G. Sardanashvily
Written with respect to an appropriate Poisson structure, a partially integrable Hamiltonian system is viewed as a completely integrable system with parameters. Then, the theorem on quasi-periodic stability in Ref. [1] (the KAM theorem) can be applied to this system.
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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.
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.
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…
In this paper we present a-posteriori KAM results for existence of $d$-dimensional isotropic invariant tori for $n$-DOF Hamiltonian systems with additional $n-d$ independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced c…
A KAM theorem of symplectic algorithms for nearly integrabel Hamiltonian systems
本文针对近可积哈密顿系统，证明了辛算法的一个类KAM定理，并将已有可积系统情形下的结果推广到更一般的近可积情形。
Given a first order dynamical system possessing a commutative algebra of dynamical symmetries, we show that, under certain conditions, there exists a Poisson structure on an open neighbourhood of its regular (not necessarily compact) invariant manifold which makes this dynamical system into a partially integrable Hamiltonian system. This Poisson structure is by no means unique. Bi-Hamiltonian partially integrable systems are described in some detail. As an outcome, we state the conditions of quasi-periodic stability (the KAM theorem) for partially integrable Hamiltonian systems.
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…
We show that ergodic affine abelian discrete actions on the torus, that have a rank-one factor in their linear part, are locally rigid in a KAM sense if and only if the rank one factor is trivial and the action is higher-rank transversally to this factor. Since it has been proved by Damjanovic and Katok that affine actions with higher-rank linear part are locally rigid, our result completes the local rigidity picture for affine actions on the torus.
Motivated by the Lagrange top coupled to an oscillator, we consider the quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal linear stability theory of an invariant torus with a generic (i.e., non-semisimple) normal $1:-1$ resonance. This theory guarantees the persistence of the invariant torus in the Diophantine case and makes possible a further quasi-periodic normal form, necessary for investigation of the non-linear dynamics. As a consequence, we find Cantor families of invariant isotropic tori of all dimensions suggested by the integrable approximation.
An exact semiclassical version of the classical KAM theorem about small perturbations of vector fields on the torus is given. Moreover, a renormalization theorem based on counterterms for some semiclassical systems that are close to being completely integrable is obtained. We apply these results to characterize the sets of semiclassical measures and quantum limits for sequences of $L^2$-eigenfunctions of these systems.
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…
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.
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.
We consider the equation $Δ_x u+u_{yy}+f(u)=0,\ x=(x_1,\dots,x_N)\in\mathbb{R}^N,\ y\in \mathbb{R},$ where $N\geq 2$ and $f$ is a sufficiently smooth function satisfying $f(0)=0$, $f'(0)<0$, and some natural additional conditions. We prove that the equation possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in $x'=(x_1,\dots,x_{N-1})$ and decaying as $|x'|\to\infty$, periodic in $x_N$, and quasiperiodic in $y$. Related theorems for more general equations are included in our analysis as well. Our method is based on center manifold and KAM-type…
For an integrable Hamiltonian with $d\ (d\geq 2)$ degrees of freedom, we show the conditions on perturbations, for which invariant tori can be destructed.
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 $C^{l+2}$.
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.…
We prove that there is an invariant torus with given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for application, we prove that a finite network of Duffing oscillators with periodic exterior forces possesses Lagrangian stability for almost all initial data.
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 $Λ$.
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。
In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-Rüssmann condition, in real-analytic non-degenerate Hamiltonian systems close to integrable. The proof, which uses rational approximations instead of small divisors estimates, is an adaptation to the Hamiltonian setting of the method we introduced in a previous work for perturbations of constant vector fields on the torus.
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.
Lyapunov instability in KAM stable Hamiltonians with two degrees of freedom
本文证明存在任意接近可积KAM非退化解析哈密顿函数的Gevrey光滑哈密顿函数，其具有频率向量满足$\omega_1\omega_2<0$的李雅普诺夫不稳定但KAM稳定的椭圆平衡点，并给出近可积情形下的类似例子。
In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds.
We present an abstract KAM theorem, adapted to space-multidimensional hamiltonian PDEs with smoothing non-linearities. The main novelties of this theorem are that: $\bullet$ the integrable part of the hamiltonian may contain a hyperbolic part and as a consequence the constructed invariant tori may be unstable. $\bullet$ It applies to singular perturbation problem. In this paper we state the KAM-theorem and comment on it, give the main ingredients of the proof, and present three applications of the theorem .
