Consider an ODE X'=F(X) where F is a polynomial in the Xi, with coefficients in Q or an algebraic extension of Q. n>=2 is the dimension of the space of X.

Then the period T of any periodic non-constant solution X(t)  is a transcendent number.

I think this is close to a Kontsevich-Zagier conjecture .... but I didn't find any paper on this question.