Quasi-Periodic Frequency Analysis Using Averaging-Extrapolation Methods

Abstract

We present a new approach to the numerical computation of the basic frequencies of a quasi-periodic signal. Although a complete toolkit for frequency analysis is presented, our methodology is better understood as a refinement process for any of the frequencies, provided we have a rough approximation of the frequency that we wish to compute. The cornerstone of this work is a recently developed method for the computation of Diophantine rotation numbers of circle diffeomorphisms, based on suitable averages of the iterates and Richardson extrapolation. This methodology was successfully extended to compute 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 the construction of a new quasi-periodic signal by appropriate averages of phase-shifted iterates of the original signal. This allows us to define a quasi-periodic orbit on the circle in such a way that the target frequency is the rotation frequency 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 (RTBP), and we consider the effect of additional planets using a multicircular model.