KAM quasi-periodic tori for the dissipative spin-orbit problem

TL;DR

This paper numerically demonstrates the existence of KAM quasi-periodic attractors in a dissipative spin-orbit celestial mechanics model, approaching the breakdown threshold, and discusses the potential for rigorous proof.

Contribution

It provides numerical evidence of KAM tori in a dissipative celestial model near breakdown, with detailed condition number analysis and near-critical parameter computations.

Findings

Existence of KAM quasi-periodic attractors close to breakdown threshold.

Numerical methods used to estimate condition numbers near critical parameters.

Results suggest potential for rigorous computer-assisted proofs.

Abstract

We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin-orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity. Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter. The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in [CCGdlL20a]. The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing that standard numerics. We hope that our work could stimulate a computer-assisted proof.