辛映射低维不变环面的保持性
摘要
本文主要研究带参数的辛映射(symplectic mapping)低维不变环面的保持性，我们证明了对大部分参数，扭转辛映射的椭圆型低维不变环面和双曲型低维不变环面在小的辛扰动下保持下来。
上世纪六十年代，著名数学家Kolmogorov，Arnold和Moser建立了KAM理论，该理论的建立具有划时代的意义，它给出了太阳系运行机制的一个合理解释，使得这一困扰人们很长时间的问题得到解决，也使人们对Hamilton系统有了新的认识，在KAM理论建立之前，人们一直认为几乎所有的Hamilton系统的轨道在其能量面上是遍历的，然而由KAM定理，典型的2n维近可积Hamilton系统的大部分轨道只在n维不变环面上遍历，并不在能量面(2n-1维)上遍历，同时，KAM理论的建立为人们研究近可积Hamilton系统和近扭转映射的动力学行为提供了一套系统的方法，并且在许多实际问题中得到了应用。
辛映射的KAM理论研究的是近扭转辛映射的动力学行为，所谓近扭转辛映射是指扭转映射加上小的扰动所得到的辛映射，在作用-角坐标下，扭转映射将相空间分成一个个不变环面，映射产生的轨道在不变环面上呈现拟周期运动，其频率互不相同，辛映射的KAM理论告诉我们：如果扭转项满足一定的条
吉林大学博士学位论文辛侠射低维不变环面的保持性H
件，在小扰动下近扭转辛映射的绝大部分不变环面将保持下来.辛映射的KAM
理论在数值计算已经中得到广泛的应用.CI，annel和Scovel，冯康，Sanz一Serna
和Calvo，尚在久等将它应用到可积Halnil七on系统的辛算法中，证明了在该算
法下不变环面的存在性和算法的收敛性.因此，对辛映射KAM理论的研究具
有重要的理论意义和实际应用价值.
以往研究近扭转辛映射得到的都是最高维数的不变环面的存在性.本文中，
我们对近扭转辛映射低维不变环面的存在性进行了研究，得到了几个KAM型
结果.
考虑近扭转辛映射且:T几xR”又R“‘义R”，x口。弓Tr‘又R“X Rm xR”‘
示二.:+、0(荟)十
且:刀=刀_
:，分，，‘，公，苟)，
句动动
了‘..、/‘.吸、、厂产r.、
丛即丛。幽。迅。
=U十
人2。(石)、‘+
92〔)(石)公一
，万，了不，?少
军，?必，，l)
(1)
一
月t
一一
入，t)
其中O。是Rl()中的有界闭集.当P(，三。时，映射(l)为如下扭转映射
示=T+、〔，(石)，
口二岁，
方=(I+公2。(石))，，，
公=(I+92。(石))一‘，:
它具有低维不变环面
，T(岁)=Tx{万}X{()}X{O}
(3)
当兑(，的形式不同时，这些低维不变环面的类型不同，在它们附近的动力学行
为也不同，研究在扰动下不变环面的保持性所需的条件和所用的方法也不同.
首先考虑椭圆型低维不变环面的保持性.假设
‘2()=(liag{犷2。、、…，‘2。，了:}，1+几。，=〔·2万门入。2〔￡).(4)
//t
」吉林大学博士学位论文.辛映射低维不变环面的保持性IH
则映射(2)的轨道在法空间中围绕原点旋转，既不向内收缩，也不向外扩张，
这时我们称(2)的低维不变环面，T(功为椭圆型低维不变环面.我们对近扭转
辛映射，进行KAM迭代时法向频率也将是产生小除数的因素，因此在迭代中
不光要考虑切向频率，还要考虑法向频率.所以迭代步骤中需要更加精细的计
算.由于多出了法向变量，使得我们要估算的量成倍增长.经过细致的计算，
我们证明了:当扭转辛映射满足某种条件时，加上小的扰动后，大部分不变环
面保持下来.另外，在这些不变环面上，映射的轨道呈现拟周期运动，频率与
原频率相差很小.表达成定理就是本文的第一个主要结果:
定理1考虑映射(l)，其中、。二记，法向频率具(封的形式，且满足条件
}挚}_:，，、粤，1:、:。，::、:。
!，a勺”口任
(5)
并且几在复邻域D(:。，s0)x口，‘上实解析，且对任意石任O。，有尸任只(:，.e)，
则对给定的参数守。，存在一个Cant，or集O，co(，和充分小的赵。，使得对任意
石任O，，当刀 ){PO 11。(，·，、)又。<、若:。拼含
时，在低维不变环面丁勿)=T“x{刀}又{O}x{O}上，映射(2)的大部分轨道
在扰动映射(l)保持下来.确切地说，对若任口*，存在T”又R”xRmx卿“
的一个Lagl’ange子流形几，(翻和一个微分同胚中Po(创使得
A(，TPo(石))=瓜(石)，
小讨(()。几。小P0({)=又(、oo(()).
其中见(、的(动)是环面丁(功上频率为、OO(动的旋转映射，即
几(、co(石))二(x+、co(石)，对，O，O)
此外，下面的估计成立
11、‘(石)一、。(著).1。，=O(z;合)，
，，leas(口。\口，)=O(守。)，
吉林大学博士学位论文辛映射低维不变环面的保持性工V
其次考虑双曲型低维不变环面的保持性，假设
(祝2)
I+几。二E。(动具有饥个互不相同的特征值，并且每个特征值的模大于1.
