I have the following system of differential equations on $\mathbb{R}^2$ in polar coordinates: \begin{cases} \dot{\rho}=\rho(1-\rho) & \text{ (i)}\\ \dot{\theta}=\sin^{2}\left ( \frac{\theta}{2} \right ) & \text{ (ii) } \end{cases} My question is: how can I study the equation $(i)$ on $\mathbb{R}^{+}$, and the equation $(ii)$ on $[0,2\pi]$? More precisely: how can I study the asymptotic behavior of the solutions and their orbits?
Thanks in advance for your help.