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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?

Mike Flynn
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  • You seem to believe that the Fourier transform of $a/(1+a)$ is $A/(1+A)$. This is wrong. – Julián Aguirre Jul 24 '18 at 16:39
  • Well there is a difference between $A/(1 + A)$ and $\int A e^{i\omega } d \omega/ \left [ 2\pi + \int A e^{i\omega t} d \omega \right ]$ right? I'm trying to substitute in the inverse Fourier transform. Though I agree that the forward direction is probably better. – Mike Flynn Jul 24 '18 at 18:31

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