This is my first time using MathStackExchange, so let me know if I'm doing anything wrong. I'm struggling to solve this question: Find a polar differential equation (i.e. $$\frac{dr}{dt}=f(r, \alpha)$$ such that it satisfies the bifurcation diagram linked: Bifurcation Diagram
I guess it isn't strictly polar, just that the variable is confined to $\geq 0$. I've played around with some powers of $r$ in the function, but am unable to figure it out. Any suggestions?
I would advise to study the different types of local bifurcations (in one parameter), all of which can be described by their normal form. The bifurcation diagram in your question corresponds to one (maybe more?) of these local bifurcations, and the functional expression $f(r,\alpha)$ that gives rise to this is given by the corresponding normal form. Note that you may have to combine several local bifurcations to construct the bifurcation diagram from your question.