## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
- Conference Publications
- AIMS Mathematics

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 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.

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.

## Year of publication

## Related Authors

## Related Keywords

[Back to Top]