 βˆ— , there exists an analytic quasi-periodic symplectic transformation π‘₯ = πœ™ βˆ— ( 𝑑 ) 𝑦 + πœ“ βˆ— ( 𝑑 ) on 𝐷 𝜌 / 2 with the frequencies πœ” , which changes (2.1) into the Hamiltonian system Μ‡ 𝑦 = 𝐽 βˆ‡ 𝑦 𝐻 βˆ— , where 𝐻 βˆ— ( 𝑦 , 𝑑 , πœ€ ) = 1 / 2 𝛽 βˆ— ( πœ€ ) ( 𝑦 2 1 + 𝑦 2 2 ) + 𝐹 βˆ— ( 𝑦 , 𝑑 , πœ€ ) , where 𝐹 βˆ— ( 𝑦 , 𝑑 , πœ€ ) = 𝑂 ( 𝑦 3 ) as 𝑦 β†’ 0 . Moreover, m e a s ( ( 0 , πœ€ βˆ— ) ⧡ 𝐸 βˆ— ) = π‘œ ( πœ€ βˆ— ) as πœ€ βˆ— β†’ 0 . Furthermore, 𝛽 βˆ— ( πœ€ ) = 𝛽 + 𝑂 ( πœ€ ) and β€– πœ™ βˆ— βˆ’ 𝐼 𝑑 β€– 𝜌 / 2 + β€– πœ“ βˆ— β€– 𝜌 / 2 = 𝑂 ( πœ€ ) , where 𝐼 𝑑 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 πœ€ 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 = ( π‘Ÿ 𝑖 𝑗 ) 1 ≀ 𝑖 , 𝑗 ≀ 2 be a Hamiltonian matrix. Then we have π‘Ÿ 1 1 + π‘Ÿ 2 2 = 0 . Define a matrix 𝑅 𝐴 = ( 1 / 2 ) 𝑑 𝐽 with 𝑑 = π‘Ÿ 1 2 βˆ’ π‘Ÿ 2 1 . Let 1 𝐡 = √ 2  √ 1 1 √ βˆ’ 1 βˆ’ ξƒͺ βˆ’ 1 . ( 3 . 1 ) It is easy to verify 𝐡 βˆ’ 1 𝑅 𝐴 1 𝐡 = 2 ξ‚€ √ d i a g √ βˆ’ 1 𝑑 , βˆ’  , 𝐡 βˆ’ 1 𝑑 βˆ’ 1 ξ€· 𝑅 βˆ’ 𝑅 𝐴 ξ€Έ 1 𝐡 = 2 βŽ› ⎜ ⎜ ⎝ 0 𝜎 ξ…ž βˆ’ √ βˆ’ 1 πœ… ξ…ž 𝜎 ξ…ž + √ βˆ’ 1 πœ… ξ…ž 0 ⎞ ⎟ ⎟ ⎠ , ( 3 . 2 ) where 𝜎 ξ…ž = 2 π‘Ÿ 1 1 and πœ… ξ…ž = π‘Ÿ 2 1 + π‘Ÿ 1 2 .

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: Μ‡ 𝑃 = 𝐴 𝑃 βˆ’ 𝑃 𝐴 + 𝑅 ( 𝑑 ) , ( 3 . 3 ) where 𝐴 = 𝛽 ( πœ€ ) 𝐽 with | 𝛽 ( πœ€ ) | > πœ‡ , πœ‡ > 0 is a constant, and 𝑅 ( 𝑑 ) = ( π‘Ÿ 𝑖 𝑗 ( 𝑑 ) ) 1 ≀ 𝑖 , 𝑗 ≀ 2 is a real analytic quasi-periodic Hamiltonian matrix on 𝐷 𝜌 with frequencies πœ” . Suppose 𝛽 ( πœ€ ) and 𝑅 are smooth with respect to πœ€ and | πœ€ 𝛽 ξ…ž ( πœ€ ) | ≀ 𝑐 0 for πœ€ ∈ 𝐸 βŠ‚ ( 0 , πœ€ βˆ— ) , where 𝑐 0 is a constant. Note that here and below the dependence of πœ€ 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 πœ€ ∈ 𝐸 , the small divisors conditions (2.3) and the following small divisors conditions hold: | | | | β‰₯ 𝛼 ⟨ π‘˜ , πœ” ⟩ βˆ’ 2 𝛽 ( πœ€ ) | | π‘˜ | | 𝜏 β€² , ( 3 . 4 ) where 𝜏 ξ…ž > 2 𝜏 + 𝑙 . Let 0 < 𝑠 < 𝜌 and 𝜌 1 = 𝜌 βˆ’ 𝑠 . Then there exists a unique real analytic quasi-periodic Hamiltonian matrix P(t) with frequencies πœ” , which solves the homological linear equation (3.3) and satisfies β€– 𝑃 β€– 𝜌 1 ≀ 𝑐 𝛼 𝑠 𝑣 β€– 𝑅 β€– 𝜌 , β€– β€– πœ€ πœ• πœ€ 𝑃 β€– β€– 𝜌 1 ≀ 𝑐 𝛼 2 𝑠 𝑣 β€² ξ€· β€– 𝑅 β€– 𝜌 + β€– β€– πœ€ πœ• πœ€ 𝑅 β€– β€– 𝜌 ξ€Έ , ( 3 . 5 ) where 𝑣 = 𝜏 ξ…ž + 𝑙 , 𝑣 ξ…ž = 2 𝜏 ξ…ž + 𝑙 and 𝑐 > 0 is a constant.

Remark 3.2. The subset 𝐸 of ( 0 , πœ€ βˆ— ) is usually a Cantor set and so the derivative with respect to πœ€ should be understood in the sense of Whitney [9].

Proof. Let 𝑃 = 𝐡 βˆ’ 1 𝑃 𝐡 , where 𝐡 is defined by (3.1). Similarly, define 𝐴 , 𝑅 , 𝑅 𝐴 . Then (3.3) becomes Μ‡ β€Œ 𝑃 = 𝐴 𝑃 βˆ’ 𝑃 𝐴 + 𝑅 ( 𝑑 ) , ( 3 . 6 ) where ξ‚€ √ 𝐴 = d i a g √ βˆ’ 1 𝛽 , βˆ’  βˆ’ 1 𝛽 . ( 3 . 7 ) Moreover, 𝑅 𝐴 and 𝑅 βˆ’ 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 𝑖 𝑗 ) 𝑖 , 𝑗 and 𝑅 = ( π‘Ÿ 𝑖 𝑗 ) 𝑖 , 𝑗 . Obviously, we have π‘Ÿ 1 1 = βˆ’ π‘Ÿ 2 2 with [ π‘Ÿ 𝑖 𝑖 ] = 0 .
Insert the Fourier series of 𝑃 and 𝑅 into (3.6). Then it follows that 𝑝 0 𝑖 𝑖 = 0 , 𝑝 π‘˜ 𝑖 𝑖 = π‘Ÿ π‘˜ 𝑖 𝑖 √ / ( ⟨ π‘˜ , πœ” ⟩ βˆ’ 1 ) for π‘˜ β‰  0 , and 𝑝 π‘˜ 𝑖 𝑗 = π‘Ÿ π‘˜ 𝑖 𝑗 √ βˆ’ 1 ( ⟨ π‘˜ , πœ” ⟩ Β± 2 𝛽 ) f o r 𝑖 β‰  𝑗 . ( 3 . 8 ) Since 𝑅 is analytic on 𝐷 𝜌 , we have | 𝑅 π‘˜ | ≀ β€– 𝑅 β€– 𝜌 𝑒 βˆ’ | π‘˜ | 𝜌 . So it follows β€– β€– 𝑃 β€– β€– 𝜌 βˆ’ 𝑠 ≀  π‘˜ ∈ 𝑍 𝑙 | | 𝑃 π‘˜ | | 𝑒 | π‘˜ | ( 𝜌 βˆ’ 𝑠 ) ≀ 𝑐 𝛼 𝑠 𝑣 β€– 𝑅 β€– 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 ( 𝑑 ) are real matrices, it is easy to obtain that 𝑃 ( 𝑑 ) is also a real matrix. Obviously, it follows that 𝑝 1 1 = βˆ’ 𝑝 2 2 and the trace of the matrix 𝑃 is zero. So is the trace of 𝑃 . Thus, 𝑃 is a Hamiltonian matrix.
Now we estimate β€– πœ€ πœ• 𝑃 / πœ• πœ€ β€– 𝜌 1 . We only consider 𝑝 1 2 and 𝑝 2 1 since 𝑝 1 1 and 𝑝 2 2 are easy.
For 𝑖 β‰  𝑗 we have 𝑑 𝑝 π‘˜ 𝑖 𝑗 ( πœ€ ) = 𝑑 πœ€ Β± 2 𝛽 ξ…ž ( πœ€ ) π‘Ÿ π‘˜ 𝑖 𝑗 βˆ’ ( ⟨ π‘˜ , πœ” ⟩ Β± 2 𝛽 ) π‘Ÿ π‘˜ 𝑖 𝑗 ' ( πœ€ ) βˆ’ √ βˆ’ 1 ( ⟨ π‘˜ , πœ” ⟩ Β± 2 𝛽 ) 2 . ( 3 . 1 0 )
Then, in the same way as above we obtain the estimate for β€– πœ€ ( πœ• 𝑃 / πœ• πœ€ ) β€– 𝜌 1 .

