Higher Order Averaging and Related Methods for Perturbed Periodic and Quasi-Periodic Systems

Abstract

for 0 ? t 0. Following the basic work of Krylov and Bogoliubov [9] in 1934, the first systematic and comprehensive account of the method of averaging was given by Bogoliubov and Mitropolski [1] in 1958. In this account, the authors were primarily concerned with first and second order approximations. They laid the mathematical foundation for the method which established the validity of the first order approximation over a time interval of 0(1/E). Several authors have subsequently considered higher order averaging, e.g., Volosov [20]-[22], but have only explicitly treated first and second order averaging. In this way they establish an algorithm for the general Nth order method; but they do not establish it as an Nth order asymptotic method on 0 0. In 1964, Mitropolski [12] indicated how to establish Nth order averaging as an Nth order asymptotic method on 0 ? t 2 simply by conjecturing the general structure of the fundamental differential inequality [12, p. 339] from which the Nth order error estimate follows. More recently, Zabreiko and Ledovskaja [23] gave a set of equations defining the general Nth order method of averaging and a set of conditions which they claim are sufficient to establish Nth order averaging as an Nth order asymptotic method on 0 ? t < Llr as E -* 0. In particular, they state [23, p. 1455] that these conditions are satisfied if the right-hand side f(t, x, E) is an almost periodic function of t (and has sufficiently many derivatives in x and ? which are bounded). This statement is in general not true, the reason being that the almost periodicity off(t, x, e) in t does not, in general, imply the boundedness of the function p1(t, x) or of its first partial derivative with respect to x, where