Accurate computations up to break-down of quasi-periodic attractors in the dissipative spin-orbit problem
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In recent papers, 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 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. We also study several scale invariant observables of the tori. The behavior at breakdown 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.
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