The following lemma will be used for the zero order term in KAM steps.

Lemma 3.3. Consider the equation Μ‡ π‘₯ = 𝐴 π‘₯ + 𝑔 ( 𝑑 ) , ( 3 . 1 1 ) where 𝐴 is the same as in Lemma 3.1, and 𝑔 is real analytic quasi-periodic in 𝑑 on 𝐷 𝜌 with frequencies πœ” and smooth with respect to πœ€ . Suppose that the small divisors conditions (3.4) hold. Then there exists a unique real analytic quasi-periodic solution x(t) with frequencies πœ” , which satisfies β€– π‘₯ β€– 𝜌 1 ≀ 𝑐 𝛼 𝑠 𝑣 β€– 𝑔 β€– 𝜌 , β€– β€– β€– πœ€ πœ• π‘₯ β€– β€– β€– πœ• πœ€ 𝜌 1 ≀ 𝑐 𝛼 2 𝑠 𝑣 β€² ξ‚΅ β€– 𝑔 β€– 𝜌 + β€– β€– β€– πœ€ πœ• 𝑔 β€– β€– β€– πœ• πœ€ 𝜌 ξ‚Ά , ( 3 . 1 2 ) where 𝑠 , 𝜌 1 , 𝑣 , 𝑣 ξ…ž are defined in Lemma 3.1.

Proof. Similarly, let π‘₯ = 𝐡 βˆ’ 1 π‘₯ , 𝐴 = 𝐡 βˆ’ 1 𝐴 𝐡 and 𝑔 ( 𝑑 ) = 𝐡 βˆ’ 1 𝑔 ( 𝑑 ) . Then (3.11) becomes Μ‡ β€Œ π‘₯ = 𝐴 π‘₯ + 𝑔 ( 𝑑 ) , ( 3 . 1 3 ) where √ 𝐴 = d i a g ( √ βˆ’ 1 𝛽 , βˆ’ βˆ’ 1 𝛽 ) . Expanding π‘₯ = ( π‘₯ 1 , π‘₯ 2 ) and 𝑔 = ( 𝑔 1 , 𝑔 2 ) into Fourier series and using (3.13), we have π‘₯ π‘˜ 𝑖 = 𝑔 π‘˜ 𝑖 √ ξ€· βˆ’ 1 ⟨ π‘˜ , πœ” ⟩ + ( βˆ’ 1 ) 𝑖 𝛽 ξ€Έ . ( 3 . 1 4 ) Using 2 π‘˜ in place of π‘˜ in (3.4), we have | | | | β‰₯ 𝛼 ⟨ π‘˜ , πœ” ⟩ βˆ’ 𝛽 ( πœ€ ) 2 | | π‘˜ | | 𝜏 β€² . ( 3 . 1 5 ) Thus, in the same way as the proof of Lemma 3.1, we can estimate β€– π‘₯ β€– 𝜌 1 and β€– πœ€ πœ• πœ€ π‘₯ β€– 𝜌 1 . We omit the details.

The following lemma is used in the estimate of Lebesgue measure for the parameter πœ€ in the case of non-degeneracy.

Lemma 3.4. Let πœ“ ( πœ€ ) = 𝜎 πœ€ 𝑁 + πœ€ 𝑁 𝑓 ( πœ€ ) , where 𝑁 is a positive integer and 𝑓 satisfies that 𝑓 ( πœ€ ) β†’ 0 as πœ€ β†’ 0 and | 𝑓 ξ…ž ( πœ€ ) | ≀ 𝑐 for πœ€ ∈ ( 0 , πœ€ βˆ— ) . Let πœ™ ( πœ€ ) = ⟨ π‘˜ , πœ” ⟩ βˆ’ 2 𝛽 βˆ’ πœ“ ( πœ€ ) . Let ξƒ― ξ€· 𝑂 = πœ€ ∈ 0 , πœ€ βˆ— ξ€Έ ∣ | | | | β‰₯ 𝛼 πœ™ ( πœ€ ) | | π‘˜ | | 𝜏 β€² ξƒ° , βˆ€ π‘˜ β‰  0 , ( 3 . 1 6 ) where 𝜏 ξ…ž β‰₯ 2 𝜏 + 𝑙 , 𝛼 ≀ ( 1 / 2 ) 𝛼 0 , 𝜎 β‰  0 . Suppose that the small condition (2.4) holds. Then when πœ€ βˆ— is sufficiently small, one has ξ€· m e a s 0 , πœ€ βˆ— ξ€Έ 𝛼 ⧡ 𝑂 ≀ 𝑐 𝛼 2 0 πœ€ βˆ— 𝑁 + 1 , ( 3 . 1 7 ) where 𝑐 is a constant independent of 𝛼 0 , 𝛼 , πœ€ βˆ—

Proof. Let 𝑂 π‘˜ = ξƒ― ξ€· πœ€ ∈ 0 , πœ€ βˆ— ξ€Έ ∣ | | | | < 𝛼 πœ™ ( πœ€ ) | | π‘˜ | | 𝜏 β€² ξƒ° . ( 3 . 1 8 ) By assumption, if πœ€ βˆ— is sufficient small, we have that | πœ“ ( πœ€ ) | ≀ 2 𝜎 πœ€ 𝑁 and | πœ“ ξ…ž ( πœ€ ) | β‰₯ ( 𝜎 / 2 ) πœ€ 𝑁 βˆ’ 1 for πœ€ ∈ ( 0 , πœ€ βˆ— ) . If πœ€ 𝑁 ≀ 𝛼 0 / ( 4 𝜎 | π‘˜ | 𝜏 ) , by (2.4) we have | | πœ™ | | β‰₯ | | | | βˆ’ | | πœ“ | | β‰₯ 𝛼 ( πœ€ ) ⟨ π‘˜ , πœ” ⟩ βˆ’ 2 𝛽 ( πœ€ ) | | π‘˜ | | 𝜏 β€² . ( 3 . 1 9 )
Thus, we only consider the case that πœ€ 𝑁 βˆ— β‰₯ πœ€ 𝑁 β‰₯ ( 𝛼 0 / ( 4 𝜎 | π‘˜ | 𝜏 ) ) . We have | π‘˜ | β‰₯ ( 𝛼 0 / ( 4 𝜎 πœ€ 𝑁 βˆ— ) ) 1 / 𝜏 = 𝐾 . Since | | πœ™ ξ…ž | | = | | πœ“ ( πœ€ ) ξ…ž | | β‰₯ 𝜎 ( πœ€ ) 2 πœ€ 𝑁 βˆ’ 1 β‰₯ 𝛼 0 8 | | π‘˜ | | 𝜏 πœ€ βˆ— , ( 3 . 2 0 ) we have m e a s ( 𝑂 π‘˜ ) ≀ ( ( 2 𝛼 ) / | π‘˜ | 𝜏 β€² ) Γ— ( ( 8 | π‘˜ | 𝜏 πœ€ βˆ— ) / 𝛼 0 ) = ( 1 6 𝛼 πœ€ βˆ— ) / ( | π‘˜ | 𝜏 β€² βˆ’ 𝜏 𝛼 0 ) . So m e a s ξ€· ξ€· 0 , πœ€ βˆ— ξ€Έ ξ€Έ ≀  ⧡ 0 | | π‘˜ | | β‰₯ 𝐾 ξ€· 𝑂 m e a s π‘˜ ξ€Έ ≀ 1 6 𝛼 𝛼 0 πœ€ βˆ—  | | π‘˜ | | β‰₯ 𝐾 1 | | π‘˜ | | 𝜏 β€² βˆ’ 𝜏 ≀ 𝑐 𝛼 𝛼 0 πœ€ βˆ— 𝐾 𝑙 βˆ’ 𝜏 β€² + 𝜏 ≀ 𝑐 𝛼 𝛼 2 0 πœ€ βˆ— 𝑁 + 1 , ( 3 . 2 1 ) where 𝑐 is a constant independent of 𝛼 0 , 𝛼 , and πœ€ βˆ— .

