I'm delving into multiscale methods, specifically homogenization, and I'm confused a bit about a notational convention.
I often see written something to the effect of $u_{\epsilon} = u(x, x/\epsilon)$, where $y = x/\epsilon$ is intended (I believe), as the parameter describing behavior at the macroscale (as $\epsilon$ is small, I assume $y$ would be much larger than $x$.
I suspect that this is wrong, though, because in other segments, when discussing dynamical systems with highly oscillatory behavior, I see systems written as :
$\begin{align} \frac{dx}{dt} = f(x, y)~~~~\\ \frac{dy}{dt} = \frac{1}{\epsilon}g(x, y)\\ \end{align}$
This suggests that the behavior of the $y$ parameter is changing more quickly, which suggests a small spatial scale.
Obviously, this doesn't mean the notation is wrong, just that I don't quite get what the convention is. Could someone familiar with the multiscale literature please clarify how the different scales are generally denoted?