Bifurcation point of $\theta' =\frac{\sin \theta}{r+\sin\theta}$

Question

Find the values of $r$ for which a bifurcation occurs and classify them as saddle node, transcritical, supercritical pitchfork, or supercritical pitchfork. $$\theta'=\frac{\sin\theta}{r+\sin\theta}$$ By graphing it on Desmos it seems that the only bifurcation point would be $r^{*}=0$. I'm not sure how to find this mathematically, since it can't figure out a way to get it in a normal form. What I have thought of is setting the denominator equal to zero and plugging in one of the two fixed points $0$ or $\pi$ and getting that at $r=0$ the fixed points are destroyed.
My question is wondering if this is enough or if there is something else I need to show or a better a way to find the bifurcation points. In general what options do you have for finding bifurcation points besides getting it into a normal form? Any help is appreciated, thanks!

Answer 1

First of all, we restrict the analysis to $\theta \in [-\pi/2, 3\pi/2]$, since the flow is periodic. If $r=0$, there is no equilibrium point at all. Otherwise, there are two equilibrium points, which are $\theta = 0$ and $\theta = \pi$. The derivative of the flow $f : \theta\mapsto \sin\theta/(r+ \sin\theta)$ is $$ f': \theta\mapsto \frac{r\cos\theta}{(r+ \sin\theta)^2} . $$ Its values at the equilibrium points are $f'(0) = 1/r = -f'(\pi)$. For $r>0$, the equilibrium $0$ is unstable, while the equilibrium $\pi$ is asymptotically stable. The stability of the equilibrium points changes as the sign of $r$ changes.