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 $V(x,\tau)$ 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 $X\Big(\frac{S(\tau)+\varepsilon \phi(\tau))}{\varepsilon}, I(\tau),\tau\Big)$, 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.