Consider the autonomous system below with the parameters $a\neq 0$ and $mn \neq 0$:
$$\begin{cases} \displaystyle\frac{dx}{dt} &=-y+mxy+ny^2\\ \displaystyle\frac{dy}{dt} &=x+ax^2 \end{cases}$$ How to prove that there is no periodic orbit in this autonomous system? I think maybe Dulac's criterion is the tips of this question, but I haven't found the Dulac's Function yet.