Renato Calleja, Alessandra Celletti, Joan Gimeno, Rafael de la Llave
Accurate computations
up to break-down of quasi-periodic attractors in the dissipative
spin-orbit problem
(11011K, PDF)
ABSTRACT. In recent papers [CCGdlL22a, CCGdlL22b], we developed extremely accurate methods to compute quasi-periodic attractors in a model of Celestial Mechanics: the spin-orbit problem with a dissipative tidal torque. This problem 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 mechanism of breakdown of these quasi-periodic attractors.
The result is obtained by implementing a method that uses at the same time numerical and rigorous improvements, in particular, (i) the time-one map of the spin-orbit problem (so that the invariant objects we seek for have less dimensions), (ii) very accurate computations of the time-one map (high order methods with extended precision arithmetic), (iii) very efficient KAM methods for maps (they are quadratically convergent, the step has low storage requirements and low operation count), (iv) the algorithms are backed by a rigorous a-posteriori KAM Theorem, that establishes
that, if the algorithm is successful and produces a small residual, then there is a
true solution nearby, and (v) The algorithms are guaranteed to reach arbitrarily
close to the border of existence, given enough computer resources. Indeed, monitoring
the properties of the solution, we obtain very effective criteria to compute the parameters for the breakdown to happen. We do not know of any other method
which can compute even heuristically this level of accurate and reliable values for
this model.
As a byproduct of the accuracy we maintain till breakdown, we study several scale invariant observables of the tori. In contrast with previously studied simple models, the behavior at breakdown of the spin-orbit problem does not satisfy standard scaling relations. Hence, the breakdown phenomena of the spin-orbit problem, are not described by a hyperbolic fixed point of the renormalization operator. In fact, it seems that the renormalization operator has a more complicated behavior.