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

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, τ) depending on the slow time τ = ɛt and with a small nonconservative term ɛg($\dot x$, x, τ), ɛ ≪ 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 ϕ are found from the system of “averaged” equations. The pragmatic result is that one can take into account the phase shift ϕ by considering it as a part of S and by simultaneously changing the initial data for the equation for I in an appropriate way.