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I am trying to apply perturbation theory to this system to approximate $y_1(t)$, which is oscillatory when $y_1(0)$ is complex:

$y'_1(t)=\epsilon y_2(t)+2y_1^2(t)$

$y'_2(t)=y_1(t)$

But the solution I obtain using first order regular perturbation is non-oscillatory, only valid for short time scales. The fact that oscillations disappear when $\epsilon=0$ suggests to me that singular perturbation might be needed but this does not look like a typical singular problem. Can you suggest an approach?

Note: I am not looking for an exact solution as I want to apply perturbation to similar, more complex problems.

Here is an example solution (real and complex parts) when $\epsilon=0.008^2$, $y_1(0)=0.2+0.3i$, $y_2(0)=400$. The regular perturbation approximation is accurate until $t\approx5$ only.

Example solution

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  • $\begingroup$ The method of multiple scales is likely going to be the answer. If you have equations for the first and second terms in the regular expansion it might help diagnose the problem. $\endgroup$
    – David
    Sep 4, 2018 at 17:34
  • $\begingroup$ I have actually tried multiple scales but didn't get anywhere. But thank you for the suggestion. $\endgroup$
    – Frimousse
    Sep 5, 2018 at 11:30

1 Answer 1

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There is a second scale involved that is established by the magnitudes of the initial conditions. $y_1(0)$ is small against $y_2(0)$ and by the second equation that stays that way for some time. So in first order one can consider $y_2$ constant.

Then apply the usual substitution for Riccati equations $y_1=-\frac{u'}{2u}$ to obtain $$u''+2ϵy_2u=0$$ to get a harmonic oscillator with frequency $\omega=\sqrt{2ϵy_2}$ which for the given data is $\sqrt2\cdot 0.16=0.22627$. The fraction $u'/u$ oscillates twice in every period of $u$ which gives a period for $y_1$ of $\pi/ω=13.8840$ which coincides with the period you can read off of your graph.

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  • $\begingroup$ Thank you that's very helpful. I'm happy to accept your answer but I'm just wondering if you can think of a way to improve the approximation further? I can see that on longer time scales the oscillations are damped by the change in $y_2$. Or since $y_2$is the integral of $y_1$, this is an integro-differential equation and maybe perturbation theory is not the right tool? $\endgroup$
    – Frimousse
    Sep 5, 2018 at 12:08

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