\begin{align*} \dot{r}&=-r^3\cos^4(\theta)\\ \dot{\theta}&=1+r^2\cos^3(\theta)\sin(\theta). \end{align*}
I need to show that the fixed point of the system in cartesian coordinates $(0,0)$ is a stable spiral in polar coordinates. I don't really know what to do from here, but I have the Jacobian
\begin{align*} Df(r,\theta)&=\begin{pmatrix} -3r^2\cos^4(\theta) & 4r^3\cos^3(\theta)\sin(\theta)\\ 2r\cos^3(\theta)\sin(\theta) & r^2[-3\cos^2(\theta)\sin^2(\theta)+\cos^4(\theta)] \end{pmatrix}.\\ \end{align*}
The origin is an equilibrium point. Let us focus around this equilibrium point.
Since your $\frac {dr}{dt} <0$, your $r$ is decreasing with time and since your $ \frac {d\theta}{dt} \approx 1 $ you have a rotation around the equilibrium point.
Thus the trajectories are spiraling in around the origin.