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### Open Access Journals

DCDS-B

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.

DCDS-B

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.

DCDS

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.

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