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。
Adiabatic Invariant Actions for Partially Integrable Systems
本文为部分可积哈密顿系统引入作用量积分，在遍历性假设下证明其成为平均动力学的运动常数，并进而成为绝热不变量。
Action-angle coordinates and KAM theory for singular symplectic manifolds
本专著研究$b^m$-Poisson流形上可积系统的分类与摄动问题，证明了其作用-角坐标定理与KAM定理，结合去奇异化技术推广得到折叠辛流形的KAM定理，还提出了带特异超曲面摄动的辛流形新KAM定理，为一般Poisson流形摄动理论研究奠定基础。
Given $l>2ν>2d\geq 4$, we prove the persistence of a Cantor--family of KAM tori of measure $O(\varepsilon^{1/2-ν/l})$ for any non--degenerate nearly integrable Hamiltonian system of class $C^l(\mathscr D\times\mathbb{T}^d)$, where $\mathscr D\subset \mathbb{R}^d$ is a bounded domain, provided that the size $\varepsilon$ of the perturbation is sufficiently small. This extends a result by D. Salamon in \cite{salamon2004kolmogorov} according to which we do have the persistence of a single KAM torus in the same framework. Moreover, it is well--known that, for the persistence of a single torus, the…
The Cohomological Equation for Jointly Integrable Partially Hyperbolic Diffeomorphisms on 3-Manifolds
研究了3-流形上满足特定条件的联合可积部分双曲微分同胚的上同调方程，证明了连续解存在当且仅当周期循环泛函平凡。
Phonon gap analysis for equilibria of perturbed almost-periodic Frenkel-Kontorova models
本文研究了扰动近周期弗伦克尔-康托罗瓦模型的平衡态，发现KAM方法得到的平衡态无声子间隙，而反可积极限方法得到的平衡态有声子间隙。
Singular KAM Theory
针对近可积 Hamiltonian 系统不变环面测度的经典猜想，提出奇异 KAM 理论，证明初级与次级不变环面测度下界，除对数修正外与猜想吻合。
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…
Towards continuity: Universal frequency-preserving KAM persistence and remaining regularity
本文针对有限可微哈密顿系统中单个保频率KAM环面的持续性及剩余正则性展开研究，通过改进Jackson逼近定理、提出抽象正则性定理及渐近分析，得到具有精确可微性假设的KAM定理，明确了不变环面及共轭映射的非Hölder正则性。
We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two-dimensional nonlinear Schrödinger equation $$iu_t-\triangle u +|u|^2u+\frac{\partial{f(x,u,\bar u)}}{\partial{\bar u}}=0, \quad t\in\Bbb R, x\in\Bbb T^2$$ with periodic boundary conditions, where the nonlinearity $\displaystyle f(x,u,\bar u)=\sum_{j,l,j+l\geq6}a_{jl}(x)u^j\bar u^l$, $a_{jl}=a_{lj}$ is a real analytic function in a neighborhood of the origin. We obtain for the equation a Whitney smooth family of small--amplitude quasi--periodic solutions which are partially hyperbolic.
The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for quasi-periodic solutions with Diophantine frequency vector converges. If one studies the Lindstedt series, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where KAM theorem is usually formulated, by exploiting the analogy between Lindstedt series and p…
In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover, in the same spirit as the notion of KAM stable integrable Hamiltonians, we will introduce a notion of effectively stable integrable Hamiltonians, conjecture a characterization of these Hamiltonians and show that our result prove this conjecture in the linear case.
We consider a nearly integrable, non-isochronous, a-priori unstable Hamiltonian system with a (trigonometric polynomial) $O(μ)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $T_d = O((1/ μ) \log (1/ μ))$ by a variational method which does not require the existence of ``transition chains of tori'' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d$ is optimal as a consequence of a general stability result proved via classical perturbation theory.