Below we give a lemma with the non-degeneracy conditions.

Lemma 3.5. Consider the real nonlinear Hamiltonian system Μ‡ π‘₯ = 𝐽 βˆ‡ π‘₯ 𝐻 , where 1 𝐻 ( π‘₯ , 𝑑 , πœ€ ) = 2 𝛽 ξ€· π‘₯ 2 1 + π‘₯ 2 2 ξ€Έ + 𝐹 ( π‘₯ , 𝑑 , πœ€ ) w i t h 𝛽 β‰  0 . ( 3 . 2 2 ) Suppose that 𝐹 ( π‘₯ , 𝑑 , πœ€ ) is analytic quasi-periodic with respect to t with frequencies πœ” and real analytic with respect to x and πœ€ on 𝐷 ( π‘Ÿ , 𝜌 , πœ€ 0 ) . Let 𝑓 ( π‘₯ , 𝑑 , πœ€ ) = 𝐽 βˆ‡ π‘₯ 𝐹 ( π‘₯ , 𝑑 , πœ€ ) . Assume that 𝑓 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ 2 π‘š 0 ) and πœ• π‘₯ 𝑓 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ π‘š 0 ) as πœ€ β†’ 0 , where π‘š 0 is a positive integer. Let 𝑄 ( 𝑑 , πœ€ ) = πœ• π‘₯ βˆ‘ 𝑓 ( 0 , 𝑑 , πœ€ ) = π‘˜ β‰₯ π‘š 0 𝑄 π‘˜ ( 𝑑 ) πœ€ π‘˜ . Suppose there exists π‘š 0 ≀ π‘˜ ≀ 2 π‘š 0 βˆ’ 1 such that [ 𝑄 π‘˜ ] 𝐴 β‰  0 and the nonresonance conditions (2.3) and (2.4) hold. Then, for sufficiently small πœ€ βˆ— > 0 , there exists a nonempty Cantor subset 𝐸 βˆ— βŠ‚ ( 0 , πœ€ βˆ— ) , such that for πœ€ ∈ 𝐸 βˆ— , there exists a quasi-periodic symplectic transformation π‘₯ = πœ™ βˆ— ( 𝑑 ) 𝑦 + πœ“ βˆ— ( 𝑑 ) with the frequencies πœ” , which changes the Hamiltonian system to Μ‡ 𝑦 = 𝐽 βˆ‡ 𝑦 𝐻 βˆ— , where 𝐻 βˆ— 1 ( 𝑦 , 𝑑 , πœ€ ) = 2 𝛽 βˆ— ξ€· 𝑦 ( πœ€ ) 2 1 + 𝑦 2 2 ξ€Έ + 𝐹 βˆ— ( 𝑦 , 𝑑 , πœ€ ) , ( 3 . 2 3 ) where 𝐹 βˆ— ( 𝑦 , 𝑑 , πœ€ ) = 𝑂 ( 𝑦 3 ) as 𝑦 β†’ 0 . Moreover, m e a s ( ( 0 , πœ€ βˆ— ) ⧡ 𝐸 βˆ— ) = 𝑂 ( πœ€ π‘š 0 βˆ— + 1 ) as πœ€ βˆ— β†’ 0 . Furthermore, 𝛽 βˆ— ( πœ€ ) = 𝛽 + 𝑂 ( πœ€ π‘š 0 ) and β€– πœ™ βˆ— βˆ’ 𝐼 𝑑 β€– 𝜌 / 2 + β€– πœ“ βˆ— β€– 𝜌 / 2 = 𝑂 ( πœ€ π‘š 0 ) .

