Abstract:
The author obtains an unimprovable estimate of the averaging method for a two-frequency problem with analytic right-hand sides under condition $\overline A$, which means a nonzero rate of change of the frequency ratio along trajectories of the averaged system. It turns out to be of order $\varepsilon^{\frac14+\frac1{2(l+1)}}$ for initial data outside a set of measure of order $\varepsilon^{\frac12}$, where $\varepsilon$ is a small parameter of the problem and $l$ is an upper bound for the maximal multiplicity of the roots of a certain finite set of equations (it is assumed that $l>1$).
Bibliography: 8 titles.
\Bibitem{Pro87}
\by V.~E.~Pronchatov
\paper On an error estimate for the averaging method in a~two-frequency problem
\jour Math. USSR-Sb.
\yr 1989
\vol 62
\issue 1
\pages 29--40
\mathnet{http://mi.mathnet.ru/eng/sm2643}
\crossref{https://doi.org/10.1070/SM1989v062n01ABEH003224}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=912409}
\zmath{https://zbmath.org/?q=an:0666.34047|0641.34046}
Linking options:
https://www.mathnet.ru/eng/sm2643
https://doi.org/10.1070/SM1989v062n01ABEH003224
https://www.mathnet.ru/eng/sm/v176/i1/p28
This publication is cited in the following 5 articles:
A. I. Neishtadt, “Averaging, passage through resonances, and capture into resonance in two-frequency systems”, Russian Math. Surveys, 69:5 (2014), 771–843
A. I. Neishtadt, “Capture into Resonance and Scattering on Resonances in Two-Frequency Systems”, Proc. Steklov Inst. Math., 250 (2005), 183–203
Arnold's Problems, 2005, 181
Bakhtin V., “Averaging in a General-Position Single-Frequency System”, Differ. Equ., 27:9 (1991), 1051–1061
A. M. Samoilenko, R. I. Petrishin, “On integral manifolds of multifrequency oscillatory systems”, Math. USSR-Izv., 36:2 (1991), 391–409
Citation in format AMSBIB
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