I have a coupled system of ODEs:
$$\cases{ i\frac{\text{d}y_1}{\text{d}t}=A f(t)y_2(t)+E_1 y_1(t)\\ i\frac{\text{d}y_2}{\text{d}t}=A f(t)y_1(t)+E_2 y_2(t) }\tag1$$
Here $f(t)$ is a periodic function with frequency $\omega$.
This is a system of equations governing energy levels population in a two-level system under perturbation of $A f(t)$. Here $E_1$ and $E_2$ are energies, $y_1$ and $y_2$ are probability amplitudes for levels 1 and 2 ($|y_1|^2+|y_2|^2=1$). For small perturbation amplitudes $A$, $|y_1|^2$ and $|y_2|^2$ go up and down almost periodically with frequency $\nu=\sqrt{A^2+(\lambda-\omega)^2}$ which is known as generalized Rabi frequency (here $\lambda=E_2-E_1$). But they don't change really periodically - something highly oscillatory at frequency of $\omega$ disturbs them. For higher $A$ these solutions no longer resemble any Rabi cycles.
I've tried setting $y_1(t)=p(t)e^{-i E_1 t}$, $y_1(t)=q(t)e^{-i E_2 t}$ so that for $A=0$ the result was as for stationary states: $p(t)=q(t)=1$. Now the system transforms to something more symmetric and more explicitly showing independence of the system of $E_1+E_2$; here $E_2-E_1$ is denoted by $\lambda$:
$$\cases{ i\frac{\text{d}p}{\text{d}t}=f(t)e^{-i\lambda t}q(t)\\ i\frac{\text{d}q}{\text{d}t}=f(t)e^{i\lambda t}p(t) \tag2 }$$
I've then tried decoupling this system to get the following independent equations:
$$q(t)=i\frac{e^{i\lambda t}}{f(t)}\frac{\text{d}p}{\text{d}t} \tag3$$ $$-\frac{e^{-i\lambda t}}{f(t)}\frac{\text{d}}{\text{d}t}\left(\frac{e^{i\lambda t}}{f(t)}\frac{\text{d}p}{\text{d}t}\right)=p(t) \tag4$$
I've successfully solved numerically the system $(1)$, and even semi-analytically taking $f(t)$ as piecewise-constant, and then taking pieces to have very small length, and this gave me results close to numerical solution, but this still doesn't give any good understanding of the analytical form of the solution.
Here're examples of how the solutions look ($|y_1|^2$, $\Re y_1 \& \Im y_1$, $\Re y_2 \& \Im y_2$):
My question is: is there any way to further simplify this problem, e.g. extract that (quasi)periodic Rabi cycle part? I'm thinking of some analogue of Bloch theorem, but don't really see how exactly to do this.
Or even better, maybe this could be solved completely analytically? If yes, how?