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 π‘˜ ] 𝐴 β‰  0 for some π‘š 0 ≀ π‘˜ ≀ 2 π‘š 0 βˆ’ 1 is a non-degeneracy condition.
Consider the following Hamiltonian system Μ‡ π‘₯ = 𝐴 π‘₯ + 𝑓 ( π‘₯ , 𝑑 , πœ€ ) , ( 3 . 2 4 ) where 𝐴 = 𝛽 ( πœ€ ) 𝐽 and 𝑓 is analytic quasi-periodic with respect to 𝑑 with frequencies πœ” and real analytic with respect to π‘₯ and πœ€ on 𝐷 = 𝐷 ( π‘Ÿ , 𝜌 , πœ€ βˆ— ) .
Let β€– 𝑓 β€– 𝐷 ≀ 𝛼 π‘Ÿ Μƒ πœ€ and β€– πœ€ πœ• πœ€ 𝑓 β€– 𝐷 ≀ 𝛼 π‘Ÿ Μƒ πœ€ . Let 𝑄 ( 𝑑 , πœ€ ) = πœ• π‘₯ 𝑓 ( 0 , 𝑑 , πœ€ ) , 𝑔 ( 𝑑 , πœ€ ) = 𝑓 ( 0 , 𝑑 , πœ€ ) and β„Ž ( π‘₯ , 𝑑 , πœ€ ) = 𝑓 ( π‘₯ , 𝑑 , πœ€ ) βˆ’ 𝑔 ( 𝑑 , πœ€ ) βˆ’ 𝑄 ( 𝑑 , πœ€ ) π‘₯ . ( 3 . 2 5 ) Then β„Ž is the higher-order term of 𝑓 . Moreover, the matrix 𝑄 ( 𝑑 , πœ€ ) is Hamiltonian. Let [ 𝑄 ] 𝐴 = Μ‚ 𝛽 ( πœ€ ) 𝐽 .
The system (3.24) is written as ξ€· 𝐴 Μ‡ π‘₯ = + ξ€Έ + 𝑅 ( 𝑑 , πœ€ ) π‘₯ + 𝑔 ( 𝑑 , πœ€ ) + β„Ž ( π‘₯ , 𝑑 , πœ€ ) , ( 3 . 2 6 ) where 𝐴 + = 𝐴 + [ 𝑄 ] 𝐴 = 𝛽 + ( πœ€ ) 𝐽 and 𝑅 = 𝑄 βˆ’ [ 𝑄 ] 𝐴 . By assumption we have β€– 𝑔 β€– 𝜌 ≀ 𝛼 π‘Ÿ Μƒ πœ€ , β€– 𝑄 β€– 𝜌 ≀ 𝛼 Μƒ πœ€ , β€– β„Ž β€– 𝐷 ≀ 3 𝛼 π‘Ÿ Μƒ πœ€ . ( 3 . 2 7 ) Moreover, we have β€– β€– πœ€ πœ• πœ€ 𝑔 β€– β€– 𝜌 β€– β€– ≀ 𝛼 π‘Ÿ Μƒ πœ€ , πœ€ πœ• πœ€ 𝑄 β€– β€– 𝜌 β€– β€– ≀ 𝛼 Μƒ πœ€ , πœ€ πœ• πœ€ β„Ž β€– β€– 𝐷 ≀ 3 𝛼 π‘Ÿ Μƒ πœ€ . ( 3 . 2 8 )
Now we want to construct the symplectic change of variables π‘₯ = 𝑇 ξ…ž 𝑦 = 𝑒 𝑃 ( 𝑑 ) 𝑦 to (3.26), where 𝑃 is a Hamiltonian matrix to be defined later. Then we have ξ‚€ 𝑒 Μ‡ 𝑦 = βˆ’ 𝑃 ξ€· 𝐴 + Μ‡ 𝑃 ξ€Έ 𝑒 + 𝑅 βˆ’ 𝑃 + 𝑒 βˆ’ 𝑃 ξ‚€ Μ‡ 𝑃 𝑒 𝑃 βˆ’ 𝑑 𝑒 𝑑 𝑑 𝑃 ( 𝑑 ) 𝑦   + 𝑒 βˆ’ 𝑃 𝑔 ( 𝑑 , πœ€ ) + 𝑒 βˆ’ 𝑃 β„Ž ξ€· 𝑒 𝑃 ξ€Έ . 𝑦 , 𝑑 , πœ€ ( 3 . 2 9 ) Let π‘Š = 𝑒 𝑃 βˆ’ 𝐼 βˆ’ 𝑃 and ξ‚‹ π‘Š = 𝑒 βˆ’ 𝑃 βˆ’ 𝐼 βˆ’ 𝑃 . Then the system (3.29) becomes ξ€· 𝐴 Μ‡ 𝑦 = + Μ‡ + 𝑅 βˆ’ 𝑃 + 𝐴 + 𝑃 βˆ’ 𝑃 𝐴 + ξ€Έ 𝑦 + 𝑄 ξ…ž 𝑦 + 𝑒 βˆ’ 𝑃 𝑔 ( 𝑑 , πœ€ ) + 𝑒 βˆ’ 𝑃 β„Ž ξ€· 𝑒 𝑃 ξ€Έ 𝑦 , 𝑑 , πœ€ , ( 3 . 3 0 ) where ξ€· Μ‡ 𝑃 ξ€Έ + ξ€· Μ‡ 𝑃 ξ€Έ ξ€· 𝐴 𝑄 β€² = βˆ’ 𝑃 𝑅 βˆ’ 𝑅 βˆ’ 𝑃 βˆ’ 𝑃 + Μ‡ 𝑃 ξ€Έ 𝑃 ξ€· 𝐴 + 𝑅 βˆ’ βˆ’ 𝑃 + Μ‡ 𝑃 ξ€Έ ξ€· 𝐴 + 𝑅 βˆ’ π‘Š + + Μ‡ 𝑃 ξ€Έ π‘Š + ξ‚‹ π‘Š ξ€· 𝐴 + 𝑅 βˆ’ + Μ‡ 𝑃 ξ€Έ 𝑒 + 𝑅 βˆ’ 𝑃 + 𝑒 βˆ’ 𝑃 ξ‚€ Μ‡ 𝑃 𝑒 𝑃 βˆ’ 𝑑 𝑒 𝑑 𝑑 𝑃  . ( 3 . 3 1 )
We would like to have Μ‡ 𝑃 βˆ’ 𝐴 + 𝑃 + 𝑃 𝐴 + = 𝑅 , ( 3 . 3 2 ) where 𝑅 = 𝑄 βˆ’ [ 𝑄 ] 𝐴 . Suppose the small divisors conditions (2.3) hold. Let 𝐸 + βŠ‚ ( 0 , πœ€ βˆ— ) be a subset such that for πœ€ ∈ 𝐸 + the small divisors conditions hold: | | ⟨ π‘˜ , πœ” ⟩ βˆ’ 2 𝛽 + | | β‰₯ 𝛼 ( πœ€ ) + | | π‘˜ | | 𝜏 β€² , βˆ€ π‘˜ ∈ 𝑍 𝑙 ⧡ { 0 } , ( 3 . 3 3 ) where 𝜏 ξ…ž > 2 𝜏 + 𝑙 . By Lemma 3.1, we have a quasi-periodic Hamiltonian matrix 𝑃 ( 𝑑 ) with frequencies πœ” to solve the above equation with the following estimates: β€– 𝑃 β€– 𝜌 βˆ’ 𝑠 ≀ 𝑐 β€– 𝑄 β€– 𝜌 𝛼 + 𝑠 𝑣 ≀ 𝑐 Μƒ πœ€ 𝑠 𝑣 , β€– β€– β€– πœ€ πœ• 𝑃 β€– β€– β€– πœ• πœ€ 𝜌 βˆ’ 𝑠 ≀ 𝑐 𝛼 2 + 𝑠 𝑣 β€² ξ‚΅ β€– 𝑄 β€– 𝜌 + β€– β€– β€– πœ€ πœ• 𝑄 β€– β€– β€– πœ• πœ€ 𝜌 ξ‚Ά ≀ 𝑐 Μƒ πœ€ 𝛼 + 𝑠 𝑣 β€² , ( 3 . 3 4 ) where 𝑣 = 𝜏 ξ…ž + 𝑙 , 𝑣 ξ…ž = 2 𝜏 ξ…ž + 𝑙 and 𝑐 > 0 is a constant. Then the system (3.30) becomes Μ‡ 𝑦 = 𝐴 + 𝑦 + 𝑓 ξ…ž ( 𝑦 , 𝑑 , πœ€ ) , ( 3 . 3 5 ) where 𝑓 ξ…ž = 𝑄 ξ…ž 𝑦 + 𝑒 βˆ’ 𝑃 𝑔 ( 𝑑 , πœ€ ) + 𝑒 βˆ’ 𝑃 β„Ž ( 𝑒 𝑃 𝑦 , 𝑑 , πœ€ ) .
By Lemma 3.3, let us denote by π‘₯ the solution of Μ‡ π‘₯ = 𝐴 + π‘₯ + 𝑔 β€² ( 𝑑 , πœ€ ) on 𝐷 𝜌 βˆ’ 2 𝑠 , where 𝑔 β€² = 𝑒 βˆ’ 𝑃 𝑔 ( 𝑑 , πœ€ ) . Then, by Lemma 3.3 we have β€– β€– π‘₯ β€– β€– 𝜌 βˆ’ 2 𝑠 ≀ 𝑐 β€– 𝑔 β€– 𝜌 βˆ’ 𝑠 𝛼 + 𝑠 𝑣 ≀ 𝑐 π‘Ÿ Μƒ πœ€ 𝑠 𝑣 , β€– β€– β€– πœ€ πœ• π‘₯ β€– β€– β€– πœ• πœ€ 𝜌 βˆ’ 2 𝑠 ≀ 𝑐 𝛼 2 + 𝑠 𝑣 β€² ξ‚΅ β€– 𝑔 β€– 𝜌 βˆ’ 𝑠 + β€– β€– β€– πœ€ πœ• 𝑔 β€– β€– β€– πœ• πœ€ 𝜌 βˆ’ 𝑠 ξ‚Ά ≀ 𝑐 π‘Ÿ Μƒ πœ€ 𝛼 + 𝑠 𝑣 β€² . ( 3 . 3 6 )
