Bendixson’s negative criterion

Let

$\dot{\text{𝐱}}=\text{𝐟}(\text{𝐱})$

be a planar system where $\text{𝐟}={(\text{𝐗},\text{𝐘})}^t$ and $\text{𝐱}={(x,y)}^t$. Furthermore $\text{𝐟}\in C^1(E)$ where $E$ is a simply connected region of the plane. If $\frac{\partial \text{𝐗}}{\partial x}+\frac{\partial \text{𝐘}}{\partial y}$ (the divergence of the vector field f, $\mathrm{\nabla }\mathrm{\cdot }\text{𝐟}$) is always of the same sign but not identically zero then there are no periodic solution in the region $E$ of the planar system.