Singular perturbation theory in non-standard form

Question

Singular perturbation theory in ODE's is a well treated and highly studied subject. Most of the references I can find take the form,

\begin{align} \dot{x} &=f\left( x,z,\varepsilon \right) \\ \varepsilon \dot{z}&=g\left( x,z,\varepsilon \right) \\ \end{align}

where there is an explicit separation of the state variables into slow mode $x \in \mathbb{R}^m$ and fast mode $z \in \mathbb{R}^n$. I was wondering if anyone can point me to a complete treatment of the more general case:

1. $$ \varepsilon \dot{x}=f\left( x,\varepsilon \right) $$ where there is no explicit separation or
2. $$ \varepsilon \dot{x}=f\left( x,\varepsilon \right)+g\left(x \right) $$

I believe in the second case a reduction to the standard form can be made by some coordinate change; but how would one go about constructing such a coordinate change? Can such a coordinate change be found in the first case?

A treatment of the linear case:

$$ \varepsilon \dot{x}=\left( A+\varepsilon B\left( \varepsilon \right) \right)x $$

can be found here but I can't seem to find any material on the most general non-linear case.

Answer 1

A general treatment does not exist at this moment; however, I would advise to keep a close eye on this upcoming title. In the mean time, geometric singular perturbation theory analysis of specific systems in non-standard form can be found here, here, here or here, to name just a few.