Under the symplectic change of variables 𝑦 = 𝑇 ξ…ž ξ…ž π‘₯ + = π‘₯ + π‘₯ + , the Hamiltonian system (3.35) is changed to Μ‡ π‘₯ + = 𝐴 + π‘₯ + + 𝑓 + ξ€· π‘₯ + ξ€Έ , 𝑑 , πœ€ , ( 3 . 3 7 ) where 𝐴 + = 𝛽 + 𝐽 and 𝑓 + = 𝑄 ξ…ž β‹… 𝑇 ξ…ž ξ…ž + 𝑒 βˆ’ 𝑃 β„Ž ∘ 𝑇 ξ…ž ∘ 𝑇 ξ…ž ξ…ž . ( 3 . 3 8 )
Let the symplectic transformation 𝑇 = 𝑇 ξ…ž ∘ 𝑇 ξ…ž ξ…ž . Then π‘₯ = 𝑇 π‘₯ + = πœ™ ( 𝑑 ) π‘₯ + + πœ“ ( 𝑑 ) , where πœ™ ( 𝑑 ) = 𝑒 𝑃 ( 𝑑 ) and πœ“ ( 𝑑 ) = 𝑒 𝑃 ( 𝑑 ) π‘₯ ( 𝑑 ) . It is easy to obtain that if β€– 𝑃 β€– 𝜌 βˆ’ 2 𝑠 ≀ 1 / 2 , then β€– πœ™ βˆ’ 𝐼 β€– 𝜌 βˆ’ 2 𝑠 ≀ 𝑐 Μƒ πœ€ 𝑠 𝑣 , β€– β€– πœ€ πœ• πœ€ πœ™ β€– β€– 𝜌 βˆ’ 2 𝑠 ≀ 𝑐 Μƒ πœ€ 𝛼 + 𝑠 𝑣 β€² , β€– πœ“ β€– 𝜌 βˆ’ 2 𝑠 ≀ 𝑐 π‘Ÿ Μƒ πœ€ 𝑠 𝑣 , β€– β€– πœ€ πœ• πœ€ πœ“ β€– β€– 𝜌 βˆ’ 2 𝑠 ≀ 𝑐 π‘Ÿ Μƒ πœ€ 𝛼 + 𝑠 𝑣 β€² . ( 3 . 3 9 ) Under the symplectic change of variables π‘₯ = 𝑇 π‘₯ + , the Hamiltonian system (3.24) becomes (3.37).
Below we give the estimates for 𝐴 + and 𝑓 + . Obviously, it follows that 𝐴 + ( πœ€ ) βˆ’ 𝐴 = [ 𝑄 ] 𝐴 = Μ‚ 𝛽 ( πœ€ ) 𝐽 and | | 𝛽 + | | = | | Μ‚ | | | | πœ€ ξ€· 𝛽 ( πœ€ ) βˆ’ 𝛽 ( πœ€ ) 𝛽 ( πœ€ ) ≀ 𝑐 𝛼 Μƒ πœ€ , ξ…ž + ( πœ€ ) βˆ’ 𝛽 ξ…ž ξ€Έ | | = | | | πœ€  𝛽 ( πœ€ ) ξ…ž | | | ( πœ€ ) ≀ 𝑐 𝛼 Μƒ πœ€ . ( 3 . 4 0 ) By (3.38) we have 𝑓 + ξ€· π‘₯ + ξ€Έ , 𝑑 , πœ€ = 𝑄 ξ…ž ξ€· π‘₯ ( 𝑑 ) + + π‘₯ ξ€Έ ( 𝑑 ) + 𝑒 βˆ’ 𝑃 ( 𝑑 ) β„Ž ξ€· 𝑒 𝑃 ( 𝑑 ) ξ€· π‘₯ + + π‘₯ ξ€Έ ξ€Έ ( 𝑑 ) , 𝑑 , πœ€ . ( 3 . 4 1 ) Let 𝜌 + = 𝜌 βˆ’ 2 𝑠 , and π‘Ÿ + = πœ‚ π‘Ÿ with πœ‚ ≀ 1 / 8 . If 𝑐 Μƒ πœ€ / 𝛼 + 𝑠 𝑣 + 𝑣 β€² ≀ πœ‚ , it follows that β€– π‘₯ β€– 𝜌 βˆ’ 2 𝑠 ≀ ( 1 / 8 ) π‘Ÿ . Let 𝐷 + = 𝐷 ( π‘Ÿ + , 𝑠 + , πœ€ βˆ— ) . Note that 𝑄 ξ…ž and β„Ž only consist of high-order terms of 𝑃 and π‘₯ , respectively. It is easy to see | 𝑒 𝑃 ( 𝑑 ) ( π‘₯ + + π‘₯ ( 𝑑 ) ) | ≀ 4 πœ‚ π‘Ÿ ≀ π‘Ÿ . By all the estimates (3.27), (3.28), (3.34), and (3.36), and using usual technique of KAM estimate, we have β€– β€– 𝑓 + β€– β€– 𝐷 + ≀ 𝑐 Μƒ πœ€ 2 𝑠 2 𝑣 πœ‚ π‘Ÿ + 𝑐 𝛼 π‘Ÿ Μƒ πœ€ πœ‚ 2 ≀ ξ‚€ 𝑐 Μƒ πœ€ 𝑠 2 𝑣  π‘Ÿ + 𝑐 𝛼 πœ‚ + β€– β€– Μƒ πœ€ , πœ€ πœ• πœ€ 𝑓 + β€– β€– 𝐷 + ≀ 𝑐 Μƒ πœ€ 2 𝛼 + 𝑠 𝑣 + 𝑣 β€² πœ‚ π‘Ÿ + 𝑐 𝛼 π‘Ÿ Μƒ πœ€ πœ‚ 2 ≀ ξ‚΅ 𝑐 Μƒ πœ€ 𝛼 𝑠 𝑣 + 𝑣 β€² ξ‚Ά π‘Ÿ + 𝑐 𝛼 πœ‚ + Μƒ πœ€ . ( 3 . 4 2 ) Let 𝛼 + = 𝛼 / 2 and πœ‚ = 𝑐 Μƒ πœ€ / ( 𝛼 2 𝑠 𝑣 + 𝑣 β€² ) . Then we have β€– β€– 𝑓 + β€– β€– 𝐷 + ≀ 𝑐 𝛼 + π‘Ÿ + πœ‚ Μƒ πœ€ = 𝛼 + π‘Ÿ + Μƒ πœ€ + , Μƒ πœ€ + = 𝑐 πœ‚ Μƒ πœ€ . ( 3 . 4 3 ) Similarly, we have β€– β€– πœ€ πœ• πœ€ 𝑓 + β€– β€– 𝐷 + ≀ 𝛼 + π‘Ÿ + Μƒ πœ€ + . ( 3 . 4 4 )
Note that KAM steps only make sense for the small parameter πœ€ satisfying small divisors conditions. However, by Whitney’s extension theorem, for convenience all the functions are supposed to be defined for πœ€ on [ 0 , πœ€ βˆ— ] .
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 𝑓 0 = 𝑓 . Let 𝑓 ( π‘₯ , 𝑑 , πœ€ ) = 𝑓 ( 0 , 𝑑 , πœ€ ) + πœ• π‘₯ 𝑓 ( 0 , 𝑑 , πœ€ ) π‘₯ + β„Ž ( π‘₯ , 𝑑 , πœ€ ) . By assumption, if πœ€ βˆ— is sufficiently small, we have that for all πœ€ ∈ [ 0 , πœ€ βˆ— ] | | | | 𝑓 ( 0 , 𝑑 , πœ€ ) ≀ 𝑐 πœ€ 2 π‘š 0 , | | πœ• π‘₯ | | 𝑓 ( 0 , 𝑑 , πœ€ ) ≀ 𝑐 πœ€ π‘š 0 , | | πœ€ πœ• πœ€ | | 𝑓 ( 0 , 𝑑 , πœ€ ) ≀ 𝑐 πœ€ 2 π‘š 0 , | | πœ€ πœ• πœ€ πœ• π‘₯ | | 𝑓 ( 0 , 𝑑 , πœ€ ) ≀ 𝑐 πœ€ π‘š 0 . ( 3 . 4 5 ) Moreover, | | | | β„Ž ( π‘₯ , 𝑑 , πœ€ ) ≀ 𝑐 | π‘₯ | 2 , | | πœ€ πœ• πœ€ | | β„Ž ( π‘₯ , 𝑑 , πœ€ ) ≀ 𝑐 | π‘₯ | 2 , βˆ€ | π‘₯ | ≀ πœ€ π‘š 0 ξ€Ί , βˆ€ πœ€ ∈ 0 , πœ€ βˆ— ξ€» . ( 3 . 4 6 ) Let π‘Ÿ 0 = πœ€ π‘š 0 , 𝜌 0 = 𝜌 , 𝑠 0 = 𝜌 0 / 8 , 𝐷 0 = 𝐷 ( π‘Ÿ 0 , 𝜌 0 , πœ€ βˆ— ) , and Μƒ πœ€ 0 = 𝑐 πœ€ π‘š 0 / 𝛼 0 . Then we have | | 𝑓 0 | | 𝐷 0 ≀ 𝛼 0 π‘Ÿ 0 Μƒ πœ€ 0 , | | πœ€ πœ• πœ€ 𝑓 0 | | 𝐷 0 ≀ 𝛼 0 π‘Ÿ 0 Μƒ πœ€ 0 . ( 3 . 4 7 ) For 𝑛 β‰₯ 1 , let 𝛼 𝑛 = 𝛼 𝑛 βˆ’ 1 2 , 𝑠 𝑛 = 𝑠 𝑛 βˆ’ 1 2 , 𝜌 𝑛 = 𝜌 𝑛 βˆ’ 1 βˆ’ 2 𝑠 𝑛 βˆ’ 1 , πœ‚ 𝑛 βˆ’ 1 = 𝑐 Μƒ πœ€ 𝑛 βˆ’ 1 𝛼 2 𝑛 βˆ’ 1 𝑠 𝑣 + 𝑣 β€² 𝑛 βˆ’ 1 , π‘Ÿ 𝑛 = πœ‚ 𝑛 βˆ’ 1 π‘Ÿ 𝑛 βˆ’ 1 , Μƒ πœ€ 𝑛 = 𝑐 πœ‚ 𝑛 βˆ’ 1 Μƒ πœ€ 𝑛 βˆ’ 1 . ( 3 . 4 8 )
