Formal methods for constructing asymptotic expansions of quasilinear vibrational systems with an aperiodic generating solution are considered. The proposed asymptotic method assumes the inclusion of damping (antidamping) terms in the generating system which leads to a non-traditional structure of the asymptotic series and of the changes of variables which are used. Although questions concerning the theoretical basis of the method are not considered, certain positive features of the proposed approach compared with the method of averaging are demonstrated by means of a simple example. The connection between the proposed method and the method of integral manifolds is discussed.