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.
Regular and chaotic motions of the fast rotating rigid body: a numerical study
Giancarlo Benettin Anna Maria Cherubini Francesco Fassò
Discrete & Continuous Dynamical Systems - B 2002, 2(4): 521-540 doi: 10.3934/dcdsb.2002.2.521
We numerically investigate the dynamics of a symmetric rigid body with a fixed point in a small analytic external potential (equivalently, a fast rotating body in a given external field) in the light of previous theoretical investigations based on Nekhoroshev theory. Special attention is posed on "resonant" motions, for which the tip of the unit vector $\mu$ in the direction of the angular momentum vector can wander, for no matter how small $\varepsilon$, on an extended, essentially two-dimensional, region of the unit sphere, a phenomenon called "slow chaos". We produce numerical evidence that slow chaos actually takes place in simple cases, in agreement with the theoretical prediction. Chaos however disappears for motions near proper rotations around the symmetry axis, thus indicating that the theory of these phenomena still needs to be improved. An heuristic explanation is proposed.
keywords: Nekhoroshev theory chaotic dynamics Rigid body symplectic integrators.
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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