In mathematical models of genetic transcriptional circuits, you get a lot of differential equations of the form:
$$ \frac{dp(t)}{dt} = k \frac{ a(t) }{1 + a(t)}$$
Is there any way to fourier analyze this differential equation in $a(t)$ and $p(t)$ - without linearizing? I'd like to examine $P(\omega)$ as a response of $A(\omega)$.
There doesn't seem to be a way.
$$ \frac{1}{2\pi} \int_{-\infty}^\infty i\omega P(\omega)e^{i\omega t} d\omega = k \frac{\frac{1}{2\pi}\int_{-\infty}^{\infty} A(\omega)e^{i\omega t} d\omega}{1 + \frac{1}{2\pi} \int_{-\infty}^{\infty}A(\omega) e^{i\omega t} d\omega} $$
Is there some kind of Fourier Analysis chain rule I could use?