  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.
