A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis

Massimiliano Guzzo; Giancarlo Benettin

doi:10.3934/dcdsb.2001.1.1

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2001, Volume 1, Issue 1: 1-28. Doi: 10.3934/dcdsb.2001.1.1

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A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis

Massimiliano Guzzo^1, and
Giancarlo Benettin^2,

1.
CNRS, Observatoire de la Côte d'Azur at Nice
2.
Dipartimento di Matematica Pura e Applicata dell'Università di Padova, Gruppo Nazionale di Fisica Matematica and Istituto Nazionale di Fisica della Materia, Via G. Belzoni 7, 35131 Padova

Revised: October 31, 2000

Published: January 2001

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Abstract

In this paper we provide an analytical characterization of the Fourier spectrum of the solutions of quasi-integrable Hamiltonian systems, which completes the Nekhoroshev theorem and looks particularly suitable to describe resonant motions. We also discuss the application of the result to the analysis of numerical and experimental data. The comparison of the rigorous theoretical estimates with numerical results shows a quite good agreement. It turns out that an observation of the spectrum for a relatively short time scale (of order $1/\sqrt{\varepsilon}$, where $\varepsilon$ is a natural perturbative parameter) can provide information on the behavior of the system for the much larger Nekhoroshev times.

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Mathematics Subject Classification: Primary: 37J40, 37M10.

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