Then we have a sequence of quasi-periodic symplectic transformations { 𝑇 𝑛 } satisfying 𝑇 𝑛 π‘₯ = πœ™ 𝑛 ( 𝑑 ) π‘₯ + πœ“ 𝑛 ( 𝑑 ) with β€– β€– πœ™ 𝑛 β€– β€– βˆ’ 𝐼 𝜌 𝑛 + 1 ≀ 𝑐 Μƒ πœ€ 𝑛 𝑠 𝑣 𝑛 , β€– β€– πœ“ 𝑛 β€– β€– 𝜌 𝑛 + 1 ≀ 𝑐 π‘Ÿ 𝑛 Μƒ πœ€ 𝑛 𝑠 𝑣 𝑛 . ( 3 . 4 9 ) Let 𝑇 𝑛 = 𝑇 0 ∘ 𝑇 1 β‹― ∘ 𝑇 𝑛 βˆ’ 1 . Then under the transformation π‘₯ = 𝑇 𝑛 𝑦 the Hamiltonian system Μ‡ π‘₯ = 𝐴 0 π‘₯ + 𝑓 0 ( π‘₯ , 𝑑 , πœ€ ) is changed to Μ‡ 𝑦 = 𝐴 𝑛 𝑦 + 𝑓 𝑛 ( 𝑦 , 𝑑 , πœ€ ) .
Moreover, 𝐴 𝑛 ( πœ€ ) = 𝛽 𝑛 ( πœ€ ) 𝐽 satisfies 𝐴 𝑛 + 1 βˆ’ 𝐴 𝑛 = [ 𝑄 𝑛 ] 𝐴 and | | 𝛽 𝑛 + 1 ( πœ€ ) βˆ’ 𝛽 𝑛 | | ( πœ€ ) ≀ 𝑐 𝛼 𝑛 Μƒ πœ€ 𝑛 , | | πœ€ ξ€· 𝛽 ξ…ž 𝑛 + 1 ( πœ€ ) βˆ’ 𝛽 ξ…ž 𝑛 ξ€Έ | | ( πœ€ ) ≀ 𝑐 𝛼 𝑛 Μƒ πœ€ 𝑛 , ( 3 . 5 0 ) β€– β€– 𝑓 𝑛 β€– β€– 𝐷 𝑛 ≀ 𝛼 𝑛 π‘Ÿ 𝑛 Μƒ πœ€ 𝑛 . ( 3 . 5 1 )
Convergence
By the above definitions we have πœ‚ 𝑛 / πœ‚ 𝑛 βˆ’ 1 = 𝑐 Μƒ πœ€ 𝑛 / Μƒ πœ€ 𝑛 βˆ’ 1 = 𝑐 πœ‚ 𝑛 βˆ’ 1 . Thus, we have πœ‚ 𝑛 ≀ 𝑐 πœ‚ 2 𝑛 βˆ’ 1 and so 𝑐 πœ‚ 𝑛 ≀ ( 𝑐 πœ‚ 𝑛 βˆ’ 1 ) 2 ≀ ( 𝑐 πœ‚ 0 ) 2 𝑛 . Note that πœ‚ 0 = 𝑐 Μƒ πœ€ 0 / ( 𝛼 2 0 𝑠 𝑣 + 𝑣 β€² 0 ) ≀ 𝑐 πœ€ π‘š 0 / ( 𝛼 2 0 𝜌 𝑣 + 𝑣 β€² 0 ) . Suppose that πœ€ βˆ— is sufficiently small such that for 0 < πœ€ < πœ€ βˆ— we have 𝑐 πœ‚ 0 ≀ 1 / 2 . 𝑇 𝑛 are affine, so are 𝑇 𝑛 with 𝑇 𝑛 π‘₯ = πœ™ 𝑛 ( 𝑑 ) π‘₯ + πœ“ 𝑛 ( 𝑑 ) . By the estimates (3.49) it is easy to prove that πœ™ 𝑛 ( 𝑑 ) and πœ“ 𝑛 ( 𝑑 ) are convergent and so 𝑇 𝑛 is actually convergent on the domain 𝐷 ( π‘Ÿ / 2 , 𝜌 / 2 ) . Let 𝑇 𝑛 β†’ 𝑇 βˆ— and 𝑇 βˆ— π‘₯ = πœ™ βˆ— ( 𝑑 ) π‘₯ + πœ“ βˆ— ( 𝑑 ) . It is easy to see that the estimates for πœ™ βˆ— and πœ“ βˆ— in Theorem 2.1 hold.
Using the estimate for 𝑓 𝑛 and Cauchy’s estimate, we have | 𝑓 𝑛 ( 0 , 𝑑 , πœ€ ) | ≀ 𝛼 𝑛 π‘Ÿ 𝑛 Μƒ πœ€ 𝑛 β†’ 0 and | πœ• π‘₯ 𝑓 𝑛 ( 0 , 𝑑 , πœ€ ) | ≀ 𝛼 𝑛 Μƒ πœ€ 𝑛 β†’ 0 as 𝑛 β†’ ∞ . Let 𝑓 𝑛 β†’ 𝑓 βˆ— . Then it follows that 𝑓 βˆ— ( π‘₯ , 𝑑 , πœ€ ) = 𝑂 ( π‘₯ 2 ) .
By the estimates (3.50) for 𝛽 𝑛 we have 𝛽 𝑛 β†’ 𝛽 βˆ— . Thus, by the quasi-periodic symplectic transformation π‘₯ = 𝑇 βˆ— 𝑦 , the original system is changed to Μ‡ 𝑦 = 𝐴 βˆ— 𝑦 + 𝑓 βˆ— ( 𝑦 , 𝑑 , πœ€ ) with 𝐴 βˆ— = 𝛽 βˆ— 𝐽 .
Estimate of Measure
Let 𝐸 𝑛 = ξƒ― ξ€· πœ€ ∈ 0 , πœ€ βˆ— ξ€Έ ∣ | | ⟨ πœ” , π‘˜ ⟩ βˆ’ 2 𝛽 𝑛 | | β‰₯ 𝛼 ( πœ€ ) 𝑛 | | π‘˜ | | 𝜏 β€² ξƒ° . ( 3 . 5 2 ) Note that 𝛽 𝑛 = 𝛽 1 + πœ“ , where βˆ‘ πœ“ = 𝑛 βˆ’ 1 𝑗 = 1 𝛽 𝑗 + 1 βˆ’ 𝛽 𝑗 , 𝛽 1 Μ‚ 𝛽 = 𝛽 + , and Μ‚ 𝛽 𝐽 = [ 𝑄 ] 𝐴 . Note that Μƒ πœ€ 1 = 𝑐 Μƒ πœ€ 2 0 / ( 𝛼 2 0 𝑠 𝑣 + 𝑣 β€² 0 ) and Μƒ πœ€ 0 = 𝑐 πœ€ π‘š 0 / 𝛼 0 . By the estimates (3.50), we have πœ“ ( πœ€ ) = 𝑂 ( πœ€ 2 π‘š 0 ) and πœ€ πœ“ β€² ( πœ€ ) = 𝑂 ( πœ€ 2 π‘š 0 ) . By assumption, [ 𝑄 ] 𝐴 is analytic with respect to πœ€ and there exists π‘š 0 ≀ 𝑁 ≀ 2 π‘š 0 βˆ’ 1 such that [ 𝑄 ] 𝐴 = 𝛿 πœ€ 𝑁 + 𝑂 ( πœ€ 𝑁 + 1 ) with 𝛿 β‰  0 . Thus, 𝛽 1 ( πœ€ ) = 𝛽 + 𝛿 πœ€ 𝑁 + 𝑂 ( πœ€ 𝑁 + 1 ) . By Lemma 3.4, we have meas ( ( 0 , πœ€ βˆ— ) βˆ’ 𝐸 𝑛 ) ≀ 𝑐 ( 𝛼 𝑛 / 𝛼 2 0 ) πœ€ βˆ— 𝑁 + 1 . Let 𝐸 βˆ— = β‹‚ 𝑛 β‰₯ 1 𝐸 𝑛 . By 𝛼 𝑛 = 𝛼 0 / 2 𝑛 , it follows that meas ( ( 0 , πœ€ βˆ— ) βˆ’ 𝐸 βˆ— ) ≀ 𝑐 πœ€ βˆ— 𝑁 + 1 / 𝛼 0 . 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 πœ€ does not involve into the small divisors and so the systems are analytic in πœ€ . 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 Μ‡ π‘₯ = 𝐴 π‘₯ + 𝑓 ( π‘₯ , 𝑑 , πœ€ ) , ( 4 . 1 ) where 𝐴 = 𝛽 𝐽 and 𝑓 = 𝐽 βˆ‡ π‘₯ 𝐹 . By Lemma 3.3, denote by π‘₯ the solution of Μ‡ π‘₯ = 𝐴 π‘₯ + 𝑓 ( 0 , 𝑑 , πœ€ ) on 𝐷 3 𝜌 / 4 . Under the change of variables π‘₯ = 𝑇 0 π‘₯ + = π‘₯ + π‘₯ + , the Hamiltonian system (2.1) becomes Μ‡ π‘₯ + = 𝐴 π‘₯ + + 𝑓 1 ξ€· π‘₯ + ξ€Έ , 𝑑 , πœ€ , ( 4 . 2 ) where 𝑓 1 ( π‘₯ + , 𝑑 , πœ€ ) = 𝑓 ( π‘₯ + π‘₯ + , 𝑑 , πœ€ ) βˆ’ 𝑓 ( 0 , 𝑑 , πœ€ ) satisfying 𝑓 1 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ 2 ) and πœ• π‘₯ + 𝑓 1 ( 0 , 𝑑 , πœ€ ) = 𝑂 ( πœ€ ) .

