So I am given a one dimensional system and I'm told that
$\dot x = f(x,\mu)$ with $f(x,\mu) = (x^2 - \mu)(\mu + 3x^2 -4x)$
I am asked to find the bifurcation points, then to draw a bifurcation diagram and also classify the bifurcation points.
Firstly, I know to be a bifurcation point, the point $(x, \mu)$ must satisfy $f(x,\mu)=0$ and also $\frac{\partial}{\partial x}f(x,\mu)=0$
So I calculated $\frac{\partial}{\partial x}f(x, \mu)$ which gave $4(x-1)(3x^2-\mu)$
From this I managed to calculate 3 bifurcation points, $(0,0)$, $(1,1)$ and $(\frac{2}{3},\frac{4}{3})$
Now this is where I got stuck, because surely I can't draw the bifurcation diagram without knowing the classification of each bifurcation point.
So I tried to get the classification for each point. I know to be a saddle-node bifurcation:
$\frac{\partial ^2}{\partial x^2}f(x,\mu) \neq 0$ and $\frac{\partial}{\partial \mu}f(x,\mu) \neq 0$
which allowed me to find that $(\frac{2}{3},\frac{4}{3})$ was a saddle node bifurcation.
And to be a pitchfork bifurcation:
$\frac{\partial ^2}{\partial x^2}f(x,\mu)=0$, $\frac{\partial}{\partial \mu}f(x,\mu)=0$, $\frac{\partial ^3}{\partial x^3}f(x,\mu) \neq 0$ and also $\frac{\partial ^2}{\partial x \partial \mu}f(x,\mu) \neq 0$
which meant that $(0,0)$ is a pitchfork bifurcation.
Now finally there are transcritical bifurcation which require: $\frac{\partial ^2}{\partial x^2}f(x,\mu) \neq 0 \\\frac{\partial ^2}{\partial x \partial \mu}f(x,\mu) \neq 0$ and $\frac{\partial}{\partial \mu}f(x,\mu) = 0$
but $(1,1)$ does not fit into any of these categories?
So my question is, how do I draw my bifurcation diagram and how do I classify these bifurcation points?
Edit: I have found a 4th bifurcation point, at $(\frac{-2}{3},\frac{4}{3})$