Non-degenerate Liouville tori are KAM stable
TLDR
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.read more
Citations
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KAM \`{a} la R
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.
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Instabilities for analytic quasi-periodic invariant tori
Gerard Farré,Bassam Fayad +1 more
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.
Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians
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.
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Generic perturbations of linear integrable Hamiltonian systems
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).
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On invariant tori in some reversible systems
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.
References
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Journal ArticleDOI
KAM à la R
TL;DR: In this paper, the authors proposed a new variant of kam-theory based on a slowly converging iteration scheme for analytic perturbations of constant vector fields on a torus.
Journal ArticleDOI
A Diophantine duality applied to the KAM and Nekhoroshev theorems
TL;DR: In this paper, a new approach to the perturbation theory for quasi-periodic solutions dealing only with periodic approximations and avoiding classical small divisors estimates was developed.
Journal ArticleDOI
Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians
TL;DR: In this paper, a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency was given, 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.
Journal ArticleDOI
Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians
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.
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