Poincaré-Bendixson theorem

Consider a system

$$ \begin{cases} \dot{x}=f(x,y)\\ \dot{y}=g(x,y) \end{cases} $$

where $f$ and $g$ have continuous partial derivatives, and such that solutions exist for all $t$. Let $R$ denote a closed, bounded region of the $xy$-plane which contains no fixed points. Suppose that no solution may leave $R$. Then the system has a periodic solution in the region $R$.