此时映射(2)的轨道在法空间中沿?，方向向内收缩，沿。方向向外扩张.我们
称(2)白勺低维不变环面丁(川为双曲型低维不变环面.通过对映射法方向系数
矩阵的特征值进行精确估算，我们证明了:在加上扰动变成近扭转辛映射时，
法方向的量并不会进入小除数.但是，由于法方向有收缩和扩张，所以在迭代
的每一步中估算映射的定义域时?
In this paper we investigate the persistence of the lower dimensional invariant tori for the symplectic mapping with parameters. We proved that for most of the parameters, the lower dimensional elliptic invariant tori and lower dimensional hyperbolic invariant tori of the twist symplectic mappings survive under the symplectic perturbation.
In 60's of the last century, famous mathematicians Kolmogorov, Arnold and Moser established KAM theory. The celebrated theory is the landmark of the research of the Hamiltonian systems. It gives a reasonable explanation to motion rule of the solar system which has puzzled scientists for a long time and shows a lot of understanding of Hamiltonian systems. Before the establishment of KAM theory, it had been believed that the orbit was ergodic on the energy surfaces for almost all Hamiltonian systems. But according to KAM theory, the orbits of the classical In dimensional nearly integrable Hamiltonian system are only ergodic on n dimensional invariant tori, rather than on the energy surfaces (2n- 1 dimensional). So far, KAM theory has become a powerful tool
in studying dynamics properties in nearly integrable Hamiltonian systems and nearly twist mappings and has been applied to many physical problems.
The KAM theory of the symplectic mappings is to descibe the dynamics properties of nearly twist symplectic mappings. The so-called nearly twist symplectic mappings are the symplectic ones which perturbat given twist mappings. In the action-angle coordinates the twist mapping split the phase space into the tori, in which the orbits of the mapping are quasi-periodic. According to the classical KAM theory of the symplectic mappings, most of invariant tori of the twist symplectic mapping survive after the small symplectic perturbation if the twist mapping satisfies some conditions. The KAM theory of symplectic mapping has been applied to the numerical computation. For conservative systems Channel and Scovel, Feng Kang, Sanz-Serna and Calvo, Shang Zaijiu etc. proposted and applied it to the symplectic logarithm of the integrable Hamiltonian systems. So the research of KAM theory of symplectic mappings are important and pratical in applications. The previous results for the nearly twist mapping were mainly focused on the existence of the highest dimensional invariant tori. In this paper, we shall investigat the existence of the lower dimensional invarint tori of nearly twist symplectic mapping and shall prove several KAM type theorems.
Consider
where is bounded closed set in R. In case of unperturbed system, i.e.
P0= 0, the mapping (1) becomes to the following twist mapping
Obviously,this mapping has lower dimensional invariant tori
The types of lower dimensional tori may be different when have different forms. Corresponding to these,there are different dynamics. In order to investigate the persistence of somewhat invariant tori, we need different more explicit analysis and different methods.
Firstly, we investigate the persistence of lower dimensional elliptic type invariant tori. Assume
Then the orbits of the mapping (2) rotate around the original point, neither shrinking to the interior nor extending to the exterior. In this case, we call the invariant tori T{y) of (2) the lower dimensional elliptic invariant tori. For the nearly twist symplectic (1), the normal frequency will become a factor that causes the small divisor. Therefore we must consider tangent frequency as well as the normal frequency using KAM iteration. Now the number of variables become much greater than that in the case of highest dimensional invariant tori because of the appearance of normal variables. By carrying out a series of tendons computing, we proved that most of the invariant tori survive under
the small syinplectic perturbation if the twist mappings satisfy some condition. Moreover, the orbits of the mappings on these invariant tori are qusi-periodic, whose frequencies are close to the original ones. We can state this result in the following theorem,
Theorem 1 Assume w0= id and the normal fre
上世纪六十年代，著名数学家Kolmogorov，Arnold和Moser建立了KAM理论，该理论的建立具有划时代的意义，它给出了太阳系运行机制的一个合理解释，使得这一困扰人们很长时间的问题得到解决，也使人们对Hamilton系统有了新的认识，在KAM理论建立之前，人们一直认为几乎所有的Hamilton系统的轨道在其能量面上是遍历的，然而由KAM定理，典型的2n维近可积Hamilton系统的大部分轨道只在n维不变环面上遍历，并不在能量面(2n-1维)上遍历，同时，KAM理论的建立为人们研究近可积Hamilton系统和近扭转映射的动力学行为提供了一套系统的方法，并且在许多实际问题中得到了应用。
辛映射的KAM理论研究的是近扭转辛映射的动力学行为，所谓近扭转辛映射是指扭转映射加上小的扰动所得到的辛映射，在作用-角坐标下，扭转映射将相空间分成一个个不变环面，映射产生的轨道在不变环面上呈现拟周期运动，其频率互不相同，辛映射的KAM理论告诉我们：如果扭转项满足一定的条
吉林大学博士学位论文辛侠射低维不变环面的保持性H
件，在小扰动下近扭转辛映射的绝大部分不变环面将保持下来.辛映射的KAM
理论在数值计算已经中得到广泛的应用.CI，annel和Scovel，冯康，Sanz一Serna
和Calvo，尚在久等将它应用到可积Halnil七on系统的辛算法中，证明了在该算
法下不变环面的存在性和算法的收敛性.因此，对辛映射KAM理论的研究具
有重要的理论意义和实际应用价值.
以往研究近扭转辛映射得到的都是最高维数的不变环面的存在性.本文中，
我们对近扭转辛映射低维不变环面的存在性进行了研究，得到了几个KAM型
结果.
