The Cartesian coordinates $(x, y)$ of a point moving in $\mathbb{R}^{2}$ are governed by the system

$$\begin{aligned} &\frac{d x}{d t}=-y+x\left(1-x^{2}-y^{2}\right), \\ &\frac{d y}{d t}=x+y\left(1-x^{2}-y^{2}\right) . \end{aligned}$$

Transform this system of equations to polar coordinates $(r, \theta)$ and hence find all periodic solutions (i.e., closed trajectories) which satisfy $r=$ constant.

Discuss the large time behaviour of an arbitrary solution starting at initial point $\left(x_{0}, y_{0}\right)=\left(r_{0} \cos \theta_{0}, r_{0} \sin \theta_{0}\right)$. Summarize the motion using a phase plane diagram, and comment on the nature of any critical points.