State the Poincaré-Bendixson Theorem for a system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$.

Prove that if $k^{2}<4$ then the system

$$\begin{aligned} &\dot{x}=x-y-x^{3}-x y^{2}-k^{2} x y^{2} \\ &\dot{y}=y+x-x^{2} y-y^{3}-k^{2} x^{2} y \end{aligned}$$

has a periodic orbit in the region $2 /\left(2+k^{2}\right) \leqslant x^{2}+y^{2} \leqslant 1$.