0
2013
Impact Factor

D. Minenkov

prosp.Vernadskogo 101/1, 119526, Moscow, Russian Federation
Institute for Problems in Mechanics, Russian Academy of Sciences

Publications:

Dobrokhotov S. Y., Minenkov D. S.
On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation
2010, vol. 15, no. 2-3, pp.  285-299
Abstract
The main aim of the paper is to compare various averaging methods for constructing asymptotic solutions of the Cauchy problem for the one-dimensional anharmonic oscillator with potential $V(x, \tau)$ depending on the slow time $\tau=\varepsilon t$ and with a small nonconservative term $\varepsilon g(\dot x, x, \tau)$, $\varepsilon \ll 1$. This problem was discussed in numerous papers, and in some sense the present paper looks like a "methodological" one. Nevertheless, it seems that we present the definitive result in a form useful for many nonlinear problems as well. Namely, it is well known that the leading term of the asymptotic solution can be represented in the form $X(\frac{S(\tau)+\varepsilon \phi(\tau))}{\varepsilon}, I(\tau), \tau)$, where the phase $S$, the "slow" parameter $I$, and the so-called phase shift $\phi$ are found from the system of "averaged" equations. The pragmatic result is that one can take into account the phase shift $\phi$ by considering it as a part of $S$ and by simultaneously changing the initial data for the equation for $I$ in an appropriate way.
Keywords: nonlinear oscillator, averaging, asymptotics, phase shift
Citation: Dobrokhotov S. Y., Minenkov D. S.,  On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 285-299
DOI:10.1134/S1560354710020152

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