Dulac’s criteria

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 there exists a function $p(x,y)\in C^1(E)$ such that $\frac{\partial (p(x,y)\text{𝐗})}{\partial x}+\frac{\partial (p(x,y)\text{𝐘})}{\partial y}$ (the divergence of the vector field $p\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}\mathit{}\text{𝐟}$, $\mathrm{\nabla }\mathrm{\cdot }p\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}\mathit{}\text{𝐟}$) is always of the same sign but not identically zero then there are no periodic solution in the region $E$ of the planar system. In addition, if $A$ is an annular region contained in $E$ on which the above condition is satisfied then there exists at most one periodic solution in $A$.