Geodesic
An error estimate for the averaging method in a two-frequency problem
Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 241-258 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. Bibliography: 5 titles. 
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V. E. Pronchatov. An error estimate for the averaging method in a two-frequency problem. Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 241-258. http://geodesic.mathdoc.fr/item/SM_1985_50_1_a15/

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[2] Neishtadt A. I., O nekotorykh rezonansnykh zadachakh v nelineinykh sistemakh, Dissertatsiya na soiskanie uch. st. kand. fiz.-matem. nauk, MGU, M., 1975

[3] Arnold V. I., Dopolnitelnye glavy teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978 | MR

[4] Neishtadt A. I., Ob osrednenii v mnogochastotnykh sistemakh. II, 226 (1976), 1295–1298 | Zbl

[5] Molchanov A. M., “The resonant structure of the solar system”, Icarus, 8 (1968), 203–215 | DOI