We develop a generalization of the drift method [Avtomat. Telemekh., No. 2,
56-69 (1988); V. P. Zhukov, Field Methods in the Analysis of Nonlinear Dyn
amical Systems [in Russian], Nauka, Moscow (1992)] which allows us to study
some dynamical properties of objects described by systems of nonautonomous
, nonlinear, ordinary differential equations of arbitrary orders. The resul
ts of research of these properties on the basis of the generalized method a
re presented.