On the proper form of the amplitude modulation equations for resonant systems

The complex amplitude modulation equations of a discrete dynamical system are derived under general conditions of simultaneous internal and external 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 specific problems, are here proven for the general case. Moreover, the drawbacks encountered in the stability analysis of incomplete motions (i.e. motions containing some 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 incomplete. Although the rotating form calls for the enlargement of the space and is not amenable to analysis of transient motions, it systematically justifies the change of variables sometimes used in literature to avoid the problems of the polar form. Third, a mixed polar-Cartesian form is presented. Starting from the hypothesis that there exists a suitable number of amplitudes which do not vanish in any motion, it is proved that the mixed form leads to standard form equations with the same dimension as the polar form. However, if such principal amplitudes do not exist, more than one standard form equation should be sought. Finally, some illustrative examples of the theory are presented.