Bifurcation, classification and diagram

Question

For the family of DEs $x'=(x^2-2x-\mu)(x-\mu),~x \in \mathbb{R}$ where $\mu$ is a parameter. The questions asked to find the bifurcation points, classify them and draw the diagram. Of course finding the bifurcation is easy; by setting the RHS=0, which yields $x=1+\sqrt{1+ \mu}, 1-\sqrt{1+ \mu}$ and $x=\mu.$ Since $\mu \in \mathbb{R}$ how can I go by and determine the two bifurcation points corresponding to the roots and answer other two questions. I'm sorry, I haven't come across bifurcation in my ODE classes. Any help is appreciated.

Answer 1

The equilibrium points are found by setting the right side to $0$ and solving. The bifurcation points are the values of $\mu$ where two or more solutions collide or start/finish being real. Thus $\mu = -1$, $0$, $3$.
By examining what happens when $\mu$ is close to these values, you'll see that $\mu = -1$ is a saddle-node bifurcation, where two equilibria appear on one side of the bifurcation but not on the other, while $\mu = 0$ and $\mu = 3$ are transcritical bifurcations, where two equilibria collide and exchange stability.