Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 1 - 28, Volume 1, Issue 1, February 2001      doi:10.3934/dcdsb.2001.1.1

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Massimiliano Guzzo - CNRS, Observatoire de la CĂ´te d'Azur at Nice, Italy (email)
Giancarlo Benettin - Dipartimento di Matematica Pura e Applicata dell'UniversitĂ  di Padova, Gruppo Nazionale di Fisica Matematica and Istituto Nazionale di Fisica della Materia, Via G. Belzoni 7, 35131 Padova, Italy (email)

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.

Keywords:  Nekhoroshev theorem, Fourier analysis, resonances, local chaotic motions.
Mathematics Subject Classification:  Primary: 37J40, 37M10.

Revised: November 2000;      Available Online: January 2001.