Efficient and accurate KAM tori construction for the dissipative spin-orbit problem using a map reduction

TL;DR

This paper introduces an efficient algorithm for constructing quasi-periodic solutions in the dissipative spin-orbit problem, enabling high-precision KAM tori computation with applications in celestial mechanics.

Contribution

The paper develops a fast, convergent Newton-based algorithm for computing invariant curves in the dissipative spin-orbit problem, utilizing high-order Taylor methods and geometric insights.

Findings

Successfully computed high-precision quasi-periodic solutions.

Demonstrated the efficiency of the $O(N \, \log N)$ algorithm.

Provided comparisons between averaged and non-averaged models.

Abstract

We consider the dissipative spin-orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and "drift". Our goal is to construct quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. With the goal of applying rigorous KAM theory, we compute such quasi-periodic solution with very high precision. To this end, we have developed a very efficient algorithm. The first step is to compute very accurately the return map to a surface of section (using a high order Taylor's method with extended precision). Then, we find an invariant curve for the return map using recent algorithms that take advantage of the geometric features of the problem. This method is based on a rapidly convergent Newton's method which is guaranteed to converge if the initial error is small enough. So, it is very suitable for a continuation algorithm. The resulting algorithm is quite efficient. We only need to deal with a one dimensional function. If this function is discretized in $N$ points, the algorithm requires $O(N \log N) $ operations and $O(N) $ storage. The most costly step (the numerical integration of the equation along a turn) is trivial to parallelize. The main goal of the paper is to present the algorithms, implementation details and several sample results of runs. We also present both a rigorous and a numerical comparison of the results of averaged and not averaged models.