Convergence or generic divergence of the Birkhoff normal form

Pérez-Marco, Ricardo

doi:10.4007/annals.2003.157.557

Cited by 52 publications

(41 citation statements)

References 23 publications

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“…See also (1.22). Since we work in the analytic category and do not wish to consider convergence questions for the Birkhoff normal forms and associated canonical transformations (in this connection, see [30]), we truncate the series at some fixed but arbitrarily large order N, and write…”

Section: Construction Of the Global Weightmentioning

confidence: 99%

Diophantine tori and spectral asymptotics for nonselfadjoint operators

Hitrik¹,

Sjöstrand²,

Ngọc³

2007

ajm

176

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We study spectral asymptotics for small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the first case, we assume that the strength ǫ of the perturbation is O(h δ ) for some δ > 0 and is bounded from below by a fixed positive power of h. In the second case, ǫ is assumed to be sufficiently small but independent of h, and we describe the eigenvalues completely in a fixed h-independent domain in the complex spectral plane.1

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Section: Construction Of the Global Weightmentioning

confidence: 99%

Diophantine tori and spectral asymptotics for nonselfadjoint operators

Hitrik¹,

Sjöstrand²,

Ngọc³

2007

ajm

176

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“…Nevertheless, we must point out that this is not always possible and some resonant terms, even of order four, cannot be canceled. It is well known that this sequence of Hamiltonians and canonical transformations produced by the Birkhoff normalization does not converge on any open neighborhood of the equilibrium point [39]. Anyway, this process is used to compute what is known as the Birkhoff normal form of the Hamiltonian, having the minimum number of monomials up to some degree.…”

Section: A Effective Computation Of Nhim and Its Stable And Unstable mentioning

confidence: 99%

Theory and computation of non-RRKM lifetime distributions and rates in chemical systems with three or more degrees of freedom

Gabern

Koon

Marsden

et al. 2005

Physica D: Nonlinear Phenomena

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The computation, starting from basic principles, of chemical reaction rates in realistic systems (with three or more degrees of freedom) has been a longstanding goal of the chemistry community. Our current work, which merges tube dynamics with Monte Carlo methods provides some key theoretical and computational tools for achieving this goal. We use basic tools of dynamical systems theory, merging the ideas of Koon et al. [Chaos 10, 427 (2000)] and De Leon et al. [J. Chem. Phys. 94, 8310 (1991)], particularly the use of invariant manifold tubes that mediate the reaction, into the start of a comprehensive theory of lifetime distributions and rates of chemical reactions and scattering phenomena, even in systems that exhibit non-statistical behavior. Previously, the main problem with the application of tube dynamics has been with the analytical evaluation of volumes in phase spaces of arbitrary dimension. The present work overcomes this hurdle with some new ideas and implements them numerically. Specifically, an efficient algorithm that uses tube dynamics to provide the initial bounding box for a Monte Carlo volume determination is used. The combination of a fine scale method for understanding the phase space structure (invariant manifold theory) with statistical methods for practical computations (Monte Carlo) is the main novel contribution of this paper. The methodology, applied here to a three degree of freedom model problem, is not restricted by dimension, and is useful for higher degree of freedom systems as well.

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“…It is based on near-identity transformations that simplify as much as possible the power expansion up to a given order. This normal form is usually divergent [PM03] and so it is only performed up to a finite order. One of its main properties is that it can be explicitly integrated, so that it provides a careful approximation of the dynamics near the fixed point.…”

Section: Introductionmentioning

confidence: 99%

On the construction of the Kolmogorov normal form for the Trojan asteroids

2005

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In this paper, we focus on the stability of the Trojan asteroids for the planar restricted three-body problem, by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on numerical determination of the frequencies of the asteroid and effective computation of the Kolmogorov normal form for the corresponding torus. This procedure has been applied to the first 34 Trojan asteroids of the IAU Asteroid Catalogue, and it has worked successfully for 23 of them.The construction of this normal form allows computer-assisted proofs of stability. To show this, we have implemented a proof of existence of families of invariant tori close to a given asteroid, for a high order expansion of the Hamiltonian. This proof has been successfully applied to three Trojan asteroids.PACS numbers: 05.10.−a, 45.20.Jj, 45.50.Pk, 95.10.Ce

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Convergence or generic divergence of the Birkhoff normal form

Abstract: We prove that the Birkhoff normal form of hamiltonian flows at a nonresonant singular point with given quadratic part is always convergent or generically divergent. The same result is proved for the normalization mapping and any formal first integral.

Cited by 52 publications

References 23 publications

Diophantine tori and spectral asymptotics for nonselfadjoint operators

Diophantine tori and spectral asymptotics for nonselfadjoint operators

Theory and computation of non-RRKM lifetime distributions and rates in chemical systems with three or more degrees of freedom

On the construction of the Kolmogorov normal form for the Trojan asteroids

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