Using the Poincaré–Bendixson theorem, how I can prove that the following system of polynomial ODEs
$$\begin{aligned} \dot{x_1}=x_1-x_2-x_1^3\\ \dot{x_2}=x_1+x_2-x_2^3 \end{aligned}$$
has one limit cycle inside the following annulus?
$$A := \left\{ x \in \mathbb R^2 \mid 1 < x_1^2 + x_2^2 < \sqrt{2} \right\}$$