Two–layer model via non–quasi–periodic normal form theory

The "two-layer model" is a 2+12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2+\frac{1}{2}$$\end{document} degrees-of-freedom non-autonomous dynamical system consisting of a massive, ellipsoidal (possibly spheric) body made of two layers - a hard core and a viscous fluid - revolving about a major planet or a star. We assume that the rotation and the two revolution periods (of core and shell) are close to a resonance, and aim to investigate, in a rigorous way, the mathematical conditions which maintain the resonant motion. In a previous article (Pinzari et al. in Celest Mech Dyn Astron 136(5):39, 2024), we discussed the phenomenon known as "capture into resonance", via qualitative arguments supported by numerical findings. In this paper, we reframe the model along the lines of a suitable version of (which we refer to as "non-quasi-periodic") normal form theory and provide an explicit amount of the resonance trapping time, which is estimated as exponentially-long, in terms of the small parameters of the system.