I have equation of form
$$ \frac{dx}{dt} = f(x), $$
and know and for some initial value $x\_0$ its solution is periodic with unknown period. What can I tell about $f(x)$ apart from non-linearity (or $f(x) \equiv 0$)? 

Also, may be I'm wrong but I think that period is somehow related to "non-linearity" of $f(x)$, being longer for "more non-linear" functions. Is there something behind this intuition?