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.
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.