考虑近扭转辛映射且:T几xR”又R“‘义R”，x口。弓Tr‘又R“X Rm xR”‘
示二.:+、0(荟)十
且:刀=刀_
:，分，，‘，公，苟)，
句动动
了‘..、/‘.吸、、厂产r.、
丛即丛。幽。迅。
=U十
人2。(石)、‘+
92〔)(石)公一
，万，了不，?少
军，?必，，l)
(1)
一
月t
一一
入，t)
其中O。是Rl()中的有界闭集.当P(，三。时，映射(l)为如下扭转映射
示=T+、〔，(石)，
口二岁，
方=(I+公2。(石))，，，
公=(I+92。(石))一‘，:
它具有低维不变环面
，T(岁)=Tx{万}X{()}X{O}
(3)
当兑(，的形式不同时，这些低维不变环面的类型不同，在它们附近的动力学行
为也不同，研究在扰动下不变环面的保持性所需的条件和所用的方法也不同.
首先考虑椭圆型低维不变环面的保持性.假设
‘2()=(liag{犷2。、、…，‘2。，了:}，1+几。，=〔·2万门入。2〔￡).(4)
//t
」吉林大学博士学位论文.辛映射低维不变环面的保持性IH
则映射(2)的轨道在法空间中围绕原点旋转，既不向内收缩，也不向外扩张，
这时我们称(2)的低维不变环面，T(功为椭圆型低维不变环面.我们对近扭转
辛映射，进行KAM迭代时法向频率也将是产生小除数的因素，因此在迭代中
不光要考虑切向频率，还要考虑法向频率.所以迭代步骤中需要更加精细的计
算.由于多出了法向变量，使得我们要估算的量成倍增长.经过细致的计算，
我们证明了:当扭转辛映射满足某种条件时，加上小的扰动后，大部分不变环
面保持下来.另外，在这些不变环面上，映射的轨道呈现拟周期运动，频率与
原频率相差很小.表达成定理就是本文的第一个主要结果:
定理1考虑映射(l)，其中、。二记，法向频率具(封的形式，且满足条件
}挚}_:，，、粤，1:、:。，::、:。
!，a勺”口任
(5)
并且几在复邻域D(:。，s0)x口，‘上实解析，且对任意石任O。，有尸任只(:，.e)，
则对给定的参数守。，存在一个Cant，or集O，co(，和充分小的赵。，使得对任意
石任O，，当刀
时，在低维不变环面丁勿)=T“x{刀}又{O}x{O}上，映射(2)的大部分轨道
在扰动映射(l)保持下来.确切地说，对若任口*，存在T”又R”xRmx卿“
的一个Lagl’ange子流形几，(翻和一个微分同胚中Po(创使得
A(，TPo(石))=瓜(石)，
小讨(()。几。小P0({)=又(、oo(()).
其中见(、的(动)是环面丁(功上频率为、OO(动的旋转映射，即
几(、co(石))二(x+、co(石)，对，O，O)
此外，下面的估计成立
11、‘(石)一、。(著).1。，=O(z;合)，
，，leas(口。\口，)=O(守。)，
吉林大学博士学位论文辛映射低维不变环面的保持性工V
其次考虑双曲型低维不变环面的保持性，假设
(祝2)
I+几。二E。(动具有饥个互不相同的特征值，并且每个特征值的模大于1.
此时映射(2)的轨道在法空间中沿?，方向向内收缩，沿。方向向外扩张.我们
称(2)白勺低维不变环面丁(川为双曲型低维不变环面.通过对映射法方向系数
矩阵的特征值进行精确估算，我们证明了:在加上扰动变成近扭转辛映射时，
法方向的量并不会进入小除数.但是，由于法方向有收缩和扩张，所以在迭代
的每一步中估算映射的定义域时?
In this paper we investigate the persistence of the lower dimensional invariant tori for the symplectic mapping with parameters. We proved that for most of the parameters, the lower dimensional elliptic invariant tori and lower dimensional hyperbolic invariant tori of the twist symplectic mappings survive under the symplectic perturbation.
In 60's of the last century, famous mathematicians Kolmogorov, Arnold and Moser established KAM theory. The celebrated theory is the landmark of the research of the Hamiltonian systems. It gives a reasonable explanation to motion rule of the solar system which has puzzled scientists for a long time and shows a lot of understanding of Hamiltonian systems. Before the establishment of KAM theory, it had been believed that the orbit was ergodic on the energy surfaces for almost all Hamiltonian systems. But according to KAM theory, the orbits of the classical In dimensional nearly integrable Hamiltonian system are only ergodic on n dimensional invariant tori, rather than on the energy surfaces (2n- 1 dimensional). So far, KAM theory has become a powerful tool
in studying dynamics properties in nearly integrable Hamiltonian systems and nearly twist mappings and has been applied to many physical problems.
The KAM theory of the symplectic mappings is to descibe the dynamics properties of nearly twist symplectic mappings. The so-called nearly twist symplectic mappings are the symplectic ones which perturbat given twist mappings. In the action-angle coordinates the twist mapping split the phase space into the tori, in which the orbits of the mapping are quasi-periodic. According to the classical KAM theory of the symplectic mappings, most of invariant tori of the twist symplectic mapping survive after the small symplectic perturbation if the twist mapping satisfies some conditions. The KAM theory of symplectic mapping has been applied to the numerical computation. For conservative systems Channel and Scovel, Feng Kang, Sanz-Serna and Calvo, Shang Zaijiu etc. proposted and applied it to the symplectic logarithm of the integrable Hamiltonian systems. So the research of KAM theory of symplectic mappings are important and pratical in applications. The previous results for the nearly twist mapping were mainly focused on the existence of the highest dimensional invariant tori. In this paper, we shall investigat the existence of the lower dimensional invarint tori of nearly twist symplectic mapping and shall prove several KAM type theorems.
Consider
where is bounded closed set in R. In case of unperturbed system, i.e.
