Accurate computations up to break-down of quasi-periodic attractors in the dissipative spin-orbit problem

Abstract

We consider a Celestial Mechanics model: the spin–orbit problem with a dissipative tidal torque, which is a singular perturbation of a conservative system. The goal of this paper is to show that it is possible to maintain the accuracy and reliability of the computation of quasi-periodic attractors for parameter values extremely close to the breakdown and, therefore, it is possible to obtain information on the breakdown mechanism of these quasi-periodic attractors. The method uses at the same time numerical and rigorous improvements to provide (i) a very accurate computation of the time-1 map of the spin–orbit problem (which reduces the dimensionality of the problem); (ii) a very efficient KAM method for maps which computes the attractor and its tangent spaces (by quadratically convergent, low storage requirements, and low operation count); (iii) explicit algorithms backed by a rigorous a posteriori KAM theorem, which establishes that if the algorithm is successful and produces a small residual, then there is a true solution nearby; and (iv) guaranteed algorithms to reach arbitrarily close to the border of existence as long as there are enough computer resources. As a by-product of the accuracy that we maintain till breakdown, we study several scale-invariant observables of the tori used in the renormalization group of infinite-dimensional spaces.