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I have some questions concerning fast slow system like the van der pol equation

say we have $\epsilon x′_1=-\frac13 x_1^3+x_1 − x_2$ and $x′_2= x_1$

Does $\epsilon x'_1$ means that $x_1$ is faster than $x_2$?

Why do we put $\epsilon = 0$?

When this happens, the graph we get means what?

I appreciate every help and thank you for your time

TravisJ
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shadow
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1 Answers1

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1) yes, because if you divide by $\epsilon$, the derivative of $x_1$ would be very large for small $\epsilon$. (note that usually one writes the equations with different signs)

2) I think nobody does it otherwise it would be stupid to write a number that is always zero. Instead one considers small $\epsilon$ (to make the difference in speed more prominent, because math like extremes)

3) graph of what?

demitau
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  • First of all thank you for your answer.

    Second when we put epsilon equal zero , we can sketch the graph of −1/3 x1^3 +x1 − x2 = 0. My question was why 0? because easier to sketch the graph?

    And this graph we get represents the "limit cycle" of this system?

    Sorry I am still a beginner in slow/fast systems. Thank you again for your help

     – shadow May 11 '15 at 07:48
  • Basically you want to say that the dynamics for small $\epsilon$ looks like (for certain time depending on the the size of $\epsilon$ the dynamics for $\epsilon=0$, which is easier (because equations become easier to solve). Limit cycle for the origianl system you mean? – demitau May 11 '15 at 13:38
  • I think i messed up two notions . I understand now.Thank you for your help and thank you for responding – shadow May 11 '15 at 19:03
  • One last question why do we care about the fast system and not the slow one? Is it because the dynamics of the fast system will cause the biggest impact because the slow is too slow to make change? And why are the trajectories attracted to this graph we obtained? I know its not 1 question and I really appreciate your help – shadow May 11 '15 at 20:13
  • To the contrary, there are cases where you'd care more about the slow system and approximate the fast system as a stochastic perturbation. Here's an example with applications to climate modeling: https://arxiv.org/abs/1110.6671v1 – Oberdada Oct 18 '16 at 23:45