Simply improved averaging for coupled oscillators and weakly nonlinear waves

TL;DR

This paper introduces an improved averaging method for coupled oscillators and nonlinear waves, providing more accurate long-term approximations through a novel coordinate transform, with theoretical and numerical validation across multiple examples.

Contribution

It proposes a new coordinate transform that enhances the classical averaging method, leading to more precise long-term dynamics approximation in high-dimensional nonlinear oscillatory systems.

Findings

Improved accuracy in coupled oscillator modeling.

Enhanced long-term approximation for nonlinear wave equations.

Numerical validation shows benefits beyond theoretical timescales.

Abstract

The long time effect of nonlinear perturbation to oscillatory linear systems can be characterized by the averaging method, and we consider first-order averaging for its simplest applicability to high-dimensional problems. Instead of the classical approach, in which one uses the pullback of linear flow to isolate slow variables and then approximate the effective dynamics by averaging, we propose an alternative coordinate transform that better approximates the mean of oscillations. This leads to a simple improvement of the averaged system, which will be shown both theoretically and numerically to provide a more accurate approximation. Three examples are then provided: in the first, a new device for wireless energy transfer modeled by two coupled oscillators was analyzed, and the results provide design guidance and performance quantification for the device; the second is a classical coupled oscillator problem (Fermi-Pasta-Ulam), for which we numerically observed improved accuracy beyond the theoretically justified timescale; the third is a nonlinearly perturbed first-order wave equation, which demonstrates the efficacy of improved averaging in an infinite dimensional setting.