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On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation. (English) Zbl 1209.34050

Summary: 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 \(X({\frac{S(\tau)+ \varepsilon \varphi (\tau)}{\varepsilon},I(\tau ),\tau } )\), where the phase \(S\), the “slow” parameter \(I\), and the so-called phase shift \(\varphi \) are found using the system of “averaged” equations. The pragmatic result is that one can take into account the phase shift \(\varphi \) by considering it as a part of \(S\) and by simultaneously changing the initial data for the equation for \(I\) in an appropriate way.

MSC:

34C29 Averaging method for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
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