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 β‰₯ 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 πœ€ . Thus, we need not delete any parameter. Moreover, the analyticity in πœ€ remains in the KAM steps, which makes the estimate easier. At the first step, we consider Μ‡ π‘₯ = 𝐴 π‘₯ + 𝑓 1 ( π‘₯ , 𝑑 , πœ€ ) . In the same way as the case of nondegenerate case, let π‘Ÿ 1 = πœ€ , 𝜌 1 = 3 𝜌 / 4 , πœ€ 1 = πœ€ 0 , 𝐷 1 = 𝐷 ( π‘Ÿ 1 , 𝜌 1 , πœ€ 1 ) , and Μƒ πœ€ 1 = 𝑐 πœ€ / 𝛼 0 . Then we have β€– 𝑓 1 β€– 𝐷 1 ≀ 𝛼 0 π‘Ÿ 1 Μƒ πœ€ 1 .
At 𝑛 t h step, we consider the Hamiltonian system Μ‡ π‘₯ = 𝐴 π‘₯ + 𝑓 𝑛 ( π‘₯ , 𝑑 , πœ€ ) , ( 4 . 3 ) where 𝑓 𝑛 is analytic quasi-periodic with respect to 𝑑 with frequencies πœ” and real analytic with respect to π‘₯ and πœ€ on 𝐷 𝑛 = 𝐷 ( π‘Ÿ 𝑛 , 𝜌 𝑛 , πœ€ 𝑛 ) . Moreover, β€– 𝑓 𝑛 β€– 𝐷 𝑛 ≀ 𝛼 0 π‘Ÿ 𝑛 Μƒ πœ€ 𝑛 . Suppose 𝑄 𝑛 ( 𝑑 , πœ€ ) = πœ• π‘₯ 𝑓 𝑛 ξ‚€ πœ€ ( 0 , 𝑑 , πœ€ ) = 𝑂 2 𝑛 βˆ’ 1  , 𝑓 𝑛 ξ€· πœ€ ( 0 , 𝑑 , πœ€ ) = 𝑂 2 𝑛 ξ€Έ . ( 4 . 4 )
Since 𝑄 𝑛 is analytic with respect to πœ€ , it follows that 𝑄 𝑛 = ∞  π‘˜ = 2 𝑛 βˆ’ 1 𝑄 π‘˜ 𝑛 πœ€ π‘˜ . ( 4 . 5 ) Truncating the above power series of πœ€ , we let 𝑅 𝑛 ( 𝑑 , πœ€ ) = 2 𝑛 βˆ’ 1  π‘˜ = 2 𝑛 βˆ’ 1 𝑄 π‘˜ 𝑛 πœ€ π‘˜ ,  𝑄 𝑛 = 𝑄 𝑛 βˆ’ 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 𝑛 π‘₯ = πœ™ 𝑛 ( 𝑑 ) π‘₯ + πœ“ 𝑛 ( 𝑑 ) satisfying (3.49). Let 𝑇 𝑛 = 𝑇 1 ∘ 𝑇 2 β‹― ∘ 𝑇 𝑛 βˆ’ 1 .
By the transformation π‘₯ = 𝑇 𝑛 𝑦 , the system (4.3) is changed to Μ‡ 𝑦 = 𝐴 𝑦 + 𝑓 𝑛 + 1 ( 𝑦 , 𝑑 , πœ€ ) , ( 4 . 7 ) where 𝑓 𝑛 + 1 =  𝑄 𝑛 β‹… 𝑇 𝑛 ξ…ž ξ…ž + 𝑄 ξ…ž 𝑛 β‹… 𝑇 𝑛 ξ…ž ξ…ž + 𝑒 βˆ’ 𝑃 𝑛 β‹… β„Ž 𝑛 ∘ 𝑇 𝑛 =  𝑄 𝑛 ( π‘₯ 𝑛 + 𝑦 ) + 𝑄 ξ…ž 𝑛 ( π‘₯ 𝑛 + 𝑦 ) + 𝑒 βˆ’ 𝑃 𝑛 β„Ž 𝑛 ( 𝑒 𝑃 𝑛 ( π‘₯ 𝑛 + 𝑦 ) ) .
The last two terms can be estimated similarly as those of (3.41). Note that  𝑄 𝑛 = 𝑄 𝑛 βˆ’ 𝑅 𝑛 =  π‘˜ β‰₯ 2 𝑛 𝑄 π‘˜ 𝑛 πœ€ π‘˜ ( 4 . 8 ) only consists of the higher order terms of πœ€ . So, in the same way as [8, 10], we use the technique of shriek of the domain interval of πœ€ to estimate the first term.
Let π‘Ÿ 1 = πœ€ , 𝜌 1 = 3 𝜌 / 4 , πœ€ 1 = πœ€ 0 and 𝑠 1 = 𝜌 / 1 6 .
Define 𝑠 𝑛 + 1 = 𝑠 𝑛 / 2 , 𝜌 𝑛 + 1 = 𝜌 𝑛 βˆ’ 2 𝑠 𝑛 , πœ‚ 𝑛 = ( 1 / 8 ) 𝑒 βˆ’ ( 4 / 3 ) 𝑛 , π‘Ÿ 𝑛 + 1 = πœ‚ 𝑛 π‘Ÿ 𝑛 , 𝛿 𝑛 = 1 βˆ’ ( 2 / 3 ) 𝑛 and πœ€ 𝑛 + 1 = 𝛿 𝑛 πœ€ 𝑛 . Let 𝐷 𝑛 + 1 = 𝐷 ( π‘Ÿ 𝑛 + 1 , 𝜌 𝑛 + 1 , πœ€ 𝑛 + 1 ) .
If 𝑐 Μƒ πœ€ 𝑛 / 𝑠 𝑛 2 𝑣 ≀ πœ‚ 𝑛 < ( 1 / 8 ) , it follows that β€– β€– 𝑓 𝑛 + 1 β€– β€– 𝐷 𝑛 + 1 ≀  𝛼 0 Μƒ πœ€ 𝑛 𝑒 βˆ’ ( 4 / 3 ) 𝑛 + ξ‚΅ 𝑐 Μƒ πœ€ 𝑛 𝑠 𝑣 𝑛 ξ‚Ά 2 ξƒͺ πœ‚ 𝑛 π‘Ÿ 𝑛 + 𝑐 𝛼 0 π‘Ÿ 𝑛 Μƒ πœ€ 𝑛 πœ‚ 2 𝑛 ≀ 𝛼 0 π‘Ÿ 𝑛 + 1 Μƒ πœ€ 𝑛 + 1 , ( 4 . 9 ) where Μƒ πœ€ 𝑛 + 1 = 𝑐 πœ‚ 𝑛 Μƒ πœ€ 𝑛 . Moreover, it is easy to see πœ• π‘₯ 𝑓 𝑛 + 1 ξ€· πœ€ ( 0 , 𝑑 , πœ€ ) = 𝑂 2 𝑛 ξ€Έ , 𝑓 𝑛 + 1 ξ‚€ πœ€ ( 0 , 𝑑 , πœ€ ) = 𝑂 2 𝑛 + 1  . ( 4 . 1 0 ) Now we verify 𝑐 Μƒ πœ€ 𝑛 / 𝑠 𝑛 2 𝑣 ≀ πœ‚ 𝑛 < 1 / 8 . Let 𝐺 𝑛 = 𝑐 Μƒ πœ€ 𝑛 / 𝑠 𝑛 2 𝑣 . By 𝐺 𝑛 = 𝑐 𝑒 βˆ’ ( 4 / 3 ) 𝑛 βˆ’ 1 1 6 𝑣 𝐺 𝑛 βˆ’ 1 , it follows that 𝐺 𝑛 = ξ€· 𝑐 1 6 𝑣 ξ€Έ 𝑛 βˆ’ 1 𝑒 βˆ’ [ ( 4 / 3 ) 𝑛 βˆ’ 1 + ( 4 / 3 ) 𝑛 βˆ’ 2 + β‹― + ( 4 / 3 ) 1 ] 𝐺 1 = ξ€· 𝑐 1 6 𝑣 ξ€Έ 𝑛 βˆ’ 1 𝑒 4 𝑒 βˆ’ 4 ( 4 / 3 ) 𝑛 βˆ’ 1 𝐺 1 . ( 4 . 1 1 ) Note that 𝐺 1 = 𝑐 Μƒ πœ€ 1 / 𝑠 1 2 𝑣 . If Μƒ πœ€ 1 is sufficiently small, we have 𝑐 Μƒ πœ€ 𝑛 / 𝑠 𝑛 2 𝑣 = 𝐺 𝑛 ≀ πœ‚ 𝑛 .
Note that ( 𝑐 π‘Ÿ 𝑛 Μƒ πœ€ 𝑛 / 𝑠 𝑣 𝑛 ) β†’ 0 and ( 𝑐 Μƒ πœ€ 𝑛 / ( πœ‚ 𝑛 𝑠 𝑣 𝑛 ) ) β†’ 0 as 𝑛 β†’ ∞ , and Μƒ πœ€ 𝑛 ≀ 𝑐 𝑠 𝑛 2 𝑣 𝐺 𝑛 . Let πœ€ βˆ— = ∏ 𝑛 β‰₯ 1 ( 1 βˆ’ ( 2 / 3 ) 𝑛 ) πœ€ 0 . Thus, in the same way as before we can prove the convergence of the KAM iteration for all πœ€ ∈ ( 0 , πœ€ βˆ— ) 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 ( π‘₯ , 𝑑 , πœ€ ) = ⟨ πœ” , 𝐼 ⟩ + ( 1 / 2 ) ( 𝛽 βˆ— + 𝜎 ( πœ€ ) ) ( π‘₯ 2 1 + π‘₯ 2 2 ) + 𝐹 ( π‘₯ , 𝑑 , πœ€ ) , where ( πœƒ , 𝐼 ) are the angle variable and the action variable and π‘₯ = ( π‘₯ 1 , π‘₯ 2 ) are a pair of normal variables. In the same way as in [11], 𝜎 ( πœ€ ) is the modified term of the normal frequency. Then by some technique as in [1113], 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).

