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 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 solution. 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.
