The following system $$\begin{matrix} \dot{x}= y,& \\ \dot{y}= Ly + \epsilon \nabla f(x), \end{matrix}$$
with $x,y : [0,T]\to \mathbb{R}^d$ is called (in many references) a "multi-scale" system or "represents multi-scale dynamics." I'd like to get some definition or explanation for the following statements:
1- $x$ is called a slow variable whereas $y$ is fast.
2- Slow/fast time scale.
3- Slow manifolds.