On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation

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, τ) depending on the slow time τ = ?t and with a small nonconservative term ?g( $ \dot x $ , x, τ), ? ? 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\left( {\frac{{S\left( \tau \right) + \varepsilon \varphi \left( \tau \right)}} {\varepsilon },I\left( \tau \right),\tau } \right) $ , where the phase S, the “slow” parameter I, and the so-called phase shift ? are found from the system of “averaged” equations. The pragmatic result is that one can take into account the phase shift ? by considering it as a part of S and by simultaneously changing the initial data for the equation for I in an appropriate way.