References

  1. N. N. Bogoljubov, J. A. Mitropoliski, and A. M. Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer, New York, NY, USA, 1976.
  2. R. A. Johnson and G. R. Sell, β€œSmoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems,” Journal of Differential Equations, vol. 41, no. 2, pp. 262–288, 1981. View at Publisher Β· View at Google Scholar
  3. A. Jorba and C. SimΓ³, β€œOn the reducibility of linear differential equations with quasiperiodic coefficients,” Journal of Differential Equations, vol. 98, no. 1, pp. 111–124, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. L. H. Eliasson, β€œFloquet solutions for the one-dimensional quasi-periodic SchrΓΆdinger equation,” Communications in Mathematical Physics, vol. 146, no. 3, pp. 447–482, 1992. View at Publisher Β· View at Google Scholar
  5. L. H. Eliasson, β€œAlmost reducibility of linear quasi-periodic systems,” in Smooth Ergodic Theory and Its Applications (Seattle, WA, 1999), vol. 69 of Proceedings of Symposia in Pure Mathematics, pp. 679–705, American Mathematical Society, Providence, RI, USA, 2001. View at Zentralblatt MATH
  6. H.-L. Her and J. You, β€œFull measure reducibility for generic one-parameter family of quasi-periodic linear systems,” Journal of Dynamics and Differential Equations, vol. 20, no. 4, pp. 831–866, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. A. Jorba and C. SimΓ³, β€œOn quasi-periodic perturbations of elliptic equilibrium points,” SIAM Journal on Mathematical Analysis, vol. 27, no. 6, pp. 1704–1737, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. X. Wang and J. Xu, β€œOn the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point,” Nonlinear Analysis, vol. 69, no. 7, pp. 2318–2329, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. H. Whitney, β€œAnalytic extensions of differentiable functions defined in closed sets,” Transactions of the American Mathematical Society, vol. 36, no. 1, pp. 63–89, 1934. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. L. Zhang and J. Xu, β€œPersistence of invariant torus in Hamiltonian systems with two-degree of freedom,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 793–802, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. J. Moser, β€œConvergent series expansions for quasi-periodic motions,” Mathematische Annalen, vol. 169, pp. 136–176, 1967. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. M. B. Sevryuk, β€œKAM-stable Hamiltonians,” Journal of Dynamical and Control Systems, vol. 1, no. 3, pp. 351–366, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. K. Soga, β€œA point-wise criterion for quasi-periodic motions in the KAM theory,” Nonlinear Analysis, vol. 73, no. 10, pp. 3151–3161, 2010. View at Publisher Β· View at Google Scholar