Anti-integrable limits for generalized Frenkel-Kontorova models on almost-periodic media
该研究针对几乎周期介质中的广义Frenkel-Kontorova模型，通过建立反可积极限理论，证明了在足够大势能下存在具有任意指定旋转数的局部唯一双曲平衡态，适用于短程和长程模型及其多维推广。
KAM Theory for almost-periodic equilibria in one dimensional almost-periodic media
本文针对一维几乎周期介质中的粒子链，提出并证明了两类后验格式的KAM定理，确立了几乎周期平衡态的存在性，无需接近可积假设，并引入逐步增加复杂度的方法辅助数值验证。
Under ceratin conditions, generalized action-angle coordinates can be introduced near non-compact invariant manifolds of completely and partially integrable Hamiltonian systems.
In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighborhood of a given $ I^{0}$ for nearly integrable and analytic Hamiltonian system $ H(I,θ) = H_{0}(I)+ εH_{1}(I,θ)$ of infinite degree of freedom and of short range. That is to say, we will give an extension of the original Kolmogorov theorem to the infinite-dimensional case of short range. The proof is based on the approximation of finite-dimensional Kolmogorov theorem and an improved KAM machinery which works for the normal form depending on initial $ I^{0}$.
Exponential Stability Estimate of Symplectic Integrators for Integrable Hamiltonian Systems
本文证明了近可积辛映射的聂克霍罗夫类型定理，推导出辛算法的指数稳定性，并给出了扰动、动作变量变化和指数时间的界限，为辛算法的非线性稳定性分析提供了新见解。
Let $\mathcal{F}$ and $\mathcal{K}$ be commuting $C^\infty$ diffeomorphisms of the cylinder $\mathbb{T}\times\mathbb{R}$ that are, respectively, close to $\mathcal{F}_0 (x, y)=(x+ω(y), y)$ and $T_α(x, y)=(x+α, y)$, where $ω(y)$ is non-degenerate and $α$ is Diophantine. Using the KAM iterative scheme for the group action we show that $\mathcal{F}$ and $\mathcal{K}$ are simultaneously $C^\infty$-linearizable if $\mathcal{F}$ has the intersection property (including the exact symplectic maps) and $\mathcal{K}$ satisfies a semi-conjugacy condition. We also provide examples showing necessity of the…
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…
In this paper we investigate periodic FPU chains with an even number of particles. We show that near the equilibrium point, any such chain admits a \emph{resonant} Birkhoff normal form of order four which is \emph{completely integrable} - an important fact which helps explain the numerical experiments of Fermi, Pasta, and Ulam. We analyze the moment map of the integrable approximation of an even FPU chain. Unlike in the case of odd FPU chains these integrable systems (generically) exhibit hyperbolic dynamics. As an application we prove that any FPU chain with Dirichlet boundary conditions admi…
We study the motion of a particle in a plane subject to an attractive central force with inverse-square law on one side of a wall at which it is reflected elastically. This model is a special case of a class of systems considered by Boltzmann which was recently shown by Gallavotti and Jauslin to admit a second integral of motion additionally to the energy. By recording the subsequent positions and momenta of the particle as it hits the wall we obtain a three dimensional discrete-time dynamical system. We show that this system has the Poncelet property: if for given generic values of the integr…
A Modified Parameterization Method for Invariant Lagrangian Tori for Partially Integrable Hamiltonian Systems
针对部分可积 Hamiltonian 系统，论文提出修正的参数化拟 Newton 方法并证明后验 KAM 收敛，改进了小性条件且适用于计算机辅助证明。
It is well known that instabilities of nearly integrable Hamiltonian systems occur around resonances. Dynamics near resonances of these systems is well approximated by the associated averaged system, called slow system. Each resonance is defined by a basis (a collection of integer vectors). We introduce a class of resonances whose basis can be divided into two well separated groups and call them dominant. We prove that the associated slow system can be well approximated by a subsystem given by one of the groups, both in the sense of the vector field and weak KAM theory. One of crucial ingredie…
Invariant Curves and the Variational Structure in Tubular Origami Dynamical Systems
本文通过将逆模数视为KAM理论中的摄动参数，分析了管状折纸拼图的动力学，证明了在大模数极限下保守动力学收敛到具有变分结构的可积映射，并利用不变曲线的持久性解释了折叠模式的控制。
We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is doubly exponentially stable in the sense that nearby solutions remain close to the torus for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the torus. We also prove that for an arbitrary small perturbation of a generic integrable Hamiltonian system, there is a set of almost full positive Lebesgue measure of KAM tori which are doubly exponentially stable. Our results hold true for real-analyti…
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