Consider the following system of ODEs:
$$ i\dot{C_1}(t)=\nu^*(\omega) e^{i(\omega-\omega_{10})t} C_2(t)\\ i\dot{C_2}(t)=\nu(\omega) e^{-i(\omega-\omega_{10})t} C_1(t) $$ with initial conditions $C_1(0)=1$ and $C_2(0)=0$.
Here $\nu(\omega)\ll1$ so I would like to perform a perturbation solution. But what confuses me is that there is not only $\nu(\omega)$, but also a complex conjugate $\nu^*(\omega)$. They clearly have the same order, but, nevertheless, they are different. Should I look at possible solutions with both of them? Namely, $$ C_1(t)\approx C_1^0+C_1^1\nu+O(\nu^2)\\ C_2(t)\approx C_2^0+C_2^1\nu+O(\nu^2) $$
and
$$ C_1(t)\approx C_1^0+C_1^1\nu^*+O(\nu^{*^2})\\ C_2(t)\approx C_2^0+C_2^1\nu^*+O(\nu^{*^2}) $$ ? And then see which one stays?