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| Frequency analysis was introduced by J. Laskar to study secular motions of the planets in the Solar system. A significant refinement of Laskar's methods, based in the simultaneous improvement of the frequencies and the amplitudes of the signal, was given given later by G. Gomez, J.M. Mondelo and C.Simo. On another front, a methodology to compute rotation numbers of invariant curves (and more general objects) has been introduced recently in several papers by A. Luque, J. Villanueva and T.M. Seara. The idea is to extrapolate the rotation number (and related quantities) from suitable averages of the iterates of an orbit.
The goal of this talk is to present a methodology to compute the frequencies of a given quasi-periodic orbit. The construction is
a generalization of the mentioned averaging-extrapolation approach to study rotation numbers and it allows us to compute with high precision the components of the frequency vector. The main advantage over other high precision methods is that we do not require to compute nor to refine the amplitudes of the signal. Moreover, using this methodology we can compute also derivatives
of the frequencies with respect to parameters, so we can use a Newton method to compute and continue numerically quasi-periodic invariant tori.
As an illustration, we will consider quasi-periodic motions close to the point L5 in an elliptic spatial restricted three-body problem. |
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