Alejandro Luque; Jordi Villanueva
Quasi-periodic frequency analysis using averaging-extrapolation method
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ABSTRACT. We present a new approach to numerically compute the basic frequencies of a quasi-periodic signal. Although a complete toolkit for frequency analysis is presented, this methodology is better understood as a refinement process for any of the frequencies, provided we have a rough
approximation of it. The cornerstone of this work is a recently developed methodology for the computation of Diophantine rotation numbers of circle diffeomorphisms, which is based on suitable averages of the iterates and Richardson extrapolation. This methodology was successfully extended later to deal with rotation numbers of quasi-periodic invariant curves of planar maps. In this paper, we address the case of a signal with an arbitrary number of frequencies. The most outstanding aspect of our approach is that frequencies can be calculated, with high accuracy at a moderate computational cost, without simultaneously computing the Fourier representation of the signal. The method consists in constructing a new quasi-periodic
signal by appropriate averages of phase-shifted iterates of the original one. This allows us to define a quasi-periodic orbit on the circle in such a way that the frequency to be computed leads the rotation of the iterates. This orbit is well-suited for the application of the aforementioned averaging-extrapolation methodology for computing rotation numbers. We illustrate the presented methodology with the study of the vicinity of the Lagrangian equilibrium points of the Restricted Three Body Problem and we consider the effect of additional planets using a multicircular model.