P0= 0, the mapping (1) becomes to the following twist mapping
Obviously,this mapping has lower dimensional invariant tori
The types of lower dimensional tori may be different when have different forms. Corresponding to these,there are different dynamics. In order to investigate the persistence of somewhat invariant tori, we need different more explicit analysis and different methods.
Firstly, we investigate the persistence of lower dimensional elliptic type invariant tori. Assume
Then the orbits of the mapping (2) rotate around the original point, neither shrinking to the interior nor extending to the exterior. In this case, we call the invariant tori T{y) of (2) the lower dimensional elliptic invariant tori. For the nearly twist symplectic (1), the normal frequency will become a factor that causes the small divisor. Therefore we must consider tangent frequency as well as the normal frequency using KAM iteration. Now the number of variables become much greater than that in the case of highest dimensional invariant tori because of the appearance of normal variables. By carrying out a series of tendons computing, we proved that most of the invariant tori survive under
the small syinplectic perturbation if the twist mappings satisfy some condition. Moreover, the orbits of the mappings on these invariant tori are qusi-periodic, whose frequencies are close to the original ones. We can state this result in the following theorem,
Theorem 1 Assume w0= id and the normal fre
引文
[1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., Ben-janmin, 1978
[2] Arnold, V. I., Mathematical methods of classical mechanics, Springer-Verlag, New York, 1978.
[3] V. I. Arnold, Proof of A. N. Kolmogorov's theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian, Russ. Math. surv. 18 (1963), 9-36
[4] V. I. Arnold, Sur la topologie des ecoulements stationaries des fluides parfaits, C. R. Acad. Sci. Paris 261(1965), 17-20
[5] V. I. Arnold, Instability of dynamical systems with several degree of freedom, Sov. Mat. Dokl., 5(1966), 581-585
[6] V. I. Arnold, A. Avez, Ergodic Problem of Classical Mechanics, New York 1968
[7] G. Benettin, L. Galgani and A. Giorgilli, A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems, Celestial Mech. 37(1985), 1-25
[8] G. Benettin, L. Galgani, A. Giorgilli, J. M. Strelsyn, A proof of Kolmogorov's theorem on invariant tori using canonical transformations defined by the Lie method, Nuovo Cimento B, 79(1984), 201-223
[9] G. Benettin and G. Gallavotti, Stability of motion near resonances in quasi-integrable Hamiltonian systems, J. Sta. Phys. 44(1986), 293-338
[10] O. I. Bogoyavlenskij, Theory of tensor invariants of integrable Hamiltonian systems:Ⅰ. Incompatible Poisson structures, Comm. Math. Phys.180(1996),529-586:Ⅱ. Theorem on symmetries and its applications, Comm. Math. Phys. 184(1997), 301-365
[11] O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math,. Phys. 186(1998), 19-51
[12] J. Bourgain, Construction of quasi-periodic solution for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int.
Math. Res. Not. 11 (1994), 475-497.
[13] H. Broer, G. Huitema and M. B. Sevryuk, Quasi-periodic motions in families of dynamical systems, Lect. Notes. Math. 1645, Springer-Verlag, 1996
[14] H. Broer, G. Huitema and F. Takens, Unfoldings of quasi-periodic tori, Mem. Amer. Math. soc. 83(1990), 13-42
[15] H. Broer, G. Huitema and M. B. Sevryuk, Families of quasi-periodic motions in dynamical systems depending on parameters, Nonlinear Dynamical Systems and Chaos (Proc dyn. syst. conf.. H. W. Broer et. al.. eds.), Birkh(?)user, Basel, 1996, 172-211
[16] J. R. Cary and R. G. Littlejohn, Hamiltonian mechanics and its application to magnetic field line flow, Ann. Phys. 151(1982), 1-34
[17] P.J. Channell, C. Scovel, Symplectic integration of Hamiltonian systems. Nonlinearity, 3 (1990), 231-259
[18] C.-Q. Cheng, Birkhoff-Arnold-Moser tori in convex Hamiltonian systems, Commun. Math. Phys. 177(1996),529-599.
[19] C. Q. Cheng and Y. S. Sun, Existence of invariant tori in three dimensional measure-preserving mappings. Celestial Mech. Dyn. Astronom. 47(1990),275-292
[20] C. Q. Cheng and Y. S. Sun, Existence of KAM tori in degenerate Hamil-tonian systems, J. Diff. Eqs. 114(1994), 288-335
[21] L. Chierchia and G. Gallavotti, Drift, and diffusion in phase space, Ann. Inst. H. Poincar(?) Phys. Thcor. 69(1994), 1-144.
[22] S. N. Chow, Y. Li and Y. F. Yi, Persistence of invariant tori on submanifolds in Hamiltonian systems, J. Nonl. Sci. 12(2002), 585-617
[23] F. Z. Cong, Y. Li and M. Y. Huang, Invariant tori for nearly twist mappings with intersection property, Northeast, Math. J. 12(1996), 280-298
[24] F. Z. Cong, Y. Li, Existence of higher dimensional invariant tori for Hamil-tonian systems, J. Math, Anal. Appl. 222(1998), 255-267
[25] J. J. Duistermaat., On global action-angle coordinates, CMAP 33(1980),687-706
[26] L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola. Norm. Sup. Pica cl. sci. Ser. Ⅳ 15(1988), 115-147
[27] L. H. Eliasson, Absolutely convergent series expansions for quasi-periodic motions, Math. Phy. Elect. J. 21(1996), 1-33.
