Non-existence of KAM torus

Cheng, Chong Qing

doi:10.1007/s10114-011-0631-5

Cited by 15 publications

(6 citation statements)

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“…The search for persistence of KAM tori and reduction problems of finitely differentiable Hamilton's systems goes back to the work of Kolmogorov-Arnold-Moser, and attracts great attention over years. For the Hamilton's equations ẋ = H y (x, y), ẏ = −H x (x, y), (x, y) ∈ R 2n , where H ∈ C ℓ , it was shown by Pöschel [30,34] and Cheng [7] that the hypotheses ℓ > 2n is optimal for the existence of KAM torus. When considering the Hamiltonian (1.5) h ǫ (I, θ, z, z) = h 0 (I, z, z) + ǫq(I, θ, z, z), (I, θ, z, z) ∈ R n × T n × C 2d , where q is C ℓ in θ, Chierchia-Qian [8] proved the persistence of lower dimensional tori with ℓ > 6n + 5 and Sun-Li-Xie [35] considered the reduction problem with ℓ ≥ 200n.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…With better estimates for the solutions of linear partial differential equations, Pöschel improved Moser's result to ℓ > 2n, see [30,34]. In [7], Cheng proved the non-existence of KAM torus if ℓ < 2n, which shows that the assumption ℓ > 2n in [30] is optimal. In this paper, the reducibility of (1.1) and (1.2) is equivalent to the reducibility of a Hamiltonian system with Hamiltonian (1.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Reducibility of the quantum harmonic oscillator in $d$-dimensions with finitely differentiable perturbations

Jian

2019

Preprint

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Reducibility of the quantum harmonic oscillator in $d$-dimensions with finitely differentiable perturbations

Jian

2019

Preprint

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“…定理 1.5 [9,27,55] [76,77] 及 Cong [78] ; 他们成功地建立了广义 Hamilton 系统的 KAM 理论, 且将一些经典的 KAM 定理推广到广义 Hamilton 系统. 定理 1.6 [76] 考虑近可积实解析广义 Hamilton 函数 [29] 得到的, 他把 Moser 的关于保面积映射的 KAM 定理的 C 333 正则条件减弱到 C 5 , 然后 Herman [34,35] [16,25,[84][85][86].…”

Section: 比起椭圆情形双曲情形会简单些设unclassified

KAM theory in finite and infinite dimensional spaces

You¹,

Geng²,

Xu³

2016

Sci. Sin.-Math.

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H(q, p) = h(p), (q, p) ∈ T n × D, 其中 p 称为作用变量, 而 q 称为角变量. 此时容易看到, 相空间被下列所有不变环面所充满: T ω = {T n × {p 0 } | p 0 ∈ D}, 且该不变环面上的拟周期流可以表示为 q(t) = q 0 + ω(p 0)t, p(t) = p 0 , 其中 ω = h p 是频率. 拟周期运动是可积 Hamilton 系统的典型 (typical) 现象, 可积 Hamilton 系统的相空间结构和动力学性质清楚而且简单. 但是, 不是所有的 Hamilton 系统都是可积的. 1892 年, Poincaré [2] 证明了著名的不可积定理, 它说明可积系统在通有的 (generic) 扰动下是不可积的 (也可参见文献 [3]). 例如, 当

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“…Roughly speaking, there is a balance among the arithmetic property of the rotation vector, the regularity of the perturbation and its topology. We mention [Her83,Her86,Mat88,For94] for destruction of invariant circles for twist maps and [Che11] for destruction of KAM torus for Hamiltonian system of multi-degrees of freedom. In particular, [Her83] considered non-existence of invariant circle with arbitrary given rotation number.…”

Section: Comparison With the Literaturesmentioning

confidence: 99%