## 2. Frequency map analysisIn this section, we provide a brief description of the frequency map analysis method. More details can be found in a recent review article written by Laskar (1996) and references therein. According to the KAM theorem (Kolmogorov 1954; Arnol'd 1963; Moser 1962), in the phase space of a sufficiently close to integrable conservative system, many invariant tori will persist. Trajectories starting on one of these tori remain on it thereafter, executing quasi-periodic motion with a fixed frequency vector depending only on the torus. The family of tori is parameterized over a Cantor set of frequency vectors, while in the gaps of the Cantor set chaotic behavior can occur. Although the frequencies are strictly speaking only defined and fixed on these tori, the frequency analysis algorithm (Laskar 1990; Laskar et al. 1992, Dumas & Laskar 1993; Laskar & Robutel 1993; Laskar et al. 1993; Laskar 1996) will numerically compute over a finite time span a frequency vector for any initial condition. On the KAM tori, this frequency vector will be a very accurate approximation of the actual frequencies, while in the chaotic regions, it will provide a natural interpolation between these fixed frequencies. Let us consider a which gives in the complex variables , where . The motion in phase space takes place on tori, products of true circles with constant radii , which are described at constant velocity . If the system is non-degenerate, that is if the frequency map is a diffeomorphism on its
image , and the tori are as well described by
the action variables or equivalently by the
frequency vector . for which the perturbed system still possesses smooth invariant tori with linear flow (the KAM tori). Moreover, according to Pöschel (1982), there exists a diffeomorphism: which is analytical with respect to and in and on transforms the Hamiltonian equations into: . For frequency vectors in , the solution lies on a torus and is given in complex form by its Fourier series where the coefficients depend smoothly on the frequencies . If we fix to some value , we obtain a frequency map on defined as: where is the projection on
(). It should be noted
that for sufficiently small , the torsion
condition (2) ensures that the frequency map is
a diffeomorphism. The frequency map analysis provides directly, in a
numerical manner, a natural frequency map Once the quasiperiodic approximation (7) is obtained, the construction of the frequency map can be made and the study of the global dynamics of the system will then be possible in a very effective way by the analysis of the regularity of this frequency map (Laskar 1990; Laskar et al. 1992, Dumas & Laskar 1993; Laskar & Robutel 1993; Laskar et al. 1993; Paper I). Let be the subset of
of the values of such that
belongs to a KAM torus of dimension The frequency map analysis relies heavily on the observation that when a quasiperiodic function in the complex domain is given numerically, it is possible to recover a quasi-periodic approximation of in a very precise way over a finite time span , several orders of magnitude more precisely than what is given by simple Fourier series. Indeed, let be a KAM quasi-periodic solution of a Hamiltonian system in , where the frequency vector satisfies a Diophantine condition (3). The frequency analysis algorithm NAFF will provide an approximation of from its numerical values over a finite time span . The frequencies and complex amplitudes are computed through an iterative scheme. In order to determine the first frequency , one searches for the maximum amplitude of where the scalar product is defined by: and where is a weight function, that is, a
positive and even function with . In all our
computations, we used the Hanning window filter, that is
. Once the first periodic term
is found, its complex amplitude
is obtained by orthogonal projection, and the
process is restarted on the remaining part of the function
. It is also necessary to orthogonalize the set
of functions , when projecting we have (Laskar 1996)
In particular, the use of the Hanning data window () ensures that, for a KAM solution, the accuracy regarding the determination of the main frequencies will be proportional to , instead of without the Hanning window (), while for an ordinary FFT method, this accuracy will only be proportional to . Thus, the frequency analysis will easily allow the recovery of the frequency vector . The dynamics of the system is then analyzed by studying the regularity of the frequency map. © European Southern Observatory (ESO) 1998 Online publication: December 8, 1997 |