# American Institute of Mathematical Sciences

February  2001, 1(1): 1-28. doi: 10.3934/dcdsb.2001.1.1

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

 1 CNRS, Observatoire de la Côte d'Azur at Nice, Italy 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, Italy

Revised  November 2000 Published  January 2001

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.
Citation: Massimiliano Guzzo, Giancarlo Benettin. A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 1-28. doi: 10.3934/dcdsb.2001.1.1
 [1] Jinxin Xue. Continuous averaging proof of the Nekhoroshev theorem. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3827-3855. doi: 10.3934/dcds.2015.35.3827 [2] Paolo Perfetti. A Nekhoroshev theorem for some infinite--dimensional systems. Communications on Pure and Applied Analysis, 2006, 5 (1) : 125-146. doi: 10.3934/cpaa.2006.5.125 [3] Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215 [4] Giancarlo Benettin, Anna Maria Cherubini, Francesco Fassò. Regular and chaotic motions of the fast rotating rigid body: a numerical study. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 521-540. doi: 10.3934/dcdsb.2002.2.521 [5] Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607 [6] Martin Bauer, Markus Eslitzbichler, Markus Grasmair. Landmark-guided elastic shape analysis of human character motions. Inverse Problems and Imaging, 2017, 11 (4) : 601-621. doi: 10.3934/ipi.2017028 [7] Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems and Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023 [8] Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure and Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1 [9] Roman Srzednicki. A theorem on chaotic dynamics and its application to differential delay equations. Conference Publications, 2001, 2001 (Special) : 362-365. doi: 10.3934/proc.2001.2001.362 [10] Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems and Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631 [11] Antonio Giorgilli, Simone Paleari, Tiziano Penati. Local chaotic behaviour in the Fermi-Pasta-Ulam system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 991-1004. doi: 10.3934/dcdsb.2005.5.991 [12] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553 [13] Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565 [14] Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934/cpaa.2020146 [15] Elena Kartashova. Nonlinear resonances of water waves. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607 [16] Darryl D. Holm, Cornelia Vizman. Dual pairs in resonances. Journal of Geometric Mechanics, 2012, 4 (3) : 297-311. doi: 10.3934/jgm.2012.4.297 [17] Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems and Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225 [18] Kazuyuki Yagasaki. Degenerate resonances in forced oscillators. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 423-438. doi: 10.3934/dcdsb.2003.3.423 [19] Erik Carlsson, John Gunnar Carlsson, Shannon Sweitzer. Applying topological data analysis to local search problems. Foundations of Data Science, 2022  doi: 10.3934/fods.2022006 [20] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41

2020 Impact Factor: 1.327