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Journal ArticleDOI

Non-degenerate Liouville tori are KAM stable

09 Apr 2016-Advances in Mathematics (Academic Press)-Vol. 292, pp 42-51
TL;DR: In this article, it was shown that a quasi-periodic torus, with a non-resonant frequency and invariant to a sufficiently regular Hamiltonian flow, is KAM stable provided it is Kolmogorov non-degenerate.
About: This article is published in Advances in Mathematics.The article was published on 2016-04-09 and is currently open access. It has received 9 citations till now. The article focuses on the topics: Kolmogorov–Arnold–Moser theorem & Hamiltonian system.
Citations
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TL;DR: In this article, a new variant of KAM theory based on a slowly converging iteration scheme is proposed for analytic perturbations of constant vector fields on a torus, which is the shortest complete KAM proof for perturbation of integrable vector fields available so far.
Abstract: Recently Russmann proposed a new new variant of KAM theory based on a slowly converging iteration scheme. It is the purpose of this note to make this scheme accessible in an even simpler setting, namely for analytic perturbations of constant vector fields on a torus. As a side effect the result may be the shortest complete KAM proof for perturbations of integrable vector fields available so far.

24 citations

Posted Content
TL;DR: In this article, the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori was shown, and the Birkhoff Normal Form at the invariant Torus can be chosen to be convergent, equal to a planar or non-planar polynomial.
Abstract: We prove the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori. Using various versions of our examples, we solve the following problems in the stability theory of analytic quasi-periodic motion: $\quad i)$ Show the existence of topologically unstable tori of arbitrary frequency. Moreover, the Birkhoff Normal Form at the invariant torus can be chosen to be convergent, equal to a planar or non-planar polynomial. $\quad ii)$ Show the optimality of the exponential stability for Diophantine tori. ' $\quad iii)$ Show the existence of real analytic Hamiltonians that are integrable on half of the phase space, and such that all orbits on the other half accumulate at infinity. $\quad iv)$ For sufficiently Liouville vectors, obtain invariant tori that are not accumulated by a positive measure set of quasi-periodic invariant tori.

8 citations

01 Dec 2012
TL;DR: In this article, the authors gave a new construction of resonant normal forms with an exponentially small remainder for near-integrable Gevrey Hamiltonians at a quasi-periodic frequency.
Abstract: 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 polynomially small upper bound on the splitting of invariant manifolds for hyperbolic tori.

6 citations

Journal ArticleDOI
TL;DR: In this paper, the authors investigated 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 action variables).
Abstract: 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. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is non-resonant is more subtle. Our second result shows that for a generic perturbation, the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long interval of time, and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation, one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).

3 citations

Posted Content
TL;DR: In this article, the authors considered a reversible system with Diophantine frequency and showed that if the Birkhoff normal form around $\Gamma_0$ is 0-degenerate, then it is accumulated by other analytic invariant tori, the Lebesgue measure of the union of these tori being positive.
Abstract: In the present paper, we consider the following reversible system \begin{equation*} \begin{cases} \dot{x}=\omega_0+f(x,y),\\ \dot{y}=g(x,y), \end{cases} \end{equation*} where $x\in\mathbf{T}^{d}$, $y\backsim0\in \mathbf{R}^{d}$, $\omega_0$ is Diophantine, $f(x,y)=O(y)$, $g(x,y)=O(y^2)$ and $f$, $g$ are reversible with respect to the involution G: $(x,y)\mapsto(-x,y)$, that is, $f(-x,y)=f(x,y)$, $g(-x,y)=-g(x,y)$. We study the accumulation of an analytic invariant torus $\Gamma_0$ of the reversible system with Diophantine frequency $\omega_0$ by other invariant tori. We will prove that if the Birkhoff normal form around $\Gamma_0$ is 0-degenerate, then $\Gamma_0$ is accumulated by other analytic invariant tori, the Lebesgue measure of the union of these tori being positive and the density of the union of these tori at $\Gamma_0$ being one. We will also prove that if the Birkhoff normal form around $\Gamma_0$ is $j$-degenerate ($1\leq j\leq d-1$) and condition (1.6) is satisfied, then through $\Gamma_0$ there passes an analytic subvariety of dimension $d+j$ foliated into analytic invariant tori with frequency vector $\omega_0$. If the Birkhoff normal form around $\Gamma_0$ is $d-1$-degenerate, we will prove a stronger result, that is, a full neighborhood of $\Gamma_0$ is foliated into analytic invariant tori with frequency vectors proportional to $\omega_0$.

2 citations


Cites background from "Non-degenerate Liouville tori are K..."

  • ...Bounemoura [4] considered the following Hamiltonian system (1....

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References
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Journal ArticleDOI
TL;DR: In this article, the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems was studied, and it was shown that 1/2nα is the optimal exponent for the time of stability and b = 1 2n as an exponent for radius of confinement of the action variables.
Abstract: . – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.¶Our stability result generalizes those by Lochak-Neishtadt and Poschel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).¶On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n−2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold’s mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system.

119 citations

Journal ArticleDOI
TL;DR: Theorems of Jackson type are given, for the simultaneous approximation of a function of class Cm and its partial derivatives, by a polynomial and the corresponding partial derivatives as mentioned in this paper.
Abstract: Theorems of Jackson type are given, for the simultaneous approximation of a function of class Cm and its partial derivatives, by a polynomial and the corresponding partial derivatives.

89 citations

Journal ArticleDOI
TL;DR: Etant donne un diffeomorphisme symplectique f en dimension 2n, n≥2, et m 0 un point fixe totalement elliptique de f en lequel f satisfait certaines conditions de non-degenerescence as mentioned in this paper.
Abstract: Etant donne un diffeomorphisme symplectique f en dimension 2n, n≥2, et m 0 un point fixe totalement elliptique de f en lequel f satisfait certaines conditions de non-degenerescence, on construit deux diffeomorphismes symplectiques f 0 et g ayant en m 0 meme serie de Taylor que f et tels que: il existe un voisinage de m 0 feuillete en tores de dimension n invariants par f 0 ; en particulier, m 0 est topologiquement stable pour f 0 ; m 0 est un point fixe non topologiquement stable de g

74 citations