In the preface to Chapter 4 ``Perturbation Theory" in Arnold's book Geometric Methods in the Theory of Ordinary Differential Equations available for preview here
``If the size of the perturbation is characterized by a small parameter $\varepsilon$, then the effect of perturbations over time of order 1 leads to a change of order $\varepsilon$ of the solution. This change can be calculated approximately by solving a variational equation along the unperturbed solution."
He then goes on to discuss the asymptotic methods necessary to discuss the validity of approximation at long times, but I am left wondering:
Question: What is the ``variational equation along the unperturbed solution'' referred to here?
I would guess that we are finding for finite time the next-order correction term in $\varepsilon$ to the leading order behavior (given by the solution of the unforced equation) but I am not sure about this and also not sure how this leads to a variational problem. Any help would be greatly appreciated!!