[28] Feng Kang, On difference schemes and symplectic geometry, Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations-Computation of Partial Differential Equations, Ed. Feng Kang, Science Press, Beijing, 1985, 42-58
[29] Feng Kang, The Hamiltonian way for computing Hamiltonian dynamics, Venice 1989, The State of the Art of Applied and Industrial Mathematics, Ed.R.Spigler, Kluwer, Dordrecht, 1990
[30] Feng Kang, Wu Hua-mo, Qin Meng-zhao, Wang Dao-liu, Construction of canonical difference schemes for Hamiltonian formalism via generating functions.J.Comput. Math., 7:1(1989), 71-96
[31] E. Fontich and P. Martin, Arnold diffusion in perturbations of analytic integrable Hamiltonian systems, Discrete Contin Dynam. Systems, 7(2001), 61-84
[32] G. Gallavotti, Twistless KAM tori, Comm. Math. Phy. 164(1994), 145-156.
[33] G. Gallavotti, G. Gentile and V. Mastropietro, Field theory and KAM tori, Math. Phy. Elect.J. 1(1995), 1-13.
[34] G. Gallavotti and L.Galgani, Formal integrals for an autonomous Hamiltonian system near an equilibrium point, Celest. Mech. 17(1978), 267-280.
[35] G. Gallavotti and L.Galgani, Rigorous estimates for the series expansions of Hamiltonian perturbation theory, Celest. Mech. 37(1985), 95-112.
[36] S. M. Graff, On the continuation of hyperbolic invariant tori for Hamiltonian systems, J. Diff. Eq. 15(1974),1-60.
[37] M. R. Herman, Topological stability of the Hamiltonian and Volume-preserving dynamical systems, Lecture at the International Conference on Dynamical Systems, Evanston, Illinois, 1991
[38] M. R. Herman, Sur les coubes invariantes par les diff'emorghisma de
I'anneau, Astersque 103-104(1983).
[39] M. R. Herman, Sur les coubes invariantes par les diff'emorghisma de l'anneau, Astersque 144(1986).
[40] M. R. Herman, Exemples de flots Hamiltonian, C. R. Acad. Sci. Paris S(?)r. I Math. 312(1991)
[41] M. R. Herman, Talk at the international conference on dynamical systems (Lyons,1990).
[42] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkh(?)user Verlag, Basel-Boston-Berlin, 1994
[43] D. D. Holm and K. B. Wolf, Lie-Poisson description of Hamiltonian ray optics, Phys. D, 51(1991), 189-199
[44] T. Kappeler and M. Makarov, On action-angle variables for the second Poisson bracket, of KDV, Comm. Math. Phys. 214(2000), 651-677
[45] A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbations of the Hamiltonian, Dokl. Akad. Nauk. USSR, 98(1954), 525-530.
[46] S. B. Kuksin, Nearly integrable infinite dimensional Hamiltonian systems. Lect. Notes in Math. 1556, Springer, Berlin, 1993
[47] A. J. Laud and K. Meyer, Canonical forms for symplectic and Hamiltonian matrices, Celestial Mech. 9(1974), No. 2, 213-238.
[48] Y. Li and Y. Yi, A quasi-periodic Poincar(?)'s theorem, Math. Ann., 326(2003) ,649-690
[49] Y. Li and Y. F. Yi, Persistence of invariant tori in generalized Hamiltonian systems, Erg. Th. Dyn. Sys, 22(2002), 1233-1261
[50] R. de la. Llave, Recent progress in classical mechanics, Mathematical Physics, X (Leipzig. 1991), 3-19, Springer, Berlin, 1992
[51] R. S. Mac Kay, A renormalisation approach to invariant circles in area-preservation maps, Physica, 7D (1983), 283-300
[52] R. S. Mac. Kay, Transport in 3D volume preserving flows, J. Nonlinear Science., 4(1994),329-354
[53] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, TAM 17, Springer-Verlag, 1994
[54] V. K. Melnikov, On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function, Sov. Math. Dokl. 6 (1965) 1592-1596
[55] I. Mezi(?). and S. Wiggis, On the integrability and perturbation of three-dimensional fluid flows with symmetry, J. Nonl. Sci. 4(1994), 157-194
[56] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. G(?)ttingen Math.-Phys, KⅠ. Ⅱ(1962), 1-20
[57] J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136-176
[58] J. Moser, Old and new applications of KAM theory, Hamiltonian systems with three or more degree of freedom (S'Agar(?), 1995), 184-192, NATO Adv. Sci. Inst, Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, 1999
[59] P..J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986
[60] [O]J. M. Ortega,, W. C. Rheinboldz, Iterative Soltion of Nonlinear Equations in Several Variables, Academic Press, New York, 1970
[61] I. O. Parasyuk, On preservation of multidimensional invariant tori of Hamiltonian systems. Ukrain Mat. Zh. 36(1984), 467-473
[62] H. Poincar(?), Les m(?)thodes nouvelles de la m(?)canique c(?)leste, Vol.Ⅰ, Vol.Ⅱ, Vol.Ⅲ, Gauthier-Villars, 1899.
[63] J. P(?)schel, On ellipitic lower dimensional tori in Hamiltonian systems, Math. Z. 202(1989), 559-608
[64] J. P(?)schel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35(1982), 653-696
[65] C. C. Pugh and C. Robinson, The C~1closing lemma including Hamiltonians, Erg. Th. Dyn. Syst. 3(1983), 261-314
[66] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2nd ed., CRC, Boca, Raton-London-New York-Washington,D. C.,
1998
[67] H. R(?)ssmann, On twist Hamiltonians, Talk on the Colloque International: M(?)canique c(?)leste et systemes hamiltoniens, Marseille, 1990
[68] H. R(?)ssmann, On the exisitence of invariant curves of twist mappings of an anulus, Geometrc Dynamics, Proceedings, Rio de Janeiro 1981, Ed. J. Palis, Lecture Note in Math. 1007, Springer-Verlag, 677-718
[69] H. R(?)ssmann, Non-degeneracy in the perturbation theory of integrable dynamical systems, Number Theory and Dynamical Systems (M. M. Dodson and J. A. G. Vickers, eds), London Math. Lecture Note Serics, Vol. 134, Cambridge University Press, Cambridge, 1989, 5-18.
