Frequency Modified Fourier Transform and its Application to Asteroids

Abstract

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 orthogonalization 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.