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 $Λ$.
We prove a new invariant torus theorem, for $α$-Gevrey smooth Hamiltonian systems, under an arithmetic assumption which we call the $α$-Bruno-Rüssmann condition, and which reduces to the classical Bruno-Rüssmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this …
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…
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 study the following nonlinear Schrödinger equation \begin{eqnarray}\label{maineq0} \textbf{i}u_{t}-u_{xx}+V*u+εf(x)|u|^4u=0,\ x\in\mathbb{T}=\mathbb{R}/2π\mathbb{Z}, \end{eqnarray} where $V*$ is the Fourier multiplier defined by $\widehat{(V* u})_n=V_{n}\widehat{u}_n, V_n\in[-1,1]$ and $f(x)$ is Gevrey smooth. It is shown that for $0\leq|ε|\ll1$, there is some $(V_n)_{n\in\mathbb{Z}}$ such that, the equation admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain \cite{BJFA2005} and Cong-Liu-Shi-Yuan \cit…
It is well known that under generic $C^r$ smooth perturbations, the phenomenon of global instability, known as Arnold diffusion, exists in a priori unstable Hamiltonian systems. In this paper, by using variational methods, we will prove that under generic Gevrey smooth perturbations, Arnold diffusion still exists in the a priori unstable Hamiltonian systems of two and a half degrees of freedom.
In this paper, we give a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency. The construction is based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging. It enables us to deal with non-analytic Hamiltonians, and in this first part we will focus on Gevrey Hamiltonians and derive normal forms with an exponentially small remainder. This extends a result which was known for analytic…
For perturbations of integrable Hamiltonians systems, the Nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time, provided the integrable part satisfies a steepness condition and the system is analytic. This fundamental result has been extended in two distinct directions. The first one is due to Niederman, who showed that under the analyticity assumption, the result holds true for a prevalent class of integrable systems which is much wider than the steep systems. The second one is due to Marco-Sauzin but it is limited to quasi-convex integrable syste…
A Gevrey symplectic normal form of an analytic and more generally Gevrey smooth Hamiltonian near a Lagrangian invariant torus with a Diophantine vector of rotation is obtained. The normal form implies effective stability of the quasi-periodic motion near the torus.
We consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. By choosing the adjustable parameter, one can ensure that the torus persists under perturbation. Such models are common in celestial mechanics. In field theory, the adjustable parameter is called the counterterm and in celestial mechanics, the drift. It is known that there are formal expansions in powers of the perturbation both for the quasi-periodic solution and the counterterm. We prove that the asym…
Introduce several KAM theorems for infinite dimensional Hamiltonian with short range and discuss the relationship between spectra of linearized operator and invariant tori. Especially, introduce a KAM theorem in the paper(Cummun. Math. Phys. 226, 61-100 (2002)) which shows that there are rich KAM tori for a class of Hamiltonian with short range and with linearized operator of pure point spectra. Here are also presented several open problems.
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), …
A wandering domain for a diffeomorphism is an open connected set whose iterates are pairwise disjoint. We endow A^n = T^n x R^n with its usual exact symplectic structure. An integrable diffeomorphism Φ^h, i.e. the time-one map of a Hamiltonian h which depends only on the action variables, has no nonempty wandering domains. The aim of this paper is to estimate the size (measure and Gromov capacity) of wandering domains in the case of an exact symplectic perturbation of Φ^h , in the analytic or Gevrey category. Upper estimates are related to Nekhoroshev theory, lower estimates are related to exa…
We prove an infinite-dimensional KAM theorem for a Hamiltonian system with sublinear growth frequencies at infinity. As an application, we prove the reducibility of the linear fractional Schrödinger equation with quasi-periodic time-dependent forcing.
This paper is a sequel to "Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians", in which we gave a new construction of resonant normal forms with an exponentially small remainder for near-integrable Gevrey Hamiltonians at a quasi-periodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable Hamiltonians, and we derive normal forms with a polynomially small remainder. As applications, we obtain a polynomially large upper bound on the stability time for the evolution of the action variables and a polynomial…
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 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.
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.
We set up a methodology for computer assisted proofs of the existence and the KAM stability of an arbitrary periodic orbit for Hamiltonian systems. We give two examples of application for systems with 2 and 3 degrees of freedom. The first example verifies the existence of tiny elliptic islands inside large chaotic domains for a quartic potential. In the 3-body problem we prove the KAM stability of the well-known figure eight orbit and two selected orbits of the so called family of rotating Eights. Some additional theoretical and numerical information is also given for the dynamics of both exam…
In this paper, we establish an abstract infinite dimensional KAM theorem dealing with normal frequencies in weaker spectral asymptotics Ω_{i}(ξ)=i^d+o(i^{d})+o(i^δ), where $d>0, δ<0$, which can be applied to a large class of Hamiltonian partial differential equations in high dimensions. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for Schrödinger equations, the Klein-Gordon equations with exponential nonlinearity and other equations of any spatial dimension.
