I'm currently taking a course in nonlinear dynamics and I am having a bit trouble drawing phase portraits on polar coordinates.
I solved systems of nonlinear equations by doing polar substitution,
\begin{align} \dot{x} &= -y + ax(x^2 + y^2) \\ \dot{y} &= x + ay(x^2 + y^2) \end{align}
I got this after.. \begin{align} \dot{r} &= ar^3 \\ \dot{\theta} &= 1 \end{align}
I have hard time understand how to draw a phase portrait for a question like this by hand. I understand the radial motion is dependent on a. For even for a simple example when a = 0, I'm not sure how to approach it.
Since \begin{align}\dot{r} = 0\end{align} this indicates that the distance from the origin is constant on time, which would indicate a circular orbit. I'm not really sure how to interpret the second equation though...
Edit: Some intuition on how the distance changes as a varies would also be really helpful!
Thanks so much for the help!