Luca Biasco, Luigi Chierchia
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.
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It is proved that the KAM tori (thus quasi-periodic solutions) are long time stable for nonlinear Schrodinger equation.
This paper is concerned with the stickiness of invariant tori obtained by KAM technics (so-called KAM tori) for higher dimensional beam equation. We prove that the KAM tori are sticky, i.e. the solutions starting in the $δ$-neighborhood of KAM torus still stay close to the KAM torus for a polynomial long time such as $|t|\leq δ^{-\mathcal{M}}$ with any $\mathcal{M}\geq 0$, by constructing a partial normal form of higher order, which satisfies $p$-tame property, around the KAM torus.
In this paper we develop some new KAM-technique to prove two general KAM theorems for nearly integrable hamiltonian systems without assuming any non-degeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some non-degeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.
It is proved that the KAM tori (thus quasi-periodic solutions) are long time stable for infinite dimensional Hamiltonian systems generated by nonlinear wave equation, by constructing a partial normal form of higher order around the KAM torus and showing $p$-tame property persists under KAM iterative procedure and normal form iterative procedure.
When a Gevrey smooth perturbation is applied to a quasi-convex integrable Hamiltonian, it is known that the KAM invariant tori that survive are sticky, that is, doubly exponentially stable. We show by examples the optimality of this effective stability.
Long time stability for KAM tori of the derivative nonlinear Schrödinger equation
针对周期边界条件下的一类导数非线性薛定谔方程，本文探讨其 KAM 环面的长时间稳定性问题。
In this paper, we establish a KAM-theorem for ordinary differential equations with finitely differentiable vector fields and multiple degeneracies. The theorem can be used to deal with the persistence of quasi-periodic invariant tori in multiple Hopf and zero-multiple Hopf bifurcations, as well as their subordinate bifurcations, of equilibrium points of continuous dynamical systems.
In this paper, it is proved that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., \textbf{229} (2005), no. 1, 62-94.] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions.
From KAM Theory it follows that the measure of phase points which do not lie on Diophantine, Lagrangian, "primary" tori in a nearly--integrable, real--analytic Hamiltonian system is $O(\sqrt{\varepsilon})$, if $\varepsilon$ is the size of the perturbation. In this paper we discuss how the constant in front of $\sqrt{\varepsilon}$ depends on the unperturbed system and in particular on the phase--space domain.
We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangean tori by glueing together local KAM conjugacies with help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly-integrable system and an integrable one. This leads to preservation of geometry, which allows us to define all the nontrivial geometric invariants like monodromy or Chern classes of an integrable system also for near integrable systems.
A generalized Benjamin-Bona-Mahony (gBBM) equation subject to the periodic boundary condition is studied in this paper. Based on a new infinite dimensional Kolomogorov-Arnold-Moser (KAM) theorem with normal frequencies of finite limit-points, it is shown that the gBBM equation admits plenty of time-quasi-periodic solutions with two frequencies of high modes.
We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector $ø_0$ is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that $ø_0$ has rationally independent coordin…
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.
The purpose of this brief note is twofold. First, we summarize in a very concise form the principal information on Whitney smooth families of quasi-periodic invariant tori in various contexts of KAM theory. Our second goal is to attract (via an informal discussion and a simple example) the experts' attention to the peculiarities of the so-called excitation of elliptic normal modes in the reversible context 2.
In this short note, we prove that a quasi-periodic torus, with a non-resonant frequency (that can be Diophantine or Liouville) and which is invariant by a sufficiently regular Hamiltonian flow, is KAM stable provided it is Kolmogorov non-degenerate. When the Hamiltonian is smooth (respectively Gevrey-smooth, respectively real-analytic), the in-variant tori are smooth (respectively Gevrey-smooth, respectively real-analytic). This answers a question raised in a recent work by Eliasson, Fayad and Krikorian ([EFK]). We also take the opportunity to ask other questions concerning the stability of no…
We prove a general theorem on the persistence of Whitney infinitely smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where dim Fix G < (codim T)/2 where Fix G is the fixed point manifold of the reversing involution G and T is the invariant torus in question. Our result is obtained as a corollary of the theorem by H.W.Broer, M.-C.Ciocci, H.Hanssmann, and A.Vanderbauwhede of 2009 concerning quasi-periodic stability of invariant tori with singular "normal" matrices in reversible systems.
