Recently a method has been suggested to analyze the chaotic behaviour of a conservative dynamical system by numerical analysis of the fundamental frequencies. Frequencies and amplitudes are determined step by step. As the frequencies are not generally orthogonal, a Gramm-Schmidt orthogonMization is made and for each new frequency the old amplitudes of previously determined frequencies are corrected. For a chaotic trajectory variations of the frequencies and amplitudes determined over different time periods are expected. The change of frequencies in such a calculation is a measure of the chaoticity of the trajectory. While amplitudes are corrected, the frequencies (once determined) are constant. We suggest here simple linear corrections of frequencies for the effect of other close frequencies. The improvement of frequency determination is demonstrated on a model case. This method is applied to the first fifty numbered asteroids.
A nonlinear theory of secular resonances is developed. Both terms corresponding to secular resonances v5 and v 6 are taken into account in the Hamiltonian. The simple overlap criterion is applied and the condition for the overlap of these resonances is found. It is shown that in given approximation the value p = (1 -e2)t/2(1 -cos I) is an integral of motion, where the mean eccentricity e and mean inclination I are obtained by eliminating short-period perturbations as well as the nonresonant terms from the planets. The overlap criterion yields a critical value of parameter p depending on the semi-major axis a of the asteroid. For p greater than the critical value, resonance overlap occurs and chaotic motion has to be expected. A mapping is presented for fast calculation of the trajectories. The results are illustrated by level curves in surfaces of section method.
This paper reviews various mapping techniques used in dynamical astronomy. It is mostly dealing with symplectic mappings. It is shown that used mappings can be usually interpreted as symplectic integrators. It is not necessary to introduce any 6 functions it is just sufficient to split Hamiltonian into integrable parts. Actually it may be shown that exact mapping with 6 function in the Hamiltonian may be non-symplectic. The application to the study of asteroid belt is emphasised but the possible use of mapping in planetary evolution studies, cometary and other problems is shortly discussed.
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