A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis
Massimiliano Guzzo Giancarlo Benettin
Discrete & Continuous Dynamical Systems - B 2001, 1(1): 1-28 doi: 10.3934/dcdsb.2001.1.1
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.
First numerical evidence of global Arnold diffusion in quasi-integrable systems
Claude Froeschlé Massimiliano Guzzo Elena Lega
Discrete & Continuous Dynamical Systems - B 2005, 5(3): 687-698 doi: 10.3934/dcdsb.2005.5.687
We provide numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We show that even if a system is sufficiently close to be integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web, and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems.
keywords: Quasi-integrable systems Arnold diffusion.
A new problem of adiabatic invariance related to the rigid body dynamics
Giancarlo Benettin Massimiliano Guzzo Anatoly Neishtadt
Discrete & Continuous Dynamical Systems - A 2008, 21(3): 959-975 doi: 10.3934/dcds.2008.21.959
We study a new problem of adiabatic invariance, namely a nonlinear oscillator with slowly moving center of oscillation; the frequency of small oscillations vanishes when the center of oscillation passes through the origin (the fast motion is no longer fast), and this can produce nontrivial motions. Similar systems naturally appear in the study of the perturbed Euler rigid body, in the vicinity of proper rotations and in connection with the 1:1 resonance, as models for the normal form. In this paper we provide, on the one hand, a rigorous upper bound on the possible size of chaotic motions; on the other hand we work out, heuristically, a lower bound for the same quantity, and the two bounds do coincide up to a logarithmic correction. We also illustrate the theory by quite accurate numerical results, including, besides the size of the chaotic motions, the behavior of Lyapunov Exponents. As far as the system at hand is a model problem for the rigid body dynamics, our results fill the gap existing in the literature between the theoretically proved stability properties of proper rotations and the numerically observed ones, which in the case of the 1:1 resonance did not completely agree, so indicating a not yet optimal theory.
keywords: rigid body dynamics Adiabatic invariants KAM theorem.

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