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A slow-fast system usually involves two kinds of dynamical variables, evolving on very different timescales. The ratio between the fast and slow timescales is measured by small parameter $\epsilon$ in which $0<\epsilon<<1$.

An (1,1) slow-fast ODE system, for instance, is written on the slow time scale $\tau$ as follow: $$ \epsilon \frac{dx}{d\tau} = f(x,y) $$ $$ \frac{dy}{d\tau} = g(x,y)$$

Or fast time scale $t=\tau/\epsilon$: $$ \frac{dx}{dt} = f(x,y) $$ $$ \frac{dy}{dt} = \epsilon g(x,y)$$

My question is:

(i) How small $\epsilon$ is so that we call this ODE a slow-fast system - 0.1, 0.01, 0.001?

(ii) Is there any mathematical definition or threshold of $\epsilon$ that we use to define slow-fast systems?

PLe
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1 Answers1

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There isn't a hard and fast rule.

The key idea is that you want $\epsilon$ to be "sufficiently small" that if you approximate $f(x,y)+\epsilon g(x,y)$ with just $f(x,y)$ you are not distorting your system "too much" on the fast scale. Just how much is "too much" will strongly depend on what kind of error you are willing to accept in your model, as well as on the specific nature of $g$ and $f$.

The "formal" way to deal with the whole issue would be to consider the behaviour of the system as $\epsilon\to 0$, and more in general to prove bounds in the form "if $\epsilon<\eta$, then ..."

Anonymous
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  • So basically we focus on the behaviors of fast and slow flow as $\epsilon \rightarrow 0$, not a specifically small value of $\epsilon$. Thanks for your explanation. – PLe Mar 26 '17 at 15:56