Vibrational resonance: a study with high-order word-series averaging

TL;DR

This paper introduces a high-order averaging technique using word series to analyze vibrational resonance, enabling precise approximations without computing change of variables.

Contribution

It presents a novel recursive method for constructing high-order averaged systems via word series, simplifying the analysis of vibrational resonance.

Findings

High-order averaged systems provide very accurate approximations.

The method simplifies calculations by avoiding change of variables.

Recursive construction of word basis functions is effective.

Abstract

We study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the tech- nique applied here, the tasks of constructing the averaged system and the associ- ated change of variables are divided into two parts. It is first necessary to build recursively a set of so-called word basis functions and, after that, all the required manipulations involve only scalar coefficients that are computed by means of sim- ple recursions. As distinct from the situation with other approaches, with word- series, high-order averaged systems may be derived without having to compute the associated change of variables. In the system considered here, the construction of high-order averaged systems makes it possible to obtain very precise approxima- tions to the true dynamics.