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 compute quasi-periodic attractors accurately and reliably for parameter values extremely close to the breakdown. Therefore, it is possible to obtain information on mathematical phenomena at breakdown. The method we use incorporates the same time numerical and rigorous improvements. Among them (i) the formalism is based on studying the time-one map of the spin-orbit problem (which reduces the dimensionality of the problem) and has mathematical advantages; (ii) very accurate integration of the ODE (high order Taylor methods implemented with extended precision) for the map at its jets; (iii) a very efficient KAM method for maps which computes the attractor and its tangent spaces ( quadratically convergent step with low storage requirements, and low operation count); (iv) the algorithms are backed by a rigorous a-posteriori KAM Theorem, which establishes that if the algorithm, produces a very approximate solution of functional equation with reasonable condition numbers. then there is a true solution nearby; and (v) the continuation algorithm is guaranteed to reach arbitrarily close to the border of existence if it is given enough computer resources. As a byproduct 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. In contrast with previously studied simple models, the behavior at breakdown of the spin-orbit problem does not satisfy standard scaling relations which implies that the spin-orbit problem is not described by a hyperbolic fixed point of a renormalization operator.