Let's have equation $$ \tag 1 \frac{d^{2}y(t)}{dt^{2}} +\omega^{2}(t)y(t) = 0, \quad t \in (t_{\text{in}}, \infty) $$ Here $$ \omega^{2}(t) = A(t) - B(t)cos(2t), $$ and functions $A(t), B(t)$ have following properties:
1) $A(t_{\text{in}}) - B(t_{\text{in}}) < 0$;
2) they are monotonically decreased with time, $\left| \frac{dB}{dt}\right| < B$, $\left|\frac{dA}{dt}\right| < A$;
3) $B(t)$ decreases faster than $A(t)$.
So there is following situation. At initial moments of time $\omega^{2}$ is alternating function, but after moment $t_{\text{final}}$, at which $A(t_{\text{final}}) - B(t_{\text{final}}) = 0$, $\omega^{2}(t)$ becomes positive only. Up to the moment $t_{\text{final}}$ $\omega^{2}$ behaves mostly as nonadiabatic function: $$ \left|\frac{d\omega^{2} (t)}{dt}\right| \approx 2B(t)sin(2t) > -\omega^{2} (t) $$
The question: how to construct approximate solution of Eq. $(1)$ on period $(t_{\text{in}}; t_{\text{final}})$?
Update. Specifically, I look for unstable solutions of Eq. $(1)$. In some approximation $A, B$ due to their properties, can be treated to be constants. Thus for period (t_{\text{in}}; t_{\text{final}}) Eq. $(1)$ is just Mathieu equation with $A < B, B >> 1$. But I haven't found the analysis for it.
This is just a sketch, you need to work out the details.
First assume $y(t)=e^{s(t)}$, where $s'' << (s')^2$. Now plug this into your ODE to get.
$$s''+(s')^2=-\omega^2$$
Now as $s'' << (s')^2$ we neglect $s''$. We get
$$(s')^2 \approx -\omega^2$$
This is "easy" to integrate. After you have found $s(x)$ proceed assuming $$y(t)=e^{s(t)+c(t)}$$ Now we assume that $c(t) << s(t)$ and $c''<(c')^2$. Now plug this into the ODE and neglect what is negligible. Then find $c(t)$, ... then assume $$y(t)=e^{s(t)+c(t)+d(t)}$$ and proceed the same manner. You will get something like a power series whicht might be divergent. This can be handled using Padé series approach.