State the Poincaré-Bendixson theorem for two-dimensional dynamical systems.

A dynamical system can be written in polar coordinates $(r, \theta)$ as

$$\begin{aligned} &\dot{r}=r-r^{3}(1+\alpha \cos \theta) \\ &\dot{\theta}=1-r^{2} \beta \cos \theta \end{aligned}$$

where $\alpha$ and $\beta$ are constants with $0<\alpha<1$.

Show that trajectories enter the annulus $(1+\alpha)^{-1 / 2}<r<(1-\alpha)^{-1 / 2}$.

Show that if there is a fixed point $\left(r_{0}, \theta_{0}\right)$ inside the annulus then $r_{0}^{2}=(\beta-\alpha) / \beta$ and $\cos \theta_{0}=1 /(\beta-\alpha)$.

Use the Poincaré-Bendixson theorem to derive conditions on $\beta$ that guarantee the existence of a periodic orbit.