A new problem of adiabatic invariance related to the rigid body dynamics

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 Enter 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.