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Astron. Astrophys. 329, 451-481 (1998)

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2. Frequency map analysis

In 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 n degrees of freedom Hamiltonian system close to integrable in the form [FORMULA], where H is real analytic for [FORMULA], [FORMULA] is a domain of [FORMULA] and [FORMULA] is the n -dimensional torus. For [FORMULA], the Hamiltonian reduces to [FORMULA] and is integrable. The equations of motion are then

[EQUATION]

which gives [FORMULA] in the complex variables [FORMULA], where [FORMULA]. The motion in phase space takes place on tori, products of true circles with constant radii [FORMULA], which are described at constant velocity [FORMULA]. If the system is non-degenerate, that is if

[EQUATION]

the frequency map [FORMULA] is a diffeomorphism on its image [FORMULA], and the tori are as well described by the action variables [FORMULA] or equivalently by the frequency vector [FORMULA].2 For a nondegenerate system (or for an isoenergetically non-degenerate system), when [FORMULA] is nonzero, the KAM theorem still asserts that for sufficiently small values of [FORMULA], there exists a Cantor set [FORMULA] of values of [FORMULA], satisfying a Diophantine condition of the form

[EQUATION]

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:

[EQUATION]

which is analytical with respect to [FORMULA] and [FORMULA] in [FORMULA] and on [FORMULA] transforms the Hamiltonian equations into: [FORMULA]. For frequency vectors [FORMULA] in [FORMULA], the solution lies on a torus and is given in complex form by its Fourier series

[EQUATION]

where the coefficients [FORMULA] depend smoothly on the frequencies [FORMULA]. If we fix [FORMULA] to some value [FORMULA], we obtain a frequency map on [FORMULA] defined as:

[EQUATION]

where [FORMULA] is the projection on [FORMULA] ([FORMULA]). It should be noted that for sufficiently small [FORMULA], the torsion condition (2) ensures that the frequency map [FORMULA] is a diffeomorphism. The frequency map analysis provides directly, in a numerical manner, a natural frequency map F, defined on the whole domain [FORMULA], which coincides, up to numerical accuracy, with [FORMULA] (Eq. 6) on the set of the KAM tori. The frequency map F is obtained by searching for quasiperiodic approximations of the solutions, over a finite time span, in the form of a finite number of terms

[EQUATION]

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 [FORMULA] be the subset of [FORMULA] of the values of [FORMULA] such that [FORMULA] belongs to a KAM torus of dimension n. In this case, we can assume that, up to the numerical accuracy of our numerical procedure, the frequency vector [FORMULA] is the true frequency vector of the considered torus. We thus assume that on [FORMULA], [FORMULA] is a very good approximation of the frequency map [FORMULA] defined in (6) and the restriction of the frequency map [FORMULA] to [FORMULA] will have the following properties: a) If [FORMULA], then [FORMULA] is constant on [FORMULA] b) For any given [FORMULA], the map [FORMULA] [FORMULA] is regular in some sense, as it coincides on [FORMULA] with the restriction to [FORMULA] of a smooth diffeomorphism. The criterion (b) ensures that when the frequency map is not regular, the corresponding KAM tori are destroyed.

The frequency map analysis relies heavily on the observation that when a quasiperiodic function [FORMULA] in the complex domain [FORMULA] is given numerically, it is possible to recover a quasi-periodic approximation of [FORMULA] in a very precise way over a finite time span [FORMULA], several orders of magnitude more precisely than what is given by simple Fourier series. Indeed, let

[EQUATION]

be a KAM quasi-periodic solution of a Hamiltonian system in [FORMULA], where the frequency vector [FORMULA] satisfies a Diophantine condition (3). The frequency analysis algorithm NAFF will provide an approximation [FORMULA] of [FORMULA] from its numerical values over a finite time span [FORMULA]. The frequencies [FORMULA] and complex amplitudes [FORMULA] are computed through an iterative scheme. In order to determine the first frequency [FORMULA], one searches for the maximum amplitude of [FORMULA] where the scalar product [FORMULA] is defined by:

[EQUATION]

and where [FORMULA] is a weight function, that is, a positive and even function with [FORMULA]. In all our computations, we used the Hanning window filter, that is [FORMULA]. Once the first periodic term [FORMULA] is found, its complex amplitude [FORMULA] is obtained by orthogonal projection, and the process is restarted on the remaining part of the function [FORMULA]. It is also necessary to orthogonalize the set of functions [FORMULA], when projecting f iteratively on these [FORMULA]. For a KAM solution, the frequency analysis algorithm allows a very accurate determination of the frequencies over the time span [FORMULA], and converges towards these frequencies as T increases. Indeed, for a weight function of the form:

[EQUATION]

we have (Laskar 1996)

Proposition 1. For a KAM solution [FORMULA] of the form (8), and using the weight function [FORMULA], the application of the frequency analysis algorithm over the time span [FORMULA], as described above, provides a determination [FORMULA] of the frequency [FORMULA] which converges towards [FORMULA] for [FORMULA], with the asymptotic expression:

[EQUATION]

[EQUATION]

In particular, the use of the Hanning data window ([FORMULA]) ensures that, for a KAM solution, the accuracy regarding the determination of the main frequencies will be proportional to [FORMULA], instead of [FORMULA] without the Hanning window ([FORMULA]), while for an ordinary FFT method, this accuracy will only be proportional to [FORMULA]. Thus, the frequency analysis will easily allow the recovery of the frequency vector [FORMULA]. The dynamics of the system is then analyzed by studying the regularity of the frequency map.

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© European Southern Observatory (ESO) 1998

Online publication: December 8, 1997
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