I have a set of variables $x_1, \ldots, x_n$ and $y_1,\ldots,y_m$ together with differential equations of the form
$$\frac{dx_i}{dt}= f_i(X,Y)$$ $$\frac{dy_i}{dt} =g_j(X,Y)$$
Is there a theory stating when the dynamics of $x$ or the dynamics of $y$ can essentially be treated as (largely) independent of the other variables? For example, this is obviously true if $f_i$ does not actually depend on $y$. I'd even argue that $f_i(X,Y)=f_i^a(X)\cdot f_i^b(Y)$ allows for a certain form of decoupling, as $f_i^b(Y)$ only changes the speed of the time-evolution of the variables $x$.