[70] H. R(?)ssmann, Non-degeneracy in the perturbation theory of integrable dynamical systems, Stochastics, Algebra and Analysis in Classical and Quantum Dynamics and its Applications (S. Albeverio, Ph. Blanchard, and D.Testard, eds.), Vol. 59, Kluwer Academic, Dordecht, 1990, 211-223.
[71] J. M. Sanz-Serna, Solving numerically Hamiltonian systems. Proceedings of the International Congress of Mathematicians, Z(?)rich 1994,1468-1472, Birkh(?)user-verlag, Besel
[72] J. M. Sanz-Serna, Numerical Hamiltonian Problems, Chapman and Hall, London
[73] M. B. Sevryuk, KAM-stable Hamiltonians, J. Dyn. Control Syst. 1(1995),351-366
[74] M. B. Scvryuk, Some problem of the KAM-theory: conditionally periodic motions in tyical systems, Russian Math. Surveys, 50(1995), 341-553
[75] M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Choas 5(1995), No.3, 552-565.
[76] Z. J. Shang, KAM theorem for symplectic mappings with the relelant estimates, Computing Center of Chinese Academy of Scinces, Ph. D. Thesis, Beijing, China, 1992
[77] Z.J. Shang, KAM theorem for mappings J. Dynamics and Diffential Equations 12:4(2000), 357-383
[78] Z.J. Shang, KAM theorem of symplectic algorithms for Hamiltonian sys-
tems, Numer. Math., 83(1999), 477-496
[79] Z. J. Shang, Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems Nonlinearity 13(2000)299-308.
[80] N. V. Svanidze, Small perturbations of an integrable dynamical syatem with an integral invariant, Proc. Stelov Inst. Math., 147(1980), 127-151
[81] D. V. Treshchev, A mechanism for the destruction of resonant tori in Hamiltonian systems, Mat. Sb. 180(1989), No. 10, 1325-1346; English trans., Math. USSR-Sb. 68(1991), No.1, 181-203.
[82] Z. H. Xia, Existence of invariant tori in volume-preserving diffeomor-phisms, Erg. Th. Dyn, Sys. 12(1992), 621-631
[83] Z. H. Xia,, Existence of invariant tori for certain non-sysmpletic diffeomorphisms, Hamiltonian dynamical systems: History, Theory and Applications (IMA Vol. Math. Appl. 63, ed H. S. Dumas, K. R. Meyer and D. S. Schmidt), Springer, New York, 1995, 373-385
[84] J. X. Xn, J. G. You and Q. J. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z. 226(1997), 375-387
[85] J. G. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian system, Commun. Math. Phys., 192(1998),145-168
[86] J-C.Yoccoz, Travaux de Herman sur les tores invariants, Asterique 206(1992), 311-344
[87] 程崇庆，孙义燧，哈密顿系统中的有序与无序运动，上海科技教育出版社，1996
[88] 从福仲，李勇，周钦德，具有退化性的辛映射的KAM定理，数学年刊18A：6(1997)，781-788
[2] Arnold, V. I., Mathematical methods of classical mechanics, Springer-Verlag, New York, 1978.
[3] V. I. Arnold, Proof of A. N. Kolmogorov's theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian, Russ. Math. surv. 18 (1963), 9-36
[4] V. I. Arnold, Sur la topologie des ecoulements stationaries des fluides parfaits, C. R. Acad. Sci. Paris 261(1965), 17-20
[5] V. I. Arnold, Instability of dynamical systems with several degree of freedom, Sov. Mat. Dokl., 5(1966), 581-585
[6] V. I. Arnold, A. Avez, Ergodic Problem of Classical Mechanics, New York 1968
[7] G. Benettin, L. Galgani and A. Giorgilli, A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems, Celestial Mech. 37(1985), 1-25
[8] G. Benettin, L. Galgani, A. Giorgilli, J. M. Strelsyn, A proof of Kolmogorov's theorem on invariant tori using canonical transformations defined by the Lie method, Nuovo Cimento B, 79(1984), 201-223
[9] G. Benettin and G. Gallavotti, Stability of motion near resonances in quasi-integrable Hamiltonian systems, J. Sta. Phys. 44(1986), 293-338
[10] O. I. Bogoyavlenskij, Theory of tensor invariants of integrable Hamiltonian systems:Ⅰ. Incompatible Poisson structures, Comm. Math. Phys.180(1996),529-586:Ⅱ. Theorem on symmetries and its applications, Comm. Math. Phys. 184(1997), 301-365
[11] O. I. Bogoyavlenskij, Extended integrability and bi-Hamiltonian systems, Comm. Math,. Phys. 186(1998), 19-51
[12] J. Bourgain, Construction of quasi-periodic solution for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Int.
Math. Res. Not. 11 (1994), 475-497.