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…
A method via the KAM technique is introduced to study the existence of invariant tori and quasiperiodic solutions for impulsive Duffing-type equations with time period 1. Basing on several planar symplectic homeomorphisms and some estimates of impulsive perturbations under each symplectic homeomorphisms, we prove via the Moser's twist theorem the boundedness (Lagrange stability) and the existence of an invariant circle for the equation with area-preserving impulsive terms. And this invariant circle having any rotation number $ω>ω_0$ with some $ω_0>0$, so we obtain also that the solutions start…
Consider an integer $n \geq 2$ and real numbers $τ>n-1$ and $l>2(τ+1)$. Using ideas of Moser, Salamon proved that individual Diophantine tori persist for Hamiltonian systems which are of class $C^l$. Under the stronger assumption that the system is a $C^{l+τ}$ perturbation of an analytic integrable system, P{ö}schel proved the persistence of a set of positive measure of Diophantine tori. We improve the last result by showing it is sufficient for the perturbation to be of class $C^{l}$ and the integrable part to be of class $C^{l+2}$.
We show that an analytic invariant torus $\cT_0$ with Diophantine frequency $ø_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at $\cT_0$ satisfies a Rüssmann transversality condition, the torus $\cT_0$ is accumulated by KAM tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least $d+1$ that is foliated by analytic invariant tori with frequency $ø_0$. For frequency vectors $ø_0$ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), …
In this paper, we investigate the existence of KAM tori for an infinite dimensional Hamiltonian system with finite number of zero normal frequencies. By constructing a constant quantity we show that, for "most" frequencies in the sense of Lebesgue measure, either if the quantity is zero, there is a KAM tori or if the quantity is not zero, there is no KAM tori in some domain. As application, we show that the nonlinear Schrödinger equation with a zero frequency possesses many quasi-periodic solutions.
We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equa…
The reversible context 2 in KAM theory refers to the situation where dim Fix G < (1/2) codim T, here Fix G is the fixed point manifold of the reversing involution G and T is the invariant torus one deals with. Up to now, the persistence of invariant tori in the reversible context 2 has been only explored in the extreme particular case where dim Fix G = 0 [M. B. Sevryuk, Regul. Chaotic Dyn. 16 (2011), no. 1-2, 24-38]. We obtain a KAM-type result for the reversible context 2 in the general situation where the dimension of Fix G is arbitrary. As in the case where dim Fix G = 0, the main technical…
In this paper, we prove the long time stability of KAM tori for the nonlinear Schrödinger equation on the torus with arbitrary dimensions.
Eliasson and Kuksin developed a KAM approach to study the persistence of the invariant tori for nonlinear Schrödinger equation on $\mathbb{T}^{d}$. In this note, we improve Eliasson and Kuksin's KAM theorem by using Kolmogorov's iterative scheme and obtain a local normal form for the transformed Hamiltonian. As a consequence, we are able to derive the time $δ^{-1}$ stability of the obtained KAM tori.
In [3] (Rend. Lincei Mat. Appl. 26 (2015), 1-10; see also arXiv:1503.08145 [math.DS]) the following result has been announced: Theorem. Consider a real-analytic nearly-integrable mechanical system with potential $f$, namely, a Hamiltonian system with real-analytic Hamiltonian $$H(y,x)=\frac12 \sum_{i=1}^n y_i^2 +εf(x)\ ,$$ $(y,x)\in{\mathbb R}^n\times{\mathbb T}^n$ being standard action--angle variables. For "general non-degenerate" potentials $f$'s there exists $ε_0,a>0$ such that, if $0<ε<ε_0$, then the Liouville measure of the complementary of $H$-invariant tori is smaller than $ε|\log ε|^a…
We present a KAM theorem for presymplectic dynamical systems. The theorem has a " a posteriori " format. We show that given a Diophantine frequency $ω$ and a family of presymplectic mappings, if we find an embedded torus which is approximately invariant with rotation $ω$ such that the torus and the family of mappings satisfy some explicit non-degeneracy condition, then we can find an embedded torus and a value of the parameter close to to the original ones so that the torus is invariant under the map associated to the value of the parameter. Furthermore, we show that the dimension of the param…
We study the planetary system of $\upsilon$~Andromedæ, considering the three-body problem formed by the central star and the two largest planets, $\upsilon$~And~\emph{c} and $\upsilon$~And~\emph{d}. We adopt a secular, three-dimensional model and initial conditions within the range of the observed values. The numerical integrations highlight that the system is orbiting around a one-dimensional elliptic torus (i.e., a periodic orbit that is linearly stable). This invariant object is used as a seed for an algorithm based on a sequence of canonical transformations. The algorithm determines the no…
Kolmogorov invariant torus theorem for weakly interacting particles I: Full dimensional tori
本文提出一个抽象KAM定理，用于研究无限多相互作用粒子系统，构建全维KAM tori以应对长程相互作用。
Stability, periodic orbits and KAM tori in the dynamics of the three fixed centers problem
本文研究三固定中心问题中无穷小质点的空间运动，通过Lie-Deprit归一化简化哈密顿量，确定圆形周期轨道与KAM环面的存在条件，揭示系统长期稳定性。
Quasi-analytic properties of the KAM curve
本文证明解析正则性下KAM曲线具有强拟解析性质，能完全刻画底层系统，并揭示共享特征对动力学的影响。
We consider Gevrey perturbations $H$ of a completely integrable Gevrey Hamiltonian $H_0$. Given a Cantor set $Ω_κ$ defined by a Diophantine condition, we find a family of KAM invariant tori of $H$ with frequencies $ω\in Ω_κ$ which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $Λ$ of the invariant tori. This leads to effective stability of the quasiperiodic motion near $Λ$.
Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory
本文扩展 MacKay 的逆 KAM 条件，提出与一维叶状结构横截的不变曲面不存在的充分条件，并通过对双波势粒子与 Beltrami 流的数值实验验证了该方法的有效性。
Constructive QP-Time-dependent KAM Algorithm for Lagrangian Tori
本文提出一种计算准周期哈密顿系统中纤维式拉格朗日环面的构造性QP时间依赖KAM算法，其收敛性已获证明，并通过Tokamak模型和涡度缺陷模型展示了应用。
Non-autonomous KAM theory for lower dimensional invariant tori (II): Normally parabolic case
本文针对受多项式衰减时变扰动的低维各向同性法向抛物不变环面哈密顿系统，证明扩展相空间中不变流形存在性并确定横向动力学渐近行为，结果适用于多类哈密顿系统。
In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs.
The existence of full dimensional KAM tori for infinite dimensional Hamiltonian systems with long range interactions
研究具有长程相互作用的无限维力学系统，在布尔甘型丢番图条件下，证明全维KAM环面存在，其不变环面半径有较慢衰减。
This paper demonstrates sufficient conditions for the existence of a positive measure set of invariant KAM tori in a singly thermostated, 1 degree-of-freedom hamiltonian vector field. This result is applied to 4 important single thermostats in the literature and it is shown that in each case, if the hamiltonian is real-analytic and well-behaved, then the thermostated system always has a positive measure set of invariant KAM tori for sufficiently weak coupling and high temperature. This extends results of Legoll, Luskin & Moeckel.
We investigate the existence of whiskered tori in some dissipative systems, called \sl conformally symplectic \rm systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family $f_μ$ of conformally symplectic maps which depend on a drift parameter $μ$. We fix a Diophantine frequency of the torus and we assume to have a drift $μ_0$ and an embedding of the torus $K_0$, which satisfy approximately the invariance equation $f_{μ_0} \circ K_0 - K_0 \circ T_ω$ (where $T_ω$ denotes the shift by $ω$). We also assume to have a splitting of the tangen…
The multi-scale KAM persistence without a scaling order for Hamiltonian systems
本文针对无序多尺度哈密顿系统，提出了一个易于验证的非退化条件，以解决现有条件过于一般、难以直接应用于天体力学中N体问题稳定性分析的问题。
The Existence of Full-Dimensional KAM tori for one-dimensional nonlinear Klein-Gordon equation
本文针对周期边界条件下一维非线性克莱因-戈登方程，采用Bourgain2005年提出的方法，建立了其具有次指数衰减的满维KAM环面的存在性与线性稳定性，研究了该方程的概周期解问题。
Non-autonomous KAM theory for lower dimensional invariant tori (I): Normally elliptic and hyperbolic cases
本文针对受随时间衰减非自治扰动的低维各向同性法椭圆/双曲不变环面哈密顿系统，证明了扩展相空间不变流形存在性并确定横向动力学渐近行为，结果适用于Hölder、光滑及解析哈密顿系统。
We discuss a Nash-Moser/ KAM algorithm for the construction of invariant tori for {\em tame} vector fields. Similar algorithms have been studied widely both in finite and infinite dimensional contexts: we are particularly interested in the second case where tameness properties of the vector fields become very important. We focus on the formal aspects of the algorithm and particularly on the minimal hypotheses needed for convergence. We discuss various applications where we show how our algorithm allows to reduce to solving only linear forced equations. We remark that our algorithm works at the…
In this paper we present and illustrate a general methodology to apply KAM theory in particular problems, based on an {\em a posteriori} approach. We focus on the existence of real-analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities we require to control the analytic norm of some functions …
What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderely, completely integrable systems, are charactrized by phase trajectories confined to low-dimensional, invariant tori. The KAM theory examines what happens to the tori when an integrable system is subjected to a small perturbation and finds that, for small enough perturbations, most of them survive. The KAM theory is mute on the suject of the disrupted tori, but, for two dimensional systems, Aubry and Mather discovered an astonis…
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…
The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. In this paper we present a Lagrangian perturbation of the geodesic flow on a flat 3-torus. The perturbation is $C^\infty$ small but the flow has a positive measure of trajectories with positive Lyapunov exponent, namely, the flow has positive metric entropy. From this result we g…
In this paper, we prove the existence of full dimensional invariant tori for a non-autonomous, almost-periodically forced nonlinear beam equation with a periodic boundary condition via KAM theory.
