I have the one-dimensional dynamical system $\dot x = f(x)$ where
$$f(x) = (x^2-1)^2-\mu^2$$
And I am asked to find the bifurcation points, draw the bifurcation diagram and classify the points.
So I know to find the bifurcation points I need to find the points $(\mu^*,x^*)$ such that
$$(x^2-1)^2-\mu^2 \\ 4x(x^2-1) = 0$$
and these points are $(1,0)$, $(-1,0)$, $(0,1)$, $(0,-1)$
Then drawing the bifurcation diagram I find something that looks like this
Now, I know one way to classify the bifurcation points is to find which conditions they satisfy
i.e $\frac{df}{du}$, $\frac{d^2f}{dx^2}$, $\frac{d^2f}{dudx}$, $\frac{d^3f}{d^3x}$
but I was wondering if there was another way to find them by just looking at the Bif diagram.
So my thinking here would be to say the bifurcation points at $(1,0)$ and $(-1,0)$ are clearly saddle nodes and at $(0,1)$ and $(0-1)$ are transcritical.
Would I be correct in thinking this? Or should I just see which conditions each point satisfies.