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 Ufimsk. Mat. Zh., 2017, Volume 9, Issue 3, Pages 63–77 (Mi ufa387)

Adiabatic approximation in a resonance capture problem

L. A. Kalyakin

Institute of Mathematics, Ufa Scientific Center, RAS, Chernyshevsky str. 112, 450008, Ufa, Russia

Abstract: By means of the averaging method, we analyze two model problems on capture into resonance that leads us to the adiabatic approximation in the leading term in the asymptotics. The main aim is an approximate (by using a small parameter) description of the domain of capture into resonance. This domain is in the phase plane and it is formed by the initial points for the resonance solutions with an unboundedly increasing energy. The capture domain depends on an additional parameter involved in the equation. We show that the adiabatic approximation fails as the capture domain becomes narrow. In this case we have to modify substantially the averaging method. As a result, a system of nonlinear differential equations arises for the leading term in the asymptotics and this system is not always integrable.

Keywords: nonlinear oscillations, small parameter, asymptotics, capture into a resonance, adiabatic approximation.

 Funding Agency Grant Number Russian Science Foundation 17-11-01004 The research is supported by the grant of Russian Science Foundation (project no. 17-11-01004).

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English version:
Ufa Mathematical Journal, 2017, 9:3, 61–75 (PDF, 837 kB); https://doi.org/10.13108/2017-9-3-61

Bibliographic databases:

UDC: 517.956.226
MSC: 34D10, 34D20, 37J25, 34E13

Citation: L. A. Kalyakin, “Adiabatic approximation in a resonance capture problem”, Ufimsk. Mat. Zh., 9:3 (2017), 63–77; Ufa Math. J., 9:3 (2017), 61–75

Citation in format AMSBIB
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