Although quantum tunneling between phase space tori occurs, it is suppressed in the semiclassical limit $\hbar\searrow 0$ for the Schrödinger equation of a particle in $\bR^d$ under the influence of a smooth periodic potential. In particular this implies that the distribution of quantum group velocities near energy $E$ converges to the distribution of the classical asymptotic velocities near $E$, up to a term of the order $\cO(1/\sqrt{E})$.
Most Probable KAM Tori in Stochastic Hamiltonian Systems Driven by Multiplicative Noise
本文研究了随机扰动，特别是乘性噪声，对哈密顿系统可积结构的影响，重点在于随机哈密顿动力学中的KAM理论。证明在适当假设下，对于同时受小确定性扰动和乘性噪声影响的可积哈密顿系统，具有迪奥菲antine频率的不变环面在最可能路径的意义下保持存在。此外，当乘性噪声强度足够小时，利用大偏差原理来表征解轨迹偏离这些不变环面的渐近概率，并推导出相应的速率函数。
We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on Nash-Moser scheme and KAM theory applied with a Hamiltonian system governing the radial deformations of the patch. Compared to the scalar case, some technical issues emerge due to the interaction between the interfaces. One of them is related to a new small divisor problem in the second order Melnikov non-resonances condition coming from the transport equations adv…
We study the existence of whiskered tori in a family $f_μ$ of conformally symplectic maps depending on parameters $μ$. Whiskered tori are tori on which the motion is a rotation, but they have as many expanding/contracting directions as allowed by the preservation of the geometric structure. Our main result is formulated in an "a-posteriori" format. We fix $ω$ satisfying Diophantine conditions. We assume that we are given 1) a value of the parameter $μ_0$, 2) an embedding of the torus $K_0$ into the phase space, approximately invariant under $f_{μ_0}$ in the sense that $f_{μ_0} \circ K_0 - K_0 …
We revisit non-autonomous systems depending quasi-periodically in time within the reversible context 2 of KAM theory and obtain Whitney smooth families of invariant tori in such systems via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question.
In this paper we consider the completely resonant beam equation on \T^2 with cubic nonlinearity on a subspace of L^2 (\T^2) which will be explained later. We establish an abstract infinite dimensional KAM theorem and apply it to the completely resonant beam equation. We prove the existence of a class of Whitney smooth small amplitude quasi-periodic solutions corresponding to finite dimensional tori.
The existence of lower dimensional KAM tori is shown for a class of nearly integrable Hamiltonian systems where the second Melnikov's conditions are eliminated. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for nonlinear wave equations, Schrödinger equations and other equations of higher spatial dimension.
Most Probable KAM Tori in Stochastic Hamiltonian Systems
本文研究了随机扰动如何影响哈密顿系统可积结构，并发展了随机哈密顿动力学的KAM理论，特别是在最可能路径的意义下。通过推导随机哈密顿系统在时间依赖噪声系数驱动下的Onsager-Machlup泛函，并识别出系统轨迹的最可能路径，建立了大偏差原理并获得了显式速率函数，定量刻画轨迹偏差，尤其是稀有事件。主要贡献是证明在随机噪声下，原始准周期不变环在最可能路径的意义下保持，从而展示了随机环境中KAM结构的稳定性。此外，证明Onsager-Machlup泛函与大偏差速率函数完全一致，从而提供了准周期运动结构保持性和随机哈密顿系统波动几何的定量描述。总体而言，本文将经典KAM框架扩展到随机环境，并为噪声下复杂动力系统的行为提供了新见解。
We present a new method for engineering the artificial capture of asteroids. Based on theories of the chaos-assisted capture of natural satellites of the giant planets, we show how an unbound asteroid that passes close to a regular region of phase space can be easily moved onto the nearby KAM tori and essentially permanently captured with the Earth's Hill sphere without closing the zero velocity curves. The method has the advantages of a relatively low delta-v requirement and no need for control strategies. An illustration of the method is given for an example asteroid trajectory, demonstratin…
Linearly Stable KAM Tori for One Dimensional Forced Kirchhoff Equations under Periodic Boundary Conditions
该研究证明了一个抽象无穷维KAM定理，可用于证明周期边界下一维受迫Kirchhoff方程小振幅拟周期解的存在性与线性稳定性，将拟线性扰动下的相关结果从单特征值推广到双特征值情形。
The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has an \emph{a-posteriori} format, i.e., we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). The method of proof is based on an iterative method to solve a functiona…
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 $k = 2n$.
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