Construct a first-order ODE with one critical point if $\left\lvert \mu \right\rvert \ge 1$ and three critical points if $\left\lvert \mu \right\rvert \lt 1$ and draw a bifurcation diagram.
Having trouble thinking up a function for this. Any help would be greatly appreciated.
From https://en.wikipedia.org/wiki/Cubic_function, the following cubic equation $$ x^3-\frac{3}{\sqrt[3]{4}}x+\frac{1}{\mu}=0\tag{1}$$ with $a=1,b=0,c=-\frac{3}{\sqrt[3]{4}},d=\mu$, has the discriminant $$ \Delta=18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2=-27(\frac1{\mu^2}-1).$$ If $\Delta\ge0$ or $|\mu|\ge 1$, (1) has three real roots and if $\Delta<0$ or $|\mu|<1$, (1) has one real root. Therefore the following ODE $$ x'=x^3-\frac{3}{\sqrt[3]{4}}x+\mu $$ will be the one you want to construct. Now you can draw bifurcation diagrams.