I'm working on this problem: Discuss the stability and bifurcation. Eg. $x'=x (\mu+x-x^2) (1-\mu+x^2).$
Equating the RHS=0: ($\epsilon$ are the critical points.) $\epsilon=0, \mu+\epsilon-\epsilon^2=0$ and $\epsilon^2=\mu-1.$ Finding where each branch intersects, three bifurcation points are found: $(0,0), (1,0) , (2,-1)$ and finding where $\frac{d\mu}{d\epsilon }=0,$ we get $(\frac{-1}{4}, \frac{1}{2}).$ I know a saddle-node bifurcation occurs at the latter. I need help in distinguishing the rest. Sketching this is harder because of solving the quadratic equation. Thank you for your time.