首页/论文/1908.02523

arXiv 1908.025232019-08math.DS

V.I. Arnold's "pointwise" KAM Theorem

Luigi Chierchia, Comlan Edmond Koudjinan

备注：: To appear in
发表信息：: Regular and Chaotic Dynamics, Volume 24, Issue 6 of 2019
DOI：: 10.1134/S1560354719060017

英文摘要：

We review V.I. Arnold's 1963 celebrated paper \cite{ARV63} {\sl Proof of A.N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian}, and prove that, optimizing Arnold's scheme, one can get "sharp" asymptotic quantitative conditions (as $\varepsilon\to 0$, $\varepsilon$ being the strength of the perturbation). All constants involved are explicitly computed.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

V.I. Arnold's "Global" KAM Theorem and geometric measure estimates

This paper continues the discussion started in [CK19] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit `global' Arnold's KAM Theorem, which yields, in particular, the Whitney conjugacy of a non-degenerate, real-analytic, nearly-integrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure subset of the phase space. Detailed measure estimates on the Kolmogorov's set are provided in the case the phase space is: (A) a uniform neighborhood of an arbitrary (bounded) set times the d-torus a…

2020-10-25math.DS

On the measure of KAM tori in two degrees of freedom

A conjecture of Arnold, Kozlov and Neishtadt on the exponentially small measure of the non-torus set in analytic systems with two degrees of freedom is discussed.

2019-12-05math.DS

A KAM Theorem for Higher Dimensional Reversible Nonlinear Schrödinger Equations

In the paper, we prove an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional reversible systems. Using this KAM theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of reversible (non-Hamiltonian) coupled nonlinear Schrödinger systems on $d-$torus $\mathbb{T}^d$.

2019-03-16math.DS

Quantitative KAM normal forms and sharp measure estimates

It is widespread since the beginning of KAM Theory that, under "sufficiently small" perturbation, of size $ε$, apart a set of measure $O(\sqrtε)$, all the KAM Tori of a non-degenerate integrable Hamiltonian system persist up to a small deformation. However, no explicit, self-contained proof of this fact exists so far. In the present Thesis, we give a detailed proof of how to get rid of a logarithmic correction (due to a Fourier cut-off) in Arnold's scheme and then use it to prove an explicit and "sharp" Theorem of integrability on Cantor-type set. In particular, we give an explicit proof of th…

2019-04-30math.DS

Orbits of nearly integrable systems accumulating to KAM tori

Consider a sufficiently smooth nearly integrable Hamiltonian system of two and a half degrees of freedom in action-angle coordinates \[ H_ε(φ,I,t)=H_0(I)+εH_1(φ,I,t), φ\in T^2,\ I\in U\subset R^2,\ t\in T=R/Z. \] Kolmogorov-Arnold-Moser Theorem asserts that a set of nearly full measure in phase space consists of three dimensional invariant tori carrying quasiperiodic dynamics. In this paper we prove that for a class of nearly integrable Hamiltonian systems there is an orbit which contains these KAM tori in its $ω$-limit set. This implies that the closure of the orbit has almost full measure in…

2014-12-22math.DS

On linear stability of KAM tori via the Craig-Wayne-Bourgain method

In this paper, we prove the Melnikov's persistency theorem by combining the traditional Kolmogorov-Arnold-Moser (KAM) technique and the Craig-Wayne-Bourgain (CWB) method. The aim of this paper is twofold. One is to establish the linear stability of the perturbed invariant tori by using the CWB method without the second Melnikov condition. The other one is to illustrate the CWB method in detail and make the CWB method more accessible.

2020-03-03math.DS

更多相关论文收起更多论文

Hamiltonian Systems: Stability and Instability Theory

This is a short survey on Nekhoroshev theory, KAM theory, and Arnold's diffusion.

