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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.

A. PI
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    You should probably just look up geometric singular perturbation theory (GSPT). Anyway, for $1)$ and $2)$, make the transformation $\epsilon dt = d \tau$. What does your new system become? How does that change the problem? – Matthew Cassell Nov 21 '16 at 14:51
  • Defining $X(\tau) = \epsilon x(\tau /\epsilon), Y(\tau) = y(\tau /\epsilon)$ and $F(X) = f(x),$ we get the following system $$\begin{matrix} \dot{X} = Y& \ \epsilon \dot{Y} = LY + \nabla F(X) \end{matrix}$$ – A. PI Nov 21 '16 at 17:04

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