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…
In present paper, from the viewpoint of physical intuition we introduce a Hamiltonian system with multiscale rotation, which describes many systems, for example, the forced pendulum with fast rotation, weakly coupled $N$-oscillators with quasiperiodic force and so on. We study the persistence of invariant tori for this Hamiltonian system, and establish some KAM type results including the isoenergetic type. As consequences, we can show that Boltzmann's ergodicity hypothesis is also not true for this Hamiltonian system.
In this article, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper upper bound on the speed of Arnold diffusion which we believe to be optimal.
In this article, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper upper bound on the speed of Arnold diffusion which we believe to be optimal.
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.
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 apply the general normal form theorems in Kolmogorov spaces to three classical cases: deformations of hypersurface singularities, normal forms of vector fields and invariant tori in Hamiltonian systems.
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.
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 $\mathbb{T}^d$.
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 .
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 consider a time independent $C^2$ Hamiltonian, sa\-tisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide.
For a convex, coercive continuous Hamiltonian on a compact closed Riemannian manifold $M$, we construct a unique forward weak KAM solution of \[ H(x, d_x u)=c(H) \] by a vanishing discount approach, where $c(H)$ is the Mañé critical value. We also discuss the dynamical significance of such a special solution.
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 proved a KAM theorem on existence of invariant tori in generalized Hamiltonian systems without action-angle variables. It is a generalization of the result of de la Llave et al. [Llave, 2005] that deals with canonical Hamiltonian system.
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.
In a previous work [Asymptotically quasiperiodic solutions for time-dependent Hamiltonians, arXiv preprint arXiv:2211.06623 (2022)], we consider time-dependent perturbations of a Hamiltonian vector field having an invariant torus supporting quasiperiodic solutions. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of an asymptotic KAM torus. An asymptotic KAM torus is a time-dependent family of embedded tori converging as time tends to infinity to the invariant torus associated with the unperturbed system. Now, it is quite natural to wonder wh…
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 [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…
The Weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton-Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In the present course, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion lecture, Albert Fathi exposes the aspects …
This paper is concerned with the study of Aubry-Mather and weak KAM theories for contact Hamiltonian systems with Hamiltonians $H(x,u,p)$ defined on $T^*M\times\mathbb{R}$, satisfying Tonelli conditions with respect to $p$ and $0<\frac{\partial H}{\partial u}\leqslant λ$ for some $λ>0$, where $M$ is a connected, closed and smooth manifold. First, we show the uniqueness of the backward weak KAM solutions of the corresponding Hamilton-Jacobi equation. Using the unique backward weak KAM solution $u_-$, we prove the existence of the maximal forward weak KAM solution $u_+$. Next, we analyse Aubry s…
We study properties of action-minimizing invariant sets for Tonelli Lagrangian and Hamiltonian systems and weak KAM solutions to the Hamilton-Jacobi equation in terms of Mather's averaging functions. Our principal discovery is that exposed points and extreme points of Mather's alpha function are closely related to disjoint properties and graph properties of the action-minimizing invariant sets, which is also related to $C^0$ integrability of the systems and the existence of smooth weak KAM solutions to the Hamilton-Jacobi equation.
A major result concerning perturbations of integrable Hamiltonian systems is given by Nekhoroshev estimates, which ensures exponential stability of all solutions provided the system is analytic and the integrable Hamiltonian not too degenerate. In the particular but important case where the latter is quasi-convex, these exponential estimates have been generalized by Marco and Sauzin if the Hamiltonian is Gevrey regular, using a method introduced by Lochak in the analytic case. In this paper, using the same approach we will investigate the situation where the Hamiltonian is assumed to be only f…
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.…
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}$.
In this paper, we study the persistence and remaining regularity of KAM invariant torus under sufficiently small perturbations of a Hamiltonian function together with its derivatives, in sense of finite smoothness with modulus of continuity, as a generalization of classical Hölder continuous circumstances. To achieve this goal, we extend the Jackson approximation theorem to the case of modulus of continuity, and establish a corresponding regularity theorem adapting to the new iterative scheme. Via these tools, we establish a KAM theorem with sharp differentiability hypotheses, which asserts th…
Non-autonomous KAM theory for lower dimensional invariant tori (II): Normally parabolic case
Universitat de Barcelona（巴塞罗那大学）；Centre de Mathématiques Laurent Schwartz (CMLS), CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院中央数学实验室）
We proved a parametrized KAM theorem in Hamiltonian system which has differentiable Hamiltonian without action-angle coordinates. It is a generalization of the result of [Llave et al. 2005] that deals with real analytic Hamiltonians.
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…