[13] H. Broer, G. Huitema and M. B. Sevryuk, Quasi-periodic motions in families of dynamical systems, Lect. Notes. Math. 1645, Springer-Verlag, 1996
[14] H. Broer, G. Huitema and F. Takens, Unfoldings of quasi-periodic tori, Mem. Amer. Math. soc. 83(1990), 13-42
[15] H. Broer, G. Huitema and M. B. Sevryuk, Families of quasi-periodic motions in dynamical systems depending on parameters, Nonlinear Dynamical Systems and Chaos (Proc dyn. syst. conf.. H. W. Broer et. al.. eds.), Birkh(?)user, Basel, 1996, 172-211
[16] J. R. Cary and R. G. Littlejohn, Hamiltonian mechanics and its application to magnetic field line flow, Ann. Phys. 151(1982), 1-34
[17] P.J. Channell, C. Scovel, Symplectic integration of Hamiltonian systems. Nonlinearity, 3 (1990), 231-259
[18] C.-Q. Cheng, Birkhoff-Arnold-Moser tori in convex Hamiltonian systems, Commun. Math. Phys. 177(1996),529-599.
[19] C. Q. Cheng and Y. S. Sun, Existence of invariant tori in three dimensional measure-preserving mappings. Celestial Mech. Dyn. Astronom. 47(1990),275-292
[20] C. Q. Cheng and Y. S. Sun, Existence of KAM tori in degenerate Hamil-tonian systems, J. Diff. Eqs. 114(1994), 288-335
[21] L. Chierchia and G. Gallavotti, Drift, and diffusion in phase space, Ann. Inst. H. Poincar(?) Phys. Thcor. 69(1994), 1-144.
[22] S. N. Chow, Y. Li and Y. F. Yi, Persistence of invariant tori on submanifolds in Hamiltonian systems, J. Nonl. Sci. 12(2002), 585-617
[23] F. Z. Cong, Y. Li and M. Y. Huang, Invariant tori for nearly twist mappings with intersection property, Northeast, Math. J. 12(1996), 280-298
[24] F. Z. Cong, Y. Li, Existence of higher dimensional invariant tori for Hamil-tonian systems, J. Math, Anal. Appl. 222(1998), 255-267
[25] J. J. Duistermaat., On global action-angle coordinates, CMAP 33(1980),687-706
[26] L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola. Norm. Sup. Pica cl. sci. Ser. Ⅳ 15(1988), 115-147
[27] L. H. Eliasson, Absolutely convergent series expansions for quasi-periodic motions, Math. Phy. Elect. J. 21(1996), 1-33.
[28] Feng Kang, On difference schemes and symplectic geometry, Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations-Computation of Partial Differential Equations, Ed. Feng Kang, Science Press, Beijing, 1985, 42-58
[29] Feng Kang, The Hamiltonian way for computing Hamiltonian dynamics, Venice 1989, The State of the Art of Applied and Industrial Mathematics, Ed.R.Spigler, Kluwer, Dordrecht, 1990
[30] Feng Kang, Wu Hua-mo, Qin Meng-zhao, Wang Dao-liu, Construction of canonical difference schemes for Hamiltonian formalism via generating functions.J.Comput. Math., 7:1(1989), 71-96
[31] E. Fontich and P. Martin, Arnold diffusion in perturbations of analytic integrable Hamiltonian systems, Discrete Contin Dynam. Systems, 7(2001), 61-84
[32] G. Gallavotti, Twistless KAM tori, Comm. Math. Phy. 164(1994), 145-156.
[33] G. Gallavotti, G. Gentile and V. Mastropietro, Field theory and KAM tori, Math. Phy. Elect.J. 1(1995), 1-13.
[34] G. Gallavotti and L.Galgani, Formal integrals for an autonomous Hamiltonian system near an equilibrium point, Celest. Mech. 17(1978), 267-280.
[35] G. Gallavotti and L.Galgani, Rigorous estimates for the series expansions of Hamiltonian perturbation theory, Celest. Mech. 37(1985), 95-112.
[36] S. M. Graff, On the continuation of hyperbolic invariant tori for Hamiltonian systems, J. Diff. Eq. 15(1974),1-60.
[37] M. R. Herman, Topological stability of the Hamiltonian and Volume-preserving dynamical systems, Lecture at the International Conference on Dynamical Systems, Evanston, Illinois, 1991
[38] M. R. Herman, Sur les coubes invariantes par les diff'emorghisma de
I'anneau, Astersque 103-104(1983).
[39] M. R. Herman, Sur les coubes invariantes par les diff'emorghisma de l'anneau, Astersque 144(1986).
[40] M. R. Herman, Exemples de flots Hamiltonian, C. R. Acad. Sci. Paris S(?)r. I Math. 312(1991)
[41] M. R. Herman, Talk at the international conference on dynamical systems (Lyons,1990).
[42] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkh(?)user Verlag, Basel-Boston-Berlin, 1994
[43] D. D. Holm and K. B. Wolf, Lie-Poisson description of Hamiltonian ray optics, Phys. D, 51(1991), 189-199
[44] T. Kappeler and M. Makarov, On action-angle variables for the second Poisson bracket, of KDV, Comm. Math. Phys. 214(2000), 651-677
[45] A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbations of the Hamiltonian, Dokl. Akad. Nauk. USSR, 98(1954), 525-530.
[46] S. B. Kuksin, Nearly integrable infinite dimensional Hamiltonian systems. Lect. Notes in Math. 1556, Springer, Berlin, 1993
[47] A. J. Laud and K. Meyer, Canonical forms for symplectic and Hamiltonian matrices, Celestial Mech. 9(1974), No. 2, 213-238.
