An error estimate for the averaging method in a two-frequency problem

Error estimates are provided for the averaging method in a two-frequency problem with analytic right-hand sides in the case where the ratio of frequencies varies monotonically along trajectories of the averaged system. When $l>1$, the estimate is of the order $\varepsilon^{\frac13+\frac2{9l+3}}$ for initial values outside a set whose measure is of the same order, where $\varepsilon$ is the small parameter in the problem, and $l$ is a nonnegative integer determined by the problem itself. This paper extends, in some aspects, the results of A. I. Neishtadt (Passing through resonances in a two-frequency problem, Dokl. Akad. Nauk SSSR, 1975, vol. 221, p. 301–304) under the assumptions $A$ and $B$, corresponding to the values $l=0$ and $l=1$ respectively.
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