Fast time in a singular perturbation ($\tau =t/ϵ$)

For a fast-slow system $ϵ\dot{x}=F(x,y)$, $\dot{y}=G(x,y)$ with $0<ϵ\ll 1$, the rescaled time $\tau =t/ϵ$ gives $x_{\tau }=F$, $y_{\tau }=ϵG$. Away from zeros of $F$, an order-one change of $x$ takes order-one fast time but only order-$ϵ$ original time, while $y$ changes little. Setting $ϵ=0$ in these rescaled equations freezes $y$ and produces the leading fast subsystem. This limit differs from the slow reduction $F=0$ in the original time; matching the two describes a relaxation oscillation.