In the proof of the following theorem it is not clear to me how the Poincare-Bendixson theorem is applied.
Theorem. If $X=\left ( X_{1},X_{2} \right )$ is a class field $C^{1}$ in $\Delta \subset \mathbb{R}^{2}$, $\Delta $ simply connected set with $divX=\frac{\partial X_{1}} {\partial x_{1}}+\frac{\partial X_{2}}{\partial x_{2}}\neq 0$,
for all points of $\Delta$, then $X$ does not have periodic orbits in $\Delta$.
Exercise. Show that
$\begin{matrix} \, \, \, {x}'=2x-x^{5}-y^{4}x\\ {y}'=y-y^{3}-yx^{2} \end{matrix}$.
it does not have periodic orbits.
Doing $X_{1}=2x-x^{5}-y^{4}x$ and $X_{2}=y-y^{3}-yx^{2}$ and applying the previous theorem to the following exercise to show that it does not have periodic orbits, I get $divX=3-5x^{4}-y^{4}-3y^{2}-x^{2}$ it seems to me that for very small $x$ and $y$ values it is not necessarily less than zero. Unless I was wrong in calculating the divergence.
Is there another way than Bendixson's criterion to solve this exercise?
Thanks for your help.