On the Proper Form of the Amplitude Modulation Equations for Resonant Systems
Résumé
The complex amplitude modulation equations of a discrete dynarnical system are derived under general conditions of simultaneous intemal and extemal resonances. Alternative forms of the real amplitude and phase equations are critically discussed. First, the most popular polar form is considered. Its properties, known in literature for a multitude of specifie problems, are here proven for the general case. Moreover, the drawbacks encountered in the stability analysis of in complete motions (i.e . motions containing sorne zero amplitudes) are discussed as a consequence of the fact the equations are not in standard normal form. Second, a so-called Cartesian rotating form is introduced, which makes it possible to evaluate periodic solutions and analyze their stability, even if they are in complete. Although the rotating form calls for the enlargernent of the space and is not amenable to analysis oftransient motions, it systematically justifies the change of variables sorne times usedin literature to a void the problems of the polar form. Third, a rnixed polar -Cartesian form is presented. Starting from the hypothesis th at there exists a suitable number of amplitudes which do not vanish in any motion, it is proved that the rnixed form leads to standard form equations with the sarne dimension as the polar form. However, if su ch principal amplitudes do not exist, more than one standard form equation should be sought. Finally, sorne illustrative examples of the theory are presented.
Domaines
Autre
Origine : Fichiers produits par l'(les) auteur(s)
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