[48] Y. Li and Y. Yi, A quasi-periodic Poincar(?)'s theorem, Math. Ann., 326(2003) ,649-690
[49] Y. Li and Y. F. Yi, Persistence of invariant tori in generalized Hamiltonian systems, Erg. Th. Dyn. Sys, 22(2002), 1233-1261
[50] R. de la. Llave, Recent progress in classical mechanics, Mathematical Physics, X (Leipzig. 1991), 3-19, Springer, Berlin, 1992
[51] R. S. Mac Kay, A renormalisation approach to invariant circles in area-preservation maps, Physica, 7D (1983), 283-300
[52] R. S. Mac. Kay, Transport in 3D volume preserving flows, J. Nonlinear Science., 4(1994),329-354
[53] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, TAM 17, Springer-Verlag, 1994
[54] V. K. Melnikov, On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function, Sov. Math. Dokl. 6 (1965) 1592-1596
[55] I. Mezi(?). and S. Wiggis, On the integrability and perturbation of three-dimensional fluid flows with symmetry, J. Nonl. Sci. 4(1994), 157-194
[56] J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. G(?)ttingen Math.-Phys, KⅠ. Ⅱ(1962), 1-20
[57] J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136-176
[58] J. Moser, Old and new applications of KAM theory, Hamiltonian systems with three or more degree of freedom (S'Agar(?), 1995), 184-192, NATO Adv. Sci. Inst, Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, 1999
[59] P..J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986
[60] [O]J. M. Ortega,, W. C. Rheinboldz, Iterative Soltion of Nonlinear Equations in Several Variables, Academic Press, New York, 1970
[61] I. O. Parasyuk, On preservation of multidimensional invariant tori of Hamiltonian systems. Ukrain Mat. Zh. 36(1984), 467-473
[62] H. Poincar(?), Les m(?)thodes nouvelles de la m(?)canique c(?)leste, Vol.Ⅰ, Vol.Ⅱ, Vol.Ⅲ, Gauthier-Villars, 1899.
[63] J. P(?)schel, On ellipitic lower dimensional tori in Hamiltonian systems, Math. Z. 202(1989), 559-608
[64] J. P(?)schel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math. 35(1982), 653-696
[65] C. C. Pugh and C. Robinson, The C~1closing lemma including Hamiltonians, Erg. Th. Dyn. Syst. 3(1983), 261-314
[66] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2nd ed., CRC, Boca, Raton-London-New York-Washington,D. C.,
1998
[67] H. R(?)ssmann, On twist Hamiltonians, Talk on the Colloque International: M(?)canique c(?)leste et systemes hamiltoniens, Marseille, 1990
[68] H. R(?)ssmann, On the exisitence of invariant curves of twist mappings of an anulus, Geometrc Dynamics, Proceedings, Rio de Janeiro 1981, Ed. J. Palis, Lecture Note in Math. 1007, Springer-Verlag, 677-718
[69] H. R(?)ssmann, Non-degeneracy in the perturbation theory of integrable dynamical systems, Number Theory and Dynamical Systems (M. M. Dodson and J. A. G. Vickers, eds), London Math. Lecture Note Serics, Vol. 134, Cambridge University Press, Cambridge, 1989, 5-18.
[70] H. R(?)ssmann, Non-degeneracy in the perturbation theory of integrable dynamical systems, Stochastics, Algebra and Analysis in Classical and Quantum Dynamics and its Applications (S. Albeverio, Ph. Blanchard, and D.Testard, eds.), Vol. 59, Kluwer Academic, Dordecht, 1990, 211-223.
[71] J. M. Sanz-Serna, Solving numerically Hamiltonian systems. Proceedings of the International Congress of Mathematicians, Z(?)rich 1994,1468-1472, Birkh(?)user-verlag, Besel
[72] J. M. Sanz-Serna, Numerical Hamiltonian Problems, Chapman and Hall, London
[73] M. B. Sevryuk, KAM-stable Hamiltonians, J. Dyn. Control Syst. 1(1995),351-366
[74] M. B. Scvryuk, Some problem of the KAM-theory: conditionally periodic motions in tyical systems, Russian Math. Surveys, 50(1995), 341-553
[75] M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Choas 5(1995), No.3, 552-565.
[76] Z. J. Shang, KAM theorem for symplectic mappings with the relelant estimates, Computing Center of Chinese Academy of Scinces, Ph. D. Thesis, Beijing, China, 1992
[77] Z.J. Shang, KAM theorem for mappings J. Dynamics and Diffential Equations 12:4(2000), 357-383
[78] Z.J. Shang, KAM theorem of symplectic algorithms for Hamiltonian sys-
tems, Numer. Math., 83(1999), 477-496
[79] Z. J. Shang, Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems Nonlinearity 13(2000)299-308.
[80] N. V. Svanidze, Small perturbations of an integrable dynamical syatem with an integral invariant, Proc. Stelov Inst. Math., 147(1980), 127-151
[81] D. V. Treshchev, A mechanism for the destruction of resonant tori in Hamiltonian systems, Mat. Sb. 180(1989), No. 10, 1325-1346; English trans., Math. USSR-Sb. 68(1991), No.1, 181-203.
[82] Z. H. Xia, Existence of invariant tori in volume-preserving diffeomor-phisms, Erg. Th. Dyn, Sys. 12(1992), 621-631
[83] Z. H. Xia,, Existence of invariant tori for certain non-sysmpletic diffeomorphisms, Hamiltonian dynamical systems: History, Theory and Applications (IMA Vol. Math. Appl. 63, ed H. S. Dumas, K. R. Meyer and D. S. Schmidt), Springer, New York, 1995, 373-385
[84] J. X. Xn, J. G. You and Q. J. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z. 226(1997), 375-387
[85] J. G. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian system, Commun. Math. Phys., 192(1998),145-168
[86] J-C.Yoccoz, Travaux de Herman sur les tores invariants, Asterique 206(1992), 311-344
[87] 程崇庆，孙义燧，哈密顿系统中的有序与无序运动，上海科技教育出版社，1996
[88] 从福仲，李勇，周钦德，具有退化性的辛映射的KAM定理，数学年刊18A：6(1997)，781-788