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**Quasi-periodic frequency analysis using averaging-extrapolation methods.**
*(English)*
Zbl 1338.34088

The authors introduce a novel approach to the numerical computation of the basic frequencies in a quasi-periodic signal. Quoting from the abstract, “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 fact, the case of a signal with an arbitrary number of frequencies is addressed and, returning to the abstract, “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.”

In fact, the case of a signal with an arbitrary number of frequencies is addressed and, returning to the abstract, “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.”

Reviewer: Christian Pötzsche (Klagenfurt)

### MSC:

34C46 | Multifrequency systems of ordinary differential equations |

37E45 | Rotation numbers and vectors |

37M10 | Time series analysis of dynamical systems |

65Txx | Numerical methods in Fourier analysis |

70K43 | Quasi-periodic motions and invariant tori for nonlinear problems in mechanics |

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

37E10 | Dynamical systems involving maps of the circle |

70F05 | Two-body problems |

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\textit{A. Luque} and \textit{J. Villanueva}, SIAM J. Appl. Dyn. Syst. 13, No. 1, 1--46 (2014; Zbl 1338.34088)

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