There is a splitting of time scales which I don't understand; it is described below:
We have the following system: $$ {\dot {\textbf x}=A({\textbf x})\textbf c\atop\dot {\textbf c}=B({\textbf x})\textbf c}$$ for $$ {\textbf x}={s\choose p}$$ and $$\textbf c=\left( \begin{array}{c} c_0 \\ c_1 \\ c_2 \end{array} \right), $$ with $A({\textbf x})$ and $B({\textbf x})$ matrices depending on ${\textbf x}$. Let $\epsilon = c_0 + c_1 + c_2$ and set $$ \textbf c=\epsilon \gamma . $$ This shows that a splitting of time-scales appears when $\epsilon$ is small because $$ \dot {\textbf x}=\epsilon A({\textbf x}) \gamma;\quad\dot{\gamma}=B({\textbf x})\gamma. $$ Introducing a new time variable $\tau=\epsilon t$, we write $$ \dot {\textbf x}=\frac{d{\textbf x}}{dt}=\epsilon\frac{d{\textbf x}}{d\tau}=\epsilon {\textbf x}',\quad \dot{ \gamma}=\frac{d \gamma}{dt}=\epsilon\frac{d \gamma}{d\tau}=\epsilon \gamma' $$ and conclude that $$ {\textbf x}'=A({\textbf x})\gamma,\,\epsilon\gamma'=B({\textbf x})\gamma. $$ For $\epsilon=0$ this reduces to $$ {\textbf x}'=A({\textbf x})\gamma$$ with $$B({\textbf x}) \gamma=0 $$ and $$\gamma_0+\gamma_1+\gamma_2=1. $$
Now my question is: what is exactly happening when we set $\epsilon = 0$? I'm looking for a rigorous explanation.
We derive the equation: $$B({\textbf x}) \gamma=0 $$ But it seems a bit fishy to me.
Also I don't know how to intepret $\gamma$ in that equation, i.e. what is the variable it depends on?
If $\epsilon = 0$, $x' = 0$, or $x$ stays constant. Usually you want to take $\epsilon$ very small, but not $0$.
The point of multiple time scale analysis is to simplify complex dynamical system that has multiple components acting on multiple time scale. By isolating the slow/fast time scale component, you either assume that the component changing on a slower time scale to be approximately constant, or the component changing on a fast time scale to reach a (quasi) steady state first. Setting $\epsilon = 0$ simply means one component is not changing at all, which is not the point of multiple time scale analysis.