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Averaging in multifrequency systems. (English. Russian original) Zbl 0611.34035

Funct. Anal. Appl. 20, 83-88 (1986); translation from Funkts. Anal. Prilozh. 20, No. 2, 1-7 (1986).
An averaging method for the system \[ \dot I=\epsilon f(I,\phi,\epsilon),\quad {\dot \phi}=\omega (I)+\epsilon g(I,\phi,\epsilon), \] where I varies in a bounded domain of the n- dimensional Euclidean space and \(\phi\) belongs to m-dimensional torus is considered. The average system has the form \(\bar I=\epsilon f_ 0(\bar I)\), where \(f_ 0(\bar I)\) is the average value of the vector f(I,\(\phi\),0) over the torus. For a typical system there exist positive constants \(C_ 1\), \(C_ 2\), \(\delta\) such that for \(\epsilon\in [0,1]\) \[ \int_{H}hdI^ 0d\phi^ 0\leq C_ 1\epsilon^{\delta},\quad \mu (G\times T^ m\setminus H)\leq C_ 2\epsilon^{\delta /r} \] where \(H(r)=\{(I^ 0,\phi^ 0)\in G\times T^ m:\| I(t)-\bar I(t)\| \leq r\quad \forall t\in [0,1/\epsilon]\}\), \(h(I^ 0,\phi^ 0)=\max_{[0,1/\epsilon]}\| I(t)-\bar I(t)\|.\)
Reviewer: I.Foltyńska

MSC:

34C29 Averaging method for ordinary differential equations
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References:

[1] D. V. Anosov, ”Averaging in systems of ordinary differential equations with fast-oscillating solutions,” Izv. Akad. Nauk SSSR, Ser. Mat.,24, No. 5, 721-742 (1960).
[2] T. Kasuga, ”On the adiabatic theorem for the Hamiltonian system of differential equations in the classical mechanics. I, II, III,” Proc. Jpn. Acad.,37, 366-382 (1961). · Zbl 0114.14903 · doi:10.3792/pja/1195577651
[3] A. I. Neishtadt, ”On averaging in multifrequency systems. II,” Dokl. Akad. Nauk SSSR,226, No. 6, 1295-1298 (1976).
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