. – 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.

/pdf/stability-and-instability-for-gevrey-quasi-convex-near-3numsmgzwj.pdf

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

In the 1970s, Nekhorochev proved that, for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Recently, Guzzo has given examples of exponentially stable integrable Hamiltonians that are non-steep but satisfy a weak condition of transversality which involves only the affine subspaces spanned by integer vectors. We generalize this notion for an arbitrary integrable Hamiltonian and prove Nekhorochev's estimates in this setting. The point in this refinement lies in the fact that it allows one to exhibit a generic class of real analytic integrable Hamiltonians which are exponentially stable with fixed exponents. Genericity is proved in the sense of measure since we exhibit a prevalent set of integrable Hamiltonians which satisfy the latter property. This is obtained by an application of a quantitative Sard theorem given by Yomdin.

Prevalence of exponential stability among nearly integrable Hamiltonian systems

In this article, we present a new approach of Nekhoroshev theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak which combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of generic stability around linearly stable tori.

pdf/generic-nekhoroshev-theory-without-small-divisors-10uxx60x7u.pdf

Generic Nekhoroshev theory without small divisors

La notion de raideur a ete introduite pour etudier la stabilite effective des systemes Hamiltoniens quasi-integrables. A l'aide de theoremes de geometrie sous-analytique, on donne une condition geometrique simple qui est equivalente a la raideur pour une fonction reelle analytique.

/pdf/hamiltonian-stability-and-subanalytic-geometry-8784ex6j8z.pdf

Hamiltonian stability and subanalytic geometry

In this paper, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper lower bound on the time of Arnold diffusion which we believe to be optimal.

Improved exponential stability for near-integrable quasi-convex Hamiltonians

In the 1970s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system with a perturbation of size $\\varepsilon$, the actions linked to the unperturbed Hamiltonian vary only by the order of $\\varepsilon^b$ over a time of the order of $\\exp (C\\varepsilon^{-a})$ for some positive constants a, b and C, provided that the unperturbed Hamiltonian meets some generic transversality condition known as steepness. Among steep systems, convex or quasiconvex systems are easier to describe since the use of energy conservation allows the proof of exponential estimates of stability to be shortened. In this case, Lochak–Neishtadt and Poschel have independently obtained the stability exponents a = b = 1/2n for systems of n degrees of freedom—especially the time exponent (a) is expected to be optimal (see P. Lochak, J.-P. Marco and D. Sauzin. Preprint. Institut de Máthematique de Jussieu, 1999; J.-P. Marco and D. Sauzin. Preprint. Publ. Math. Inst. Hautes Etudes Science, 2001). Moreover, Lochak's study relies on simultaneous Diophantine approximation which gives a very transparent proof. However, the proof in the steep case has rarely been studied since Nekhorochev's original work despite various physical examples where the model Hamiltonian is only steep. Here, we combine the original scheme with a simultaneous Diophantine approximation as in Lochak's proof. This yields significant simplifications with respect to Nekhorochev's reasoning: it also allows the exponents $a=b=(2n p_1\\dotsb p_{n-1})^{-1}$ where $(p_1\\dotsb p_{n-1})$ are the steepness indices of the considered Hamiltonian to be obtained. In the quasiconvex case, the steepness indices are all equal to one and we find the same exponents 1/2n as Lochak–Neishtadt and Poschel, whose results are thus generalized to the steep case.

Exponential stability for small pertubations of steep integrable Hamiltonian systems

/pdf/exponential-stability-for-small-perturbations-of-steep-1p3entty34.pdf

Exponential stability for small perturbations of steep integrable Hamiltonian systems – CORRIGENDUM

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Laurent Niederman

Author Tools

Chat about Author

Papers

Prevalence of exponential stability among nearly integrable Hamiltonian systems

Exponential stability for small pertubations of steep integrable Hamiltonian systems

Exponential stability for small perturbations of steep integrable Hamiltonian systems – CORRIGENDUM