Consider the two-dimensional dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ given in polar coordinates by

$$\begin{aligned} &\dot{r}=\left(r-r^{2}\right)(r-g(\theta)), \\ &\dot{\theta}=r, \end{aligned}$$

$$\begin{aligned} & \dot{\theta}=r, \end{aligned}$$

where $g(\theta)$ is continuously differentiable and $2 \pi$-periodic. Find a periodic orbit $\gamma$ for $(*)$ and, using the hint or otherwise, compute the Floquet multipliers of $\gamma$ in terms of $g(\theta)$. Explain why one of the Floquet multipliers is independent of $g(\theta)$. Give a sufficient condition for $\gamma$ to be asymptotically stable.

Investigate the stability of $\gamma$ and the dynamics of $(*)$ in the case $g(\theta)=2 \sin \theta$.

[Hint: The determinant of the fundamental matrix $\Phi(t)$ satisfies

$$\frac{d}{d t} \operatorname{det} \Phi=(\nabla \cdot \mathbf{f}) \operatorname{det} \Phi$$