I can't understand one approximation I've come across in paper on mathematical biology. I have equation:
$\frac{d\theta}{dt}=\omega+P\left(\frac{t}{\tau}\right)\Delta(\theta)$,
where $\omega$ is constant, $P\left(\frac{t}{\tau}\right)$ and $\Delta(\theta)$ are periodical with period 1.
If $P(\Phi)$ and $\Delta(\Phi)$ are smooth enough we can average this equation to:
$\frac{d\theta}{dt}=\omega+H\left(\frac{t}{\tau}-\theta\right)$,
where
$H(\Phi)=\int\limits_{0}^{1}P(s)\Delta(s-\Phi)ds$.
Do you have any idea what could be justification of this approximation? This comes from section 2 in article https://www.math.uh.edu/~zpkilpat/teaching/math4309/project/jmb91_ermentrout.pdf