According to Dulac theorem any 2D autonomous system with a periodic orbit has a region with positive and a region with negative divergence inside such orbit. Here represented by red and green regions respectively
Without loss of generality, let there exist a function such that
in simply connected region . Let be a closed trajectory of the plane autonomous system in . Let be the interior of . Then by Green's theorem,
Because of the constant sign, the left-hand integral in the previous line must evaluate to a positive number. But on , and , so the bottom integrand is in fact 0 everywhere and for this reason the right-hand integral evaluates to 0. This is a contradiction, so there can be no such closed trajectory .