2008-07-10math.DS

二维完全共振可逆 Schrödinger d 系统的拟周期解

Quasi-Periodic solutions of Two Dimensional Completely Resonant Reversible Schrödinger d Systems

本文提出无穷维可逆 Schrödinger 系统的抽象 KAM 定理，并结合偏 Birkhoff 标准形方法，证明二维环面上一类完全共振耦合非线性系统存在拟周期解。

2023-08-15math.DS

Kolmogorov–Arnold–Moser 定理

Kolmogorov-Arnold-Moser Theorem

本文沿 Kolmogorov 在 ICM 报告中的原始思路详细给出 KAM 定理的严格证明，以澄清其论证能否严谨化，并区别于 Arnold 后来采用不同方案发表的证明。

2023-04-23math.APmath.DS

A Diophantine duality applied to the KAM and Nekhoroshev theorems

In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a consequence, we develop a new approach to the perturbation theory for quasi-periodic solutions dealing only with periodic approximations and avoiding classical small divisors estimates. We obtain two results of stability, in the spirit of the KAM and Nekhoroshev theorems, in the model case of a perturbation of a constant vector field on the $n$-dimensional torus. Our first result, which is a Nekhoroshev type theorem, is t…

2012-06-20math.DSmath.NT

Quasi-periodic motions on symplectic tori

The KAM (Kolmogorov-Arnold-Moser) theorem guarantees the stability of quasi-periodic invariant tori by perturbation in some Hamiltonian systems. Michel Herman proved a similar result for quasi-periodic motions, with $k$-dimensional involutive manifolds in Hamiltonian systems with $n$ degrees of freedom $n \leq k < 2n $. In this paper, we extend this result to the case of a quasi-periodic motion on symplectic tori $k = 2n$.

2020-05-05math.DS

A strong form of Arnold diffusion for two and a half degrees of freedom

In the present paper we prove a strong form of Arnold diffusion. Let $\mathbb{T}^2$ be the two torus and $B^2$ be the unit ball around the origin in $\mathbb{R}^2$. Fix $ρ>0$. Our main result says that for a "generic" time-periodic perturbation of an integrable system of two degrees of freedom \[ H_0(p)+εH_1(θ,p,t),\quad \ θ\in \mathbb{T}^2,\ p\in B^2,\ t\in \mathbb{T}, \] with a strictly convex $H_0$, there exists a $ρ$-dense orbit $(θ_ε,p_ε,t)(t)$ in $\mathbb{T}^2 \times B^2 \times \mathbb{T}$, namely, a $ρ$-neighborhood of the orbit contains $\mathbb{T}^2 \times B^2 \times \mathbb{T}$. Our …

2018-04-08math.DS

Generic perturbations of linear integrable Hamiltonian systems

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem which does not hold true for all perturbations.…

2015-11-25math.DS

具有Kolmogorov-Arnold网络的全纯神经ODEs用于复杂动力学的可解释发现

Holomorphic Neural ODEs with Kolmogorov-Arnold Networks for Interpretable Discovery of Complex Dynamics

本文提出了一种基于Kolmogorov-Arnold网络的全纯神经ODE框架，用于在复杂动力学系统中发现可解释的 governing equations，通过可微的正则化保持全纯结构，并在多个复杂动力学系统上验证了其有效性。

2026-05-21cs.LGmath.DS

Quasi periodic Hamiltonian Motions, Scale Invariance, Harmonic Oscillators

The work of Kolmogorov, Arnold and Moser appeared just before the renormalization group approach to statistical mechanics was proposed by Wilson: it can be classified as a multiscale approach which also appeared in works on the convergence of Fourier's series, or construction of Euclidean quantum fields, or the scaling analysis of the short scale behaviour of Navier-Stokes fluids to name a few which originated a great variety of further problems. In this review the proof of the KAM theorem will be presented as a classical renormalization problem with the harmonic oscillator as a `trivial' fixe…

2020-07-19math.DS

天体力学中的微扰理论与正则坐标

Perturbation theory and canonical coordinates in celestial mechanics

本文综述天体力学中行星问题相关研究，回顾KAM理论下Arnold提出的基本定理（FT），以及50年后适配坐标的应用推动FT落地后，该问题辛评估领域的进展。

2022